For me, this issue has a wider context.

This may be a step too far. But there are many people who turn up on this forum - and elsewhere - who deeply believe that nothing is true and everything is probable. — Ludwig V

I never gave any thought to the relation of truth and probability. Probability is just a number we assign to an event. Before you roll a die there is no truth to its outcome, it hasn't happened yet. You do know with probability 1 that it will turn up 1, 2, 3, 4, 5, or 6. And that if you roll it a million times, about 1/6 of the time it will turn up each number. I don't know how you relate that to truth.

The usual basis for this is traditional (since Descartes) scepticism, and one usually tries to meet it by arguing about that.

But what if they have been introduced to probability theory and infinity? Suddenly, there is a mathematical proof.
Sometimes probability = 1 and 1 = 0.9999... So everything is probability, — Ludwig V

I don't understand what you're saying. I have 1 apple in the fridge, and there is 1 president of the United States, but there is no deep philosophy there. 1 is a number that has many applications.

You are still (I think) applying some mysticism to .999... = 1 but there really isn't any. It's a theorem of ZF. If you wanted .999... to be 47 you could make up a system in which that's a theorem.

1 is a probability and 1 is the number of stars in our solar system. I simply do not see the point you're making.

I think this is a mistake, because it neglects context. But it is new angle on the mistake. — Ludwig V

Well if I have a hammer I can use it to pound a nail or go out into the parking lot and smash everyone's windows. A hammer is just a tool with many distinct and unrelated uses; and if you think of it that way, the number 1 is also a tool with many uses.

The hardware store owner has no use for hammers, to him it's the buyer who supplies the use. Likewise the pure mathematician has no use or application for the number 1; he just makes sure all the numbers are nice and shiny and logically constructed, for others to use.

Any of this make sense? I don't get what you are trying to say.

I'm basing this on an assumption that both theses are correct - in their context. — Ludwig V

Hammers and numbers. Tools for most people, objects of interest in and of themselves to hardware store owners and mathematicians, respectively. I am baffled at where you are going with this.

I think it follows that "0.999...." does not equal 1. — Ludwig V

In what system of rules? In ZF? You are wrong. In the "point-9-repeating equals 42" system? You're right. What underlying assumptions are you making?

If you assume the axioms of ZF, then .999... = 1 can not be challenged or disputed, any more than you can argue with how the knight moves in chess. But if you make up a chess variant in which the knight goes, say, three steps vertical or horizontal and two step diagonal, then that's how the knight moves in his alternative variant.

Make sense?

Now, if you would like to chat about why .999... = 1 in ZF, I am trained to know this. I had it beaten into me by professors at some of our finest universities. But if you prefer the "point-9-repeating equals 42" system, I'm perfectly happy to work with that as well.

Sadly, my best time for philosophy is first thing in the morning... — Ludwig V

I've always been a night person. I generally post in the evenings US left coast time.

Yes. I assume you mean all the terms of the infinite sequence? — Ludwig V

Yes. "Sequence" and "infinite sequence" are basically synonymous, since finite sequences aren't of interest in this context.

And I'm puzzled why you think I'm disagreeing with you. — Ludwig V

It's late, time for bed. I don't think you're disagreeing, but possibly misunderstanding.

So it is. But what is the element of the sequence immediately preceding 1? — Ludwig V

[/quote]

There is none. Why do you think there is one or should be one? That's why I think you're misunderstanding. There's no element of the a sequence immediately preceding the limit point.

It probably saves time and energy. Actually, you mentioned it and I got curious. I'm afraid I innocently asked a question and set off a land-mine. — Ludwig V

It's typically a land mine of ignorance and confusion. Mathematically there is no question whatsoever. .999... = 1 is a theorem of ZF once the appropriate definitions of the real numbers, limits, and infinite series are made. It's like looking at a chess position and saying yes or no, is this a legally reachable position according to the rules. .999... = 1 is a legally reachable position in ZF.

Well, if I've understood how this works, there is a number that gets between each element of the sequence - the next element in the sequence - and is there is no last element of the sequence. So there is no answer to your question. — Ludwig V

There is an answer. The answer is that there is no number greater than all the terms of the sequence, and less than 1.

However, it is also true that 1 is the sum of the infinite series 0.999... - and therefore the limit. — Ludwig V

But an infinite series never reaches its limit. — Ludwig V

We had this conversation. 1 is the limit of the sequence .9, .99, .999, ... "reaching" is just something people say to confuse themselves.

To put it another way, "=" in this context (an infinite series) does not mean what it usually means. — Ludwig V

It means exactly what it usually means. The limit of .9, .99, .999, ... is 1. Or equals 1.

I think I'm a little bit puzzled that you have this confusion after I've explained it in the other thread.

The limit is equal to 1, in exactly the same sense that 1 + 1 equals 2.

In an infinite multiverse, there would be an infinite number of Boltzmann Brain universes, so what are the odds you're in one? 50-50? — RogueAI

I'm convinced I am one. Its statistically more likely than that I just showed up 13 billion years after the big bang.

The technical point is that if there are countably many universes, we can't put a probability number on it at all. Not 50-50, not 1 in a zillion, not anything. There's no way to assign such a probability. And the cosmologists are concerned.

What did Trump do that was authoritarian? Seriously? He tried to pressure the Republican Georgia secretary of state to "find" exactly the number of votes he needed to win. He tried to pressure his own VP to not certify the election (Pence had to call Dan Qualye, of all people, for moral guidance), and he spread and continues to spread lie after lie about the election he lost. You should listen to Bill Barr's testimony about the aftermath of the election. Total banana republic stuff. We dodged a serious bullet. Had Pence not certified, or had Raffensperger gone along with the attempt to steal the election (he says he felt threatened by Trump), it could have gotten a lot uglier than it was. And then there's the fake elector scheme, and of course Jan 6th. — RogueAI

Ok so it's all J6. Bunch of unarmed people are invited in by the Capitol cops, and Pelosi and the hysterical Dems whip up a national hysteria. In the end, it's J6. A Reichstag fire for our times. I find myself wondering what the left will do if Trump wins in November. I expect the left to riot, as they do whenever they don't get their way. Maybe you missed the George Floyd riots. $1-2B in damages, "the highest recorded damage from civil disorder in U.S. history" according to Wiki. And who supported a fund to bail out the violent rioters? Kamala Harris.

So if that's all you got, what about the rest of Trump's four years in office? "Republicans are authoritarians," is what you said. I list all the postwar GOP presidents and all you've got is J6. As a matter of logic, can you see that you have not made your point?

It is.

If the (unknown) theory of universe is not categorical, then the physical universe is part of a larger multiverse.

Therefore, it is a mathematical problem. — Tarskian

Oh man, you do have a unique take on things.

His policy of splitting migrant families resulted with many children being interred away from their families. — Wayfarer

I have followed southern border politics for decades. Here's how it works. I am going to explain some things to you now.

An adult shows up at the border with a kid. The adult says, "This is my child." The kid is tired, hungry, scared, and doesn't say a thing. The adult has no documentation.

You are the administration in charge of border policy. What do you do?

If you say, "Ok, you can both come in," then you turn a lot of kids over to traffickers.

So what do you do to avoid turning children over to traffickers? You separate the kids from the adults until you can contact the authorities in their claimed home country, and find out who they are. If they are a legit family, you reunite them and send them on their way. If not, you just stopped a trafficker and saved a child.

Now, what do you do with the kids? If you put them in a big dormitory, they will be assaulted by sexual predators. So you put chain link around the kids to protect them.

In 2014, Obama had a huge refugee crisis. He "put kids in cages." Photos circulated on social media of the kids in chain link enclosures, with each kid wrapped in a space blanked looking like a baked potato in foil. The images shocked people.

So what did Obama do? Well, optics are everything in politics. They started separating fewer families, stopped putting "kids in cages," and turned a hell of a lot of children over to traffickers. The Washington Post wrote a story about Obama's trafficking problem, but mostly the story got no play.

Fast forward to Trump. Trump does not like traffickers. He tried to protect the kids. He did separate families, to determine if they really were families. Photos were circulated on social media -- the same Obama kids in cages photos. Liberals were outraged till they found out those were Obama's kids in Obama's cages. More photos circulated. Again -- Obama's kids, Obama's cages.

Bad optics. "Trump put kids in cages." So fucking ignorant. A lot of liberals -- ok a lot of people in general -- lead with their emotions, especially when they are ignorant of the facts. So "kids in cages" became the attack on Trump, when in fact the whole idea is to separate traffickers from children and keep the children safe from sexual predators until the true family status can be sorted out.

Now Biden comes in, rescinds all of Trump's border policies including Remain in Mexico. Biden now has a massive immigration crisis on his hands. But the optics are the most important thing. So what does Biden do? He just lets all the adults stay with the kids and lets them in to the country.

What is the result? Biden has lost track of 85,000 children. Most likely turned over to traffickers as sex and work slaves, as you and I speak tonight. Here, read this.

https://cis.org/Arthur/Did-Joe-Biden-Lose-85000-Migrant-Kids

The House Oversight Committee’s National Security Subcommittee held a hearing this week on the Office of Refugee Resettlement’s Unaccompanied Alien Children Program. Robin Dunn Marcos, director of the office, appeared, but if you watch that hearing you’ll learn a lot more from the questions than the answers — because there weren’t many answers on key issues, such as the fate of 85,000 children the office has apparently lost contact with. Someone needs to put a up a large “Help Wanted” sign in Washington, because the American people are desperately in need of accountability on migrant children — both in the government and in the media.

I told you a few posts ago, in a post you never replied to, that Joe Biden is running the largest child trafficking operation in the world. It is true. It is a moral outrage. Nobody gives a shit.

Now you know what "separating families" and "kids in cages" are all about. You separate kids from adults until you can determine who's a parent and who's a trafficker. And you keep the kids behind chain link fences to protect them from sexual predators.

But kids in cages makes for bad optics. So Biden just turns the kids over to traffickers, and ignorant liberals know nothing about it, and STILL THEY BITCH ABOUT TRUMP'S CAGES.

Liberals still do not know that those were Obama's cages, Obama's family separation policy, and that once the optics got bad, Obama just said fuck it, and turned the kids over to traffickers. No more bad optics. And that's Biden's policy too. No cages. Just trafficked children.

Get a clue, brother. Get a moral clue. I explained this to you two weeks ago and you never acknowledged the post. Joe Biden is a child trafficker. Because the optics are better than "kids in cages," which upsets ignorant liberals.

My liking him or not is irrelevant. His danger to democracy is not a matter of opinion. He’s not only a terrible person, he’s a dreadful leader, his only policy is retribution. His speeches are horrific and contain nothing about policy as such, only threats and fear-mongering. How you can fall for his schtick beats me. — Wayfarer

Ok. I could, for sake of argument, stipulate to all that. I can still talk politics! I can still talk about Joe Biden's mental impairment. I do believe you said to me earlier that you can't even talk about the Biden pickle unless I hate Trump as you do. Some people feel that way. Myself, I'll talk politics to talk politics. I don't have to love or hate Trump to talk about the mess the Dems got themselves into this week.

I would be glad to explain to you "how I can fall for his schtick," but that's more for the Trump forum. In this thread I'm trying to focus on the topic, the 2024 US election. Or as Joe Biden puts it: "I'll beat Donald Trump again in 2020." It would be funny if it weren't so tragic, and if old age and sickness didn't eventually catch up with us all.

I'll be happy to argue the merits of Trump with you if you'd like, but it's not really all that productive. I did write you a long-assed post a couple of weeks ago about my journey from dedicated liberal to the politically homeless, reluctant Trumper than I am today. I could write more. It's been decades in the making. It started when Teddy Kennedy killed a girl and the left rallied around him. It was my first sense of a disturbance in the liberal force. There were many other such moments over the years. This Biden fiasco is just the latest.

Biden is not ‘a husk’. He’s been an effective senator and president, but he needs to pass the torch. — Wayfarer

Man even the New York Times thinks he's a husk. I don't even have to make the case. Biden's own "friends" are making that case with sharp knives. Julius Caesar never got it so bad on the floor of the Roman senate. George Stephanopoulos said today that he doesn't think Biden can make it another four years. Et tu, George.

I note today that Gavin Newsom is acting as party whip for Biden. I believe he’s totally sincere in so doing, but also that he’s ideally positioned to step up if the torch is passed. — Wayfarer

Newsom is too smart and too ambitious to touch the current mess with a ten foot pole. Whitmer too. Any Dem who's viable for 2028 is going to show loyalty to Biden and stay out of 2024. Why go down with this sinking ship, when a brand new ship is arriving in four years?

I appreciate the opportunity to chat. I really did take it to heart a few weeks ago when you expressed disappointment in my political sentiments, in light of my math-related content. I'm always willing to talk politics with people who don't share my opinions. I'm not blind to Trump's many flaws, but IMO he really is not the monster the Dems have made him out to be. I'm always happy to explain myself.

Bottom line: The Dems and the left have deeply lost their way; and Trump is the only alternative. I'm not for Trump. I'm against what the Dems and the left have become. I saw what the Dems had become in 2002, when Hillary made an impassioned speech on the Senate floor in favor of the war in Iraq. The Democrats could have stopped that war. They were looking to Hillary for leadership. She chose the path of war. So when 2016 showed up and it was Trump or Hillary, I chose Trump. And why did the Dems nominate a corrupt, warmongering, unlikable, lousy politician like Hillary? As Obama said when he destroyed her in 2008: "You're likable enough, Hillary."

And as I pointed out in my latest reply to @Mr Bee, Trump is a monster of the Dems' own making. If the Dems had (1) Totally ignored Trump starting in 2022: no lawfare, no rhetoric; and (2) run open, competitive primaries; then today, as we speak. Gavin or Gretchen would be handily beating DeSantis.

The Democrats turned Trump in to a martyr. I was sick of the guy myself before the Dems turned the apparatus of the American justice system on him. That Mar a Lago raid put Trump into the White House.

Yeah the dam didn't break for now, but clearly nobody is eager to unite behind Biden just yet. Bennet's comments recently indicate that Biden isn't gonna be able to ignore and move on from the problem as he usually likes to do. — Mr Bee

I think that even if Biden bowed out gracefully (unlikely) or the Dems forced him out (quite unseemly, also unlikely) and elevated Kamala, whose popularity is below Biden's (I haven't checked that lately) life would NOT suddenly be a bed of roses. A lot of Dem voters would be unhappy and confused. And the policies are no different. Biden had a bad approval rating and was losing the election to Trump even before the debate. Many of the Dems' policy results such as inflation, unchecked immigration causing blue cities to be overrun with a humanitarian and financial disaster; the two wars, etc etc, are quite unpopular. And Kam is to the left of Joe. I don't see how this solves the Dems' electoral problems.

Courage is a rare thing among elected officials which is why Trump wasn't banished from the GOP, despite their occasional concerns about him post Jan 6 and 2022 midterms. — Mr Bee

Verily I say unto you, and unto all the fervent Trump haters on this forum:

Trump is 100% the Democrats' fault

Back in early 2022, people were tired of Trump. I was tired of Trump. He has all his negatives, the bluster, the bullshit, the thin skin, the midnight tweeting, the lack of understanding of how the government works. I was ready for a new GOP candidate to challenge the Dem orthodoxy that's not working for a good portion of the people in this country.

If you remember, DeSantis was running neck-and-neck with Trump in the polls. Then Biden, Garland, and Wray raided Mar a Lago. The very next day, DeSantis was obliged to come out in support of Trump. From that moment onward, Trump started rising in the polls and DeSantis sank out of sight.

With every new lawfare case, Trump became more popular. Because the Trump haters see these cases as righteous applications of the law; but the other half of the country sees them as bad faith politicization of the American system of justice, one of the best things we (used to) have going for us.

And so now, Trump is all but unstoppable, and then there was the debate, and here we are.

I say this: If the Dems had done two things: (1) Totally ignored Trump; and (2) Had an open, competitive primary; then at this very moment, Gavin or Gretchen would be beating the stuffing out of DeSantis.

The Democrats created all of this. They made a martyr then a hero out of Trump; and they refused to confront reality about Biden's condition. The Dems did this. Not the GOP. Most of the GOP hate Trump, they'd love an alternative. The Democrats forced the GOP to rally around Trump.

I'd say call the bluff. Biden isn't exactly a guy who embodies strength as we saw during the debate and how he's been handling the Gaza situation. I mean sure he has alot of angry stubborn grandpa energy but Netanyahu has been crossing his red lines repeatedly and he has not done anything to stop him. — Mr Bee

I wouldn't mess with Jill and Hunter.

At this point I don't know what the congressional Dems have to lose either so they might as well try to improve their party's situation and place themselves on the right side of history in case Biden stays in, Trump wins, and he ends democracy. — Mr Bee

I truly do not understand that talking point. Trump was already president for four years and he didn't end democracy. On the contrary, he got rolled by the bureaucrats and most of the people who worked for him.

I think what people mean is that Trump is going to do to the Dems exactly what they did to him. And frankly, some of that would be a good thing. Garland and Wray are thugs. The country may never recover from their abuse of the justice system.

But Trump "ending democracy?" Nonsense. Most of that is projection on the part of the Democrats. We're having an election. That's democracy, imperfect as it is.

One interesting aspect of Biden's ABC interview was that he never really specified how he would react if the congressional Dems turned on him. He outright refused to answer the question and acted like there's no revolt going on. If he was really delusional you would've thought that he would give a non-assuring answer like, "I would sit down and tell them 'We will win'," or something to that effect. That will probably hang over the Dems minds as they contemplate what to do next. — Mr Bee

He's the president of the United States. He doesn't have to do or say a damn thing. He said something the other day I really liked. He said, "If someone wants to challenge me at the convention, let them." He's a tough old bird. I don't like the guy but this might be his finest hour!

He's the president. He has Jill and Hunter, two pit bulls. He has 3896 Democratic delegates.

What do the Dems have? A strongly worded editorial from the New York Times?

The Dems have no hand to play.

Arrogance. They thought they could probably roll with Biden into the next election and dismissed people's concerns about his age. I mean they got pretty far before we saw what happened a week ago... putting aside all those viral videos of Biden having senior moments. — Mr Bee

Cheap fakes. Like I say. The Dems are in a pickle entirely of their own making. Trump didn't make the Democrats ignore the Biden situation for the past three years. Whose bright idea was it to anoint Joe with their non-primary primary? If they'd had a real primary, Gav and Gretch would have been all over it. The 1968 Democrats had a wild primary that ultimately drove LBJ out. They could have and should have done exactly the same thing.

Arrogance, I guess that's as good a word for it as any. Short-sightedness. They tried to keep a lid on it and now it's blown up in their faces.

Sounds like great qualities to have in a leader, both for the party and the country. — Mr Bee

I'm likin' Biden more this week than I ever have in my life. I like this stubborn old coot telling the DNC and the New York Times to stuff it.

Yeah, but that would be much better than well, trying to convince the public to vote for a soon to be 82 year old man who clearly has cognitive issues to serve another 4 years in office. — Mr Bee

Wouldn't have to. He can run then turn it over to Kam in 2025. Would have made his point. Kam is not any more likely to win the election than Biden. Kam has high unfavorability. She's a lousy politician, the 2020 primaries showed that. She had to drop out in 2019. She is not the Dems' savior.

Depends on your political affiliation but as someone who doesn't want Trump winning I have no sympathy for an old man who is selfishly staying in and gambling with his party and country simply to try and get a second term in his 80s. — Mr Bee

I'm making a nonpartisan point. Say you hate Trump. I am making the point that Biden arguably has a better shot than Kamala. The party will look like a clown show if they throw over Joe after telling us he was "sharp as a tack" for three years. People will not like that. They don't have to vote for Trump, but enough of them might just stay home.

The message would be, "We said Joe is sharp as a tack but we were lying, so here, vote for highly unpopular Kamala." I don't think that's a winning message for the Dems. Not a partisan point. Biden has a better shot to win than Kamala. It doesn't matter that his mind is gone. He's not Trump, AND the DNC isn't pulling a last-minute switcheroo.

I don't think the voting public is going to like a switcheroo on top of the fraud they've already seen. Hope I made my point that I'm not talking partisanship. I think Kam's a terrible candidate. Her negatives don't go away if they elevate her.

At this point I can see way more upsides to a new candidate than running with Biden. Biden can't do anything to fix the fact that he's down in the polls but another candidate can. — Mr Bee

Ok, so that's a point we disagree on. But not a partisan point for me. If Trump didn't exist, the Dems should still run Joe. The swicheroo factor, I'll call it. People will feel that they've been played.

As Nikki Haley said, in a race between two incredibly unpopular geriatrics, the first party to get rid of their candidate wins the election. Polling seems to back that idea up, showing that a generic Dem or Rep running against either Trump or Biden respectively will easily win. It'll be interesting to see if that theory holds true. — Mr Bee

LOL. Well you know, maybe you are convincing me a little. I could be wrong. I give my own theory only about 75% credence. Maybe people are more horrified at Joe's condition than I realize. All the people who were genuinely shocked by the debate.

Yeah but they can severely harm and embarrass him, which at this point Biden frankly deserves. As a narcissist that's something he probably cares deeply about. Leverage isn't the same as having complete control over someone. — Mr Bee

I am pretty sure Biden is way beyond embarrassment at this point. And Jill and Hunter surely have no shame. But I see your point. At some point he'll cave to the political pressure of being so unliked. Could happen. Or it could just make him dig in more. He's been in politics over 50 years. Survival is an instinct. We see it all the time. His body knows how to be a politician even if his mind is gone.

Yeah doesn't seem like it so far. He's become oddly Trumpian in just about every respect since the debate happened. That being said it could all be a bluff and he may fold if his party lost faith in him. Biden's recent attempt at painting his problems as the elites trying to get rid of him as Trump usually does just isn't believable coming from him, a man who has been propped up by the elites all his life. — Mr Bee

Either way, I'm enjoying the show. I'm one who always enjoys a political show. If the GOP were having a fiasco this week I'd enjoy that just as much.

They're likely gonna coalesce if Biden lasts until the convention, and the party and the media will never bring up the age or replacement issue again. — Mr Bee

Chicago! The Palestinian wing of the party riots. "The whole world's watching!" 1968 here we come!

Or... maybe they will continue bringing up the issue of replacement if it's possible to swap him out post nomination, though at that point it'd just be Kamala who would be the nominee. Could be possible (apparently there was discussion of Pence taking over the GOP ticket in October of 2016 after the Access Hollywood tapes came out after all). Biden is likely to have a major senior moment in the next 4 months especially during the next debate which may reignite the discussion, or he could just die of old age. He's 81 after all, so it's not a possibility you can definitively rule out. — Mr Bee

Could happen. And Trump is no spring chicken either. One more Big Mac could do it.

According to Thoralf Skolem's construction, i.e. by injecting infinite cardinalities in the model's structure, which is a countable set of symbols, there is at most a countable number of models of arithmetic. — Tarskian

Whoa, the subject is multiverse cosmology in eternal inflation. Physics theory. Not math.

The strong assumption here is indeed the continuum hypothesis: — Tarskian

Nothing whatever to do with CH, though I do take your point (if this is your point) that if cosmology considers an infinite multiverse, questions of set theory arise.

https://en.m.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem

It implies that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ, and that no first-order theory with an infinite model can have a unique model up to isomorphism. As a consequence, first-order theories are unable to control the cardinality of their infinite models.

In her lecture on the subject, Victoria Gitman confirms this:

https://victoriagitman.github.io/talks/2015/04/22/an-introduction-to-nonstandard-model-of-arithmetic.html — Tarskian

I'm not sure what any of this has to to with multiverse cosmology. Nothing, actually, though I'm familiar with Hamkins's student Gitman.

An easy application of the compactness theorem shows that there are countable nonstandard models of the Peano axioms, or indeed of any collection of true arithmetic statements. — Tarskian

Wrong thread, wrong topic.

If the physical multiverse is somewhat structurally similar to the arithmetical multiverse, it should also have a countable number of physical universes. — Tarskian

Multiverse cosmology is not about set theoretic models.

If we deny the continuum hypothesis, however, then most of the then uncountable universes would be unreachable because there can still only be a countable number of infinite cardinality symbols to do so. — Tarskian

Non sequitur much?

Here, see this.

https://en.wikipedia.org/wiki/Eternal_inflation

and this.

https://en.wikipedia.org/wiki/Multiverse

I think you have conflated Hamkins's set-theoretic multiverse with the physical multiverse of speculative cosmology. Or maybe cosmetology.

You don't have a left in the US. You have a slightly left of centre Sanders who is silenced by the Democratic Party which is itself right but not as authoritarian as the Republicans (unitary theory of government Bullshit). — Benkei

I hardly see the GOP as authoritarian. Going back a ways, which of these postwar GOP presidents were authoritarians? Ike, Nixon, Ford, Reagan, Bush 41, Bush 43, Trump. Feel free to explain to me what these folks did that was authoritarian. I opposed the hell out of 43's wars in Iraq and Afghanistan, but those wars never could have happened without the enthusiastic consent of the top Dems such as Hillary, Schumer, Biden, et. al.

In theory, the GOPs should be for free enterprise. Not that they really are, but that's more of libertarian sensibility. But I'm open to understanding you observation. When Obama ruled "with a pen and phone," was that the unitary executive? Or when the Supreme Court told Biden he couldn't transfer student loans to the taxpayers and he did it anyway, was that the unitary executive?

I recall qualifying the 2020 election as a choice between two evils. One of those evils got a lot worse. — Benkei

Every election I can remember has been that way. Bush (I'll do two wars) vs Kerry (I'll just do them better). What kind of choice was that?

It clarifies once again that the USA doesn't qualify as a democracy. If the political system cannot produce choices beyond a vegetable and a criminal then quite obviously other people are in control what you get to vote on. We call that banana republics. — Benkei

YouTubers Eric Hunley and Marc Groubert of America's Untold Stories awarded the US eight bananas (out of ten) following Bragg's conviction of Trump. What of it? Some of us don't think this country's had a legitimate government since the deep state killed JFK in 1963. We have a country of, by, and for the military-industrial complex.

$200 billion to Ukraine, and barely $330 million to Maui after their fire. And that was under the Biden administration. Care to defend that? I'd send $200B to Maui and send the corrupt Nazi Zelinsky money-laundering operation straight to hell. But that's just me. Peacenik from way back, like the left used to be.

Trump started no new wars, the only president in my lifetime to have managed that. Yet he's universally hated by the "good people." Why is that? Why do the good people love the warmongers?

What is on your mind about this? Who's the authoritarian and what have they done? Biden has been quite the unitary executive.

Let's not distract from supertasks by questioning very simple mathematical facts. — Michael

Oh I see what happened. @Ludwig brought up the old .999... = 1 chestnut in the staircase thread, and it apparently got moved over here to revivify this four year old thread.

Ludwig, let me put to you a question.

Suppose that .999... is not 1. If they are different numbers, then there must be a third number strictly between them. What is it?

Put another way, and echoing the point you made in the other thread, suppose I have the sequence

.9, .99, .999, .9999, .99999, ...

What number can possibly get between ALL the terms of that sequence, and the number 1?

Well maybe it's .95. No, that's smaller than .99.

Ok maybe it's .995. No, that's smaller than .999.

Ok then maybe it's .9995. No, that's smaller than .9999.

You see how this works? You can't find any number to stick in between ALL of the elements of the sequence, and 1.

Since you can't find a number between them. the limit of the sequence is 1. Or putting it another way: .999... = 1.

I have a metaphysical probability question:
Suppose there are an infinite number of parallel Earths. Alice uses a teleporter to teleport to a random Earth. Bob tries to follow Alice, but he has to guess which Earth she teleported to. What are Bob's chances of getting it right? Is there any way for a teleporter machine to randomly select an Earth out of an infinite number of them in a finite amount of time, or is there always going to be, practically speaking, only a finite amount of Earths for Alice to teleport to because of the limitations of the machine? What if I cheat and say the teleporter pokes a hole into the universe and the universe somehow, through a mysterious process, randomly picks an Earth out of an infinitely large ensemble for Alice to teleport to? Are Bob's chances of teleporting to Alice's world zero? — RogueAI

Are there a countable or uncountable infinity of worlds?

Reason I ask is that there is no uniform probability measure on a countably infinite set. In other words if we throw all the positive integers 1, 2, 3, ... into a bag, and reach in and pull one out, there is no sensible way to assign probabilities to that scenario, if we insist that each number has the same chance of being picked as any other. It's literally impossible to do that; and the impossibility is actually very easy to prove.

But there is a uniform probability on an uncountable set.

It turns out (I'm no expert, I saw a Sean Carroll video on this) that the multiverse cosmologists are terribly concerned about this problem. In an infinite multiverse, how many universes are like this, and how many like that?

They call this the measure problem. This is not to be confused with the measurement problem of quantum physics, the question of what is a measurement that makes the wave function collapse. As Wiki says:

"The measure problem in cosmology concerns how to compute the ratios of universes of different types within a multiverse. It typically arises in the context of eternal inflation. The problem arises because different approaches to calculating these ratios yield different results, and it is not clear which approach (if any) is correct."

This is a deep, unsolved problem in multiverse cosmology.

If I said anything about that, I would be way out of my depth. So I'm afraid I shall have to ignore it - until another time, maybe. — Ludwig V

Ok forget that. But 0 and 1 are perfectly legitimate probabilities. After all if I roll a die, the probability is 1 that it will be either 1, 2, 3, 4, 5, or 6, Right? Nothing degenerate or unusual about that. And the probability is 0 that it will show 7.

.. in the context of probability theory, that may be so. But I'm interested in probability in the context of truth and falsity, which is a different context. — Ludwig V

Are you talking about credence, perhaps?

So when you say that 1 is a perfectly sensible probability, are you saying that probability = 1 means that the relevant statement is true? — Ludwig V

No. Only that the event is certain, in the finite case; or "almost certain" to happen, in the infinite case.

Can you give me an example of what you mean? What kind of statements are you applying probability to?

(I don't want to disappear down the rabbit hole, so I just want to know what you think; I have no intention of arguing about it. — Ludwig V

Never mind probability. You just startled me by denying the legitimacy or sensibility of 0 and 1 as probabilities.

I watched all three of those debates (and the numerous primary debates) and thought Biden did fine. If he was in bad shape, as you claim, he wouldn't have won any of them. — RogueAI

I recall being amazed that Biden made it alive through those debates. I was not the only one with that expectation. I never thought he'd make it to election day. I saw an unwell man. And you are right, he did surprise me by surviving. Guess it was just me. What do I know, I liked Tulsi. Still do.

An interesting point is that while we can express the indiscernibility of identicals as a first order schema, we can express the identity of indiscernibiles as a first order schema if and only if there are only finitely many operation and predicate symbols.

It's an interesting exercise to try to express the identity of indiscernibiles as a first order schema with a language of infinitely many non-logical symbols. You'd think you'd just reverse the indiscernibility of identicals. But when you try, it doesn't work! If I'm not mistaken, one of the famous logicians proved it can't be done.
— TonesInDeepFreeze

Sadly this is all over my head. Maybe I'll crack open a logic book. If I could only dispatch a clone.

Another nice thing: Identity theory can be axiomatized another way, courtesy of Wang:

For all formulas P:

Ax(P(x) <-> Ey(x=y & P(y)))

From that we can derive both the law of identity and the indiscernibility of identicals. — TonesInDeepFreeze

That looks interesting.

I only made the remark about medical factors causing Biden to retire, because I think he ought to retire. Like a lot of people, I think the public perception of him being 'too old' is a factor which might cause him to loose. If I were an American elector, and Biden was the candidate, I'd vote for him. I'm just concerned that many others won't, — Wayfarer

If you are backing off the stroke remark, I'll be glad to give you a pass on that. I did take it as representative of the massive anger that Dem suddenly feel towards Biden, when they'd been supporting him five minutes earlier.

and as I've already said, I believe the re-election of Donald Trump would be an unqualified disaster for the United States and the rest of the world. — Wayfarer

I understand that you feel that way. But Trump was already president for four years. He didn't put people in camps. He didn't do any of the bad things the Trump haters are afraid of. In fact he got rolled by the administrative state and most of the people who worked for him. Some dictator.

And no new wars started on his watch. That is something. That is a lot. And it was no accident. Trump was the peace candidate in 2016 and 2020, and he's the peace candidate today. The left used to be for peace. One of the factors in my defection from the left.

Nobody's been 'covering anything up' about Biden. — Wayfarer

That is just not true. He's been bumbling and stumbling in a frankly heartbreaking manner for several years now. It's not possible to have not seen it. The wandering off stages, the mis-statements that had to be cleaned up by aides the next day ... ok I won't go on. If you claim to have first seen Joe's infimity at the debate, I'll believe you. Because you say so; not because such a claim is credible.

When Joe wandered off at the G7, froze up at Juneteenth, and head-butted the Pope, did you believe KJP when she called those "cheap fakes?" Curious to know.

He's never been an orator, he often had verbal stumbles and gaffes throughout his career. — Wayfarer

He has always been corrupt and a rather stupid man. But he was always verbal. Nothing like the last few months and the last few years. The slurring of words. After the debate I was shocked that everyone else was shocked. He seemed to me the same as he's been for quite some time.

So what? The Washington Post kept a daily tally of Trump's lies in his first term which topped out at some number around 38,000 (correction, 30,583) so don't talk about 'deception'. Anyway Im not going to discuss it with you, if you can't see Trump's obvious malfeasance then there's obviously no point. — Wayfarer

The point is not that Orange Hitler is worse than the cognitively-impaired husk. That's a political judgment and politics is not about purity. But look what you're doing. You are denying Joe's cognitive impairment on the grounds that Trump is a terrible person. How does that even make sense? Trump is Trump, I get you don't like the guy. That has nothing to do with the fact that the Dems have indeed been covering up Biden's increasingly worse cognitive issues.

Do you even see your bad logic? You are saying that Trump is evil THEREFORE the Dems have not been covering up Biden's cognitive issues. Surely you can see the flaw in that argument.

You can't even discuss Biden's sad state of mind because you hate Trump so much. What kind of sense does that make?

"I just need 11,686 votes"..... — Wayfarer

You can't discuss Biden's cognitive issues because you hate Trump. This is exactly how the Democratic party got itself into the pickle it's in! Five years of denial, gaslighting, and coverups.

Ok let me say this another way. If I am understanding you, you claim that you cannot discuss the Democrats' current cognitively-impaired Biden pickle with me, because I don't agree with you about how evil Trump is.

Some people see politics purely in partisan terms. I can discuss a pickle whether the pickle is on one side or the other. The current situation is unprecedented in US history. It's nothing like when LBJ dropped out in 1968 over the Vietnam war, or when Truman chose not to run in 1952 due to his unpopularity.

To me, politics is a partisan affair, to be sure. But it is also a spectator sport. I don't have to love Trump or hate Trump to be enjoying the spectacle. But from your point of view, you can't even have a conversation with me about politics if I don't hate Trump the way you do.

In that case I'm sorry I troubled you. I enjoy talking politics. I don't have to love or hate the people involved. You can vote for Biden (if you were a US voter) regardless of his mental state; and in theory, you could have a conversation about politics with someone whose politics are different than yours. I've always been able to do that. Not everyone does that, sadly. Political conversation is polarized these days, but it can be otherwise.

If the GOPs were in a pickle this week I'd discuss that. I've seen GOPs and Dems in plenty of pickles over the years. I love a good political scandal. That's just me.

Those are statements in the meta-theory that describe an infinite set whose members are all axioms that are in the object-theory. — TonesInDeepFreeze

Ok. I see your point. Lately wishing I'd paid attention in logic class.

And I don’t believe that for a minute. Biden was quite capable of executing his first term, and did so with distinction. — Wayfarer

Ok. You and I can agree to disagree on many things, and this is another such.

But don't you know that he's been getting the questions ahead of time at his infrequent press conferences? And making errors and telling falsehoods for the past three years? I guess people see what they want to see.

Are you saying you were genuinely shocked at his condition at the debate? Believed everything else was "cheap fakes?" How can that be? If you say so, I believe you ... but I've seen Biden's cognitive decline since 2019. Even at the Dem primaries in 2020 Cory Booker and others were making fun of his failing memory.

Distinction? Well I am trying to focus on the politics and not the policies because we all know each other's talking points on policy. But the inflation, the direct result of the massive printing and spending? The open borders that are costing LA and Chicago and NYC billions? Two new wars? Trump had none. If you call that distinction, we can agree to disagree on that too. I'll stipulate to all well-known talking points and rebuttals on both sides, not intending to argue policy. But a lot of Americans are quite unhappy with the Biden admin totally apart from Joe's personal condition.

Which by the way, is one reason swapping out Joe for Kamala might not be the panacea the Dems think it is. Same policies with less mental confusion. Not clear that's an electoral winner.

I stated explicitly several times that that is what I mean by 'identity theory'. — TonesInDeepFreeze

You never said that LOL!

I recall having seen the term used professionally before, and so I adopted it a long time ago, but I would have to dig to find citations. I like it, because it is a first order theory about one certain predicate that is indeed the identity predicate. — TonesInDeepFreeze

Now that I know what you mean, it's helpful.

If someone says "I'm talking about blahblah theory'" and they tell me the axioms, then I don't quarrel with them about it. I know the axioms so I know precisely what is meant by 'blahblah theory'. — TonesInDeepFreeze

I see your point. But I don't always catch your meaning from your symbology.

I still have it in my queue to go back to your recent post about the indiscernibles. Maybe if you could make your point in words. I thought the = of set theory is the = from the underlying logic. But now you say it's not. So I'm confused again. If you could explain it clearly in a sentence or two I'd find it helpful

And I know that if I make this request, and you give me a response and I don't related to it, that's frustrating to you. Maybe there's a happy medium of explanatory level.

I mean Kamala was a prosecutor. I don't think either were gonna be popular with the Democrat base in 2020 due to the BLM riots, but Biden decided on Harris. — Mr Bee

Yes good point. I remember that conversation now. I thought Demings was a great choice. As a Californian I never liked Kamala. Well it's a done deal now. And she has the inside track. Nobody's talking about Gavin or Gretchen lately.

I'll wait until tomorrow since alot of other congressional Dems are kind of staying silent and clearly are not just falling in line like the president wanted. I'm assuming you meant Mark Warner there, and his statements are also very noncommittal. Nobody except one representative came out today against Biden, and my guess is that the dissenters are currently keeping quiet so they can present a united front when they meet later. — Mr Bee

They had a meeting today. A reporter asked an attendee if they were on the same page, and he said, "We're not even on the same book." Reports that some people were in tears. Lot of misery in the Dem party. Pretty much anything could happen.

Yes John Warner was the one married to Elizabeth Taylor. I always get them confused

From what I can tell the House Dems are planning a meeting in the morning to discuss the matter privately, while the Senate is also doing the same at noon. Whether they'll decide to confront the president and whether any of that will be made public is anyone's guess. — Mr Bee

Right. Rumors they're all pretty upset and no solution in sight.

That's Biden's intended play here, but given that nothing he's done in the past week has assured worried Dems about his reelection prospects, and his complete dismissal of the concerns being thrown his way, I think he's only infuriated and emboldened his critics more. He couldn't convince them that he's not senile so now he's trying to say "don't oppose me or else I'll make it ugly for all of us" to get them to fall in line. Could be a sign he really is hopelessly stubborn or it could be a last ditch attempt at keeping the dam from breaking. Whether the Dems speaking tomorrow will act or not will depend on how they read what he said, but it's clear the president is daring them to oppose him. — Mr Bee

Right again. Nate Silver has an article out implying that Biden is bluffing. Maybe he is. He was at NATO today, didn't embarrass himself. He's hanging in. A politician who's been running for office for fifty years or more isn't going to go out easily.

Biden isn't at all a beloved figure. That was why he was thrown under the bus so easily. He's doesn't command a cult like following like Trump so it's easy for them to do so. He was nominated in 2020 purely for his perceived electability and now in an election where he seems to be losing that by being down against a convicted felon the Dems have largely soured on him. I mean they'll still vote for him to stop Trump but they have no support for Biden himself. — Mr Bee

Yes good point. He was always kind of a joke, then in 2020 he seemed like the best option to beat Trump. But why didn't the Dems do something sooner? If they'd just have had a competitive primary they'd have replaced him already.

Biden's status, or lack thereof, in the Democratic party cuts both ways. They clearly don't have loyalty to him, but he also has no loyalty to them. That's another reason he's hanging in. He's not thinking of the good of the party, he's taken a lot of disrespect from his fellow Dems over the years. It's the Bidens versus the world at this point.

What are they gonna do if he stays in? It seems at this point he's dragging the entire party down for his own selfish goals. At this point they might as well try to make it untenable for him and hope he isn't gonna stubbornly let his own party collapse under his hubris. — Mr Bee

Well, he was doing badly in the polls and had a high unpopularity rating even before the debate. Just another reason for them to have dealt with this during the primaries. Dems have no good options.

It's kind of a mixed bag at this point. Alot of them have "concerns" as well. May be a civil war situation but who knows, some of the supporters may believe deep down that Biden isn't the right guy for the job. Reportedly you have folks like Don Beyer saying in private that Biden should resign and let Harris be president while openly supporting him for instance. — Mr Bee

Love to be a fly on the wall in the Dem meetings. Kamala's playing it cool, supporting Biden in public. Someone mentioned that of all the Democrats, Kamala is the only one who had a Constitutional duty to notify people that Biden wasn't all there. Especially with the Parkinson's story in play. I wonder if that will come up. A lot of people have been covering up this situation for a long time.

Like I said, I'll wait until Tuesday to see if Biden has weathered the storm. The critics have been silent until they meet and gather. Here's a Politico article from Monday evening suggesting that things aren't necessarily over. — Mr Bee

Definitely not over. This thing's just getting started. Even if they swapped in Kamala, it would not be smooth sailing. The public would have a lot of questions about "What did they know, and when did they know it," as they used to say during Watergate.

Yeah I was one of the people who noticed it back then too (comparing it unfavorably to his 2012 debate performance), but it's way worse now. He could at least debate and do a forceful interview in 2020. — Mr Bee

He's gotten much worse just in the past few months. It's heartbreaking at a human level. Especially since none of us are immune. I kind of admire his stubbornness. I'd like to see him stay in and stick it to the party. According to the polling he was losing on the issues anyway. Not clear a last-minute swap would help. Not entirely clear that Biden's condition is the only reason he's behind in the polls.

Yeah I understand that ultimately it really is on Biden to step aside unless the Dems are brave enough to take stronger measures. — Mr Bee

They'll never impeach or invoke the 25th. They won't do it. And I don't know if pressure will be enough. I don't see Jill giving in "for the good of the party."

When the GOPs came to Nixon, they told him he was certain to be impeached and convicted. The Dems have no such leverage. This really is a day-by-day situation. Next week is the GOP convention, that might take some of the media attention off the Dems.

The hope I guess is to make the situation as untenable to Biden as possible because clearly he is out of touch with the reality of the situation, and also hope that the supposed good man in Joe will make him realize how destructive his political ambitions are to a party and country that's lost faith in him. — Mr Bee

BIden is not a good man. His lunchbucket Joe act is just for the public. I've heard he's always been a very nasty guy in private. Of course you're right, if he would gracefully bow out and endorse Kamala, that's the best the Dems can hope for.

Who knows, maybe he will let the party crumble before he steps aside, but even he should realize that he can't win an election if even his team lacks any confidence in him. — Mr Bee

I think the Dems should crumble for what they've done. They had three years to deal with this. Instead they've been lying and gaslighting the country. It was all "cheap fakes" and right wing propaganda right up until the debate. By rights, the voters should punish the Dems severely for all this. But of course Trump has his negatives. People who hate Trump are not going to suddenly vote for him.

I think if the Dems coalesced behind Joe that gives them their best chance. Then Kamala can take over shortly after the inauguration if Joe should win. It's going to be a close election either way. It's very unclear if swapping out Joe actually improves the Dems' chances.

Whatever side represents the rule of law and upholds the constitution. The side which didn’t attempt the overthrow of the Government and the subversion of the election. — Wayfarer

I ask you to introspect about your sentiments regarding Biden. I'll stipulate that you have your political opinions, which are shared by many and opposed by many. No point in arguing those since as you yourself recently noted, we're not in the Trump thread. I'm more interested in the psychological reaction to Biden, the vicious backstabbing and, in your case, the hope for a terrible physical malady to befall him.

The viciousness toward Biden from his own side -- that's a psychological reaction to years of going along with the lies about his condition. No other explanation fits. Who, honestly, was shocked by his debate performance? I said to myself during the debate, "Biden's reasonably lucid tonight." I actually said that. He was no worse than he's been for months, and actually a little better. He didn't glitch out like at Juneteenth. He didn't wander off like at the G7. He didn't head-butt the Pope. He didn't raise his fist and start insulting people as he frequently did in 2019.

I was literally shocked that so many people were shocked. Biden has been like this for a long time. Dems and the media and those who hate Trump have been lying -- to themselves, mostly -- and covering it all up. And now that it's exploded, are they angry at themselves? No. They're angry at Biden. And you hope he'll stroke out, to save you the cognitive dissonance of your own years of enabling the Democrats' fraud on the American people.

It's something to behold.

If Benacerraf is not skipping the condition, then where does he recognize it? — TonesInDeepFreeze

Please accept my regrets for not engaging. I have little interest in supertask puzzles in general, and this thread has long since exhausted any points I could possibly make.

But there is more to say. — TonesInDeepFreeze

I don't think I'm going to get involved in the details at this point.

So indeed, let's go back to square one: — TonesInDeepFreeze

"Let's not and say we did," as the saying goes.

'=' is primitive in logic (first order logic with equality, aka 'identity theory'). — TonesInDeepFreeze

Aha. "Identity theory" is first order predicate logic with equality. Is that your own terminology? Nothing in the Wiki disambiguation page for identity theory refers to it

And '=' has a fixed interpretation (which is semantical, not part of the axioms) that '=' stands for identity.

So identity theory has axioms so that we can make inferences with '='.

The axioms are:

Ax x=x ... the law of identity

And the axiom schema (I'm leaving out technical details):

For all formulas P:

Axy((P(x) & x= y) -> P(y)) ... the indiscernibly of identicals

Then set theory adds its axiom:

Axy(Az(zex <-> zey) -> x=y) ... extensionality

Now we ask how we derive:

(zex & x=y) > zey

Answer: from the indiscernibility of identicals. Indeed the above is an instance of the indiscernibility of identicals, where P(x) is zex. — TonesInDeepFreeze

I'm going to pass on engaging with this. Just don't have the inclination at the moment.

But one query. If you are doing first order logic, how do you quantify over all propositions P? Maybe I shouldn't ask.

It seems to me that Benacerraf is skipping that condition. — TonesInDeepFreeze

@Michael's point, about which he and I disagree.

And so is the Cinderella example, which, if I'm not mistaken is a rewording of Benacerraf. — TonesInDeepFreeze

Don't believe so. But by expressing disagreement I invite rebuttal. I am supertasked out, really. Hope I have the strength to not get sucked in again.

Not quite. The mathematicians I knew BITD had little to no interest in discussing the distinctions between provability and truth. We were mostly in classical (complex) analysis. Mostly we are gone now. A few of us remain. — jgill

That's as true today as it was back then, logic being a niche, ignored by most math departments. But in terms of antiquity, Godel's work precedes you.

Yeah a Harris/Whitmer ticket would be more realistic. That being said Harris will more likely go for Shapiro because he's a white man. — Mr Bee

Like I say. Shapiros are not in style in this year's Democratic party. Which reminds me that in 2020, the Dems had an excellent black female VP candidate, Val Demings. But she was a cop, and cops were not in style in the Democratic party of 2020. Live by identity politics, die by same

Yep, and the delegates are not very eager to nominate him right now. We'll see if it snowballs into something. — Mr Bee

I looked this up. Biden has 3896 delegates, and everyone else has 43 combined. Biden is the overwhelming choice of Dem primary voters, and that's one of his advantages.

Just found this, which is just one article and doesn't prove anything, but it's still of interest.

Democratic convention delegates say they’re loyal to Biden and balk at other options


Democrats urging President Joe Biden to end his campaign and allow the party to select another nominee before – or during – August’s national convention are unlikely to find allies in the ranks of Chicago-bound delegates, who are increasingly closing ranks around Biden.

In fact now that Biden's dug in, some Dems are coming around. House Speaker Hakeem Jeffries is for Joe. Alexandria Ocasio-Cortez came out for Joe today. Senate majority leader Chuck Schumer came out for Joe. And surprisingly Senator John Warner, who the other day said he was gathering a group of anti-Joe Senators, today came out for Joe.

I believe that once Joe said he's staying in, people realize that he's going to be virtually impossible to dislodge. He has the power of the presidency, which is huge. He has Hunter and Jill on his team. And drug and hooker jokes aside, Hunter appears to be a capable ally at the moment. Joe has all those delegates. And the Democrats risk looking very anti-democratic if they swap in a last-minute candidate chosen by the party insiders. Joe looks stronger today than he did a few days ago when everyone thought he was toast.

And a lot of the backstabbers look unseemly, a point I've been making. You love a guy yesterday and knife him in the back today? That says more about the backstabber than it does about Biden. Looking at you, New York Times, Washington Post, Joe Scarborough, etc.

Sure, but everybody adamantly says they're in it until they aren't. I think it's too late for Biden to stop the dam from breaking within his own party. — Mr Bee

I'm on record that the Dems are not going to dislodge him as long as Jill and Hunter want him in. The Dems do not have the stones to impeach him or invoke he 25th amendment. My bet's a long shot, I'll give you that. Joe looks like toast. But what are the Dems really going to do if he refuses to step down?

There's a full-on civil war in the Democratic party. The inevitable result of decisions they took in 2020 and 2024. People were saying Biden was cognitively impaired as early as 2019. The Dems could have avoided this. Now they're stuck.

Too many different groups from the donors to the representatives to the senators are already saying he should step aside and likely this week (as congress reconvenes) this will lead to a large number of public statements for Joe to step aside. At some point such a situation becomes untenable. — Mr Bee

Like I noted, prominent Dems are also stepping up to support him. They realize that panicking right now could well come out worse than just getting behind Biden.

But who knows how he'll react. Is Joe selfish enough to stay in anyways even if it means the total collapse of his party? — Mr Bee

Oh yeah. Biden and his family are out for themselves. The stories about the family corruption are not "right wing propaganda" any more than his cognitive decline was. Biden's for Biden, and his family wants him to have pardon authority as long as possible. That dam might break too, and when it does, the family's going to want Biden in power.

Perhaps but it's clear his attempts to quiet any dissent through a mix of stubbornness and finally getting out there have been completely unsuccessful so far. — Mr Bee

Not so. The tide began to turn today (Monday evening US time as I write). Lot of Dems came out for Biden, even some who'd been against him just a day or two ago.

A normal politician would've taken drastic action immediately after that debate, doing numerous interviews, town halls, and unscripted events in order to assure people that they can do this. — Mr Bee

Of course Biden is incapable of doing any of that. But we started seeing it in 2019! They hid the guy all during the 2020 campaign. People have been talking about Biden's tragic age-related cognitive impairment for years. Media types have admitted they covered it up so as not to help Trump.

Biden has of course done what he can. He called in to Scarborough's program. He's given some teleprompter speeches. It's all he can do.

But that is not the point. The point is: Who is going to dislodge him, and how?

Biden instead went back to hiding for a week and later did a 20 minute interview where he still sounded rambling and delusional, and well we can sort of guess why. I think the video I linked to where he said he will be content with losing to Trump and ending democracy because all that matters to him is his reelection attempt will turn his critics off — Mr Bee

Agreed, of course. But again: Who is going to dislodge him? The parallel's been made with Nixon, when his advisors came to see him and told him it was all over. But he was facing certain impeachment and conviction. What leverage to the Dems have over Biden? A strongly worded letter? They have nothing. Let's see if they'll start impeachment or 25A proceedings. Of course they will not do it.

So it's advantage Biden, no matter how compromised his mental state.

This is the greatest political scandal of our lifetimes. This thing is just getting started. A full-on civil war in the Democratic party just four months from a highly consequential election. Anything can happen.

Perhaps Biden will have that stroke @Wayfarer is hoping for. Something to see, actual Biden supporters hoping for that. End stage Trump Derangement Syndrome. If the Dems had just ignored Trump and had a real primary season, Newsom or Whitmer would be beating DeSantis right now.

at least you've made clear what side you're on — Wayfarer

From a few weeks ago. I was struck by your extreme partisanship back then -- and now you are hoping your own preferred candidate will have a stroke.

What side are YOU on?

Can you not see your own moral corruption brought on by your extreme hatred of one man who was already president for four years and didn't do any of the things you claim he'll now do?

You don't even wish Trump would stroke out. With his diet and lifestyle it could happen. But no. You have worked yourself into such a state of anger and hate that you hope your own guy will have a stroke. And why? Because you are angry at yourself for going along with the lies. You and all the other Dems who are shocked, shocked that Biden's suffering the age-related cognitive impairment that was apparent in 2019. Do you have any self-awareness at all?

Your remarks got to me a few weeks ago. In case you're wondering why I'm addressing you about this.

I don't say that selecting and organizing the quotations is easy. It fits better with the fact that I tend to get slabs of time when I can pursue these discussions but in between, I'm not available at all. So the quick back and to is more difficult for me. — Ludwig V

Oh I see. That's what I like about discussion forums. You can pick up a topic weeks or even months later.

Don't get me started. What particularly annoys me is that so many people seem absolutely certain that they are right about that. I think it is just a result of thinking that you can write probability = 1, when 1 means that p cannot be assigned a probability, since it is true. — Ludwig V

Oh my. We must have a conversation about probability sometime. You're wrong about that. 1 is a perfectly sensible probability. But worse, probability 1 events may be false. For example if you randomly pick a real number in the unit interval, it will be irrational with probability 1, even though there are infinitely many rationals.

A friend once conceded to me that it was a degenerate sense of probability, which is like saying that cheese is a degenerate form of milk. — Ludwig V

1 is a perfectly sensible probability. Your friend is misinformed. As Mark Twain said, if you don't read the newspapers, you're uninformed. If you do read the newspapers, you're misinformed.

I think I shall stick to my view that defining an infinite sequence or getting a beer from the fridge is the completion of an infinite number of tasks. I don't think it gives any real basis for thinking that supertasks are possible. — Ludwig V

It's mathematically unhelpful to think of a infinite sequence as the "completion of an infinite number of tasks." It leads to confusion. It's not how mathematicians think about sequences.

You notice that maths outside time is metaphorical, right? — Ludwig V

No, it's literally true. Of course math as a human activity is historically contingent. But math itself speaks to truths that are outside of time.

I prefer to say that time does not apply to maths, meaning that the grammatical tenses (past, present and future) do not apply to the statements of mathematics. — Ludwig V

Ok, but IMO it's deeper than a semantic point.

I like "always already" for this. There is a use of language that corresponds to this - the "timeless present". "One plus one is two" makes sense, but "One plus one was two" and "One plus one will be two" don't. — Ludwig V

Right. We don't even have good words to talk about things outside of time. Timeless present is a pretty good phrase.

Yes. But there are complications. How does math apply to the physical world? — Ludwig V

As in Wigner's famous paper on the "unreasonable effectiveness" of math in the physical sciences. If math doesn't actually refer to anything, why's it so useful?

We have a choice between insisting that Non-Euclidean geometries are not created but discovered and insisting that they are not discovered but created - though they exist, presumably, forever. But if we create them, what happens if and when we forget them? — Ludwig V

I have no idea. I have myself argued from time to time that 5 was not a prime number before there were intelligent beings to observe that fact. I don't actually believe that, but I've argued it.

As I said before there are a number of ways to describe this. They're all a bit weird. — Ludwig V

This was in reference to 0, 1, 2, ... existing "all at once" in PA. What ways are there to describe this? Is it a timeless present? That's a great locution.

It sounds as if you are saying that "approach" is a simply two different senses of the same word, like "bank" as in rivers and "bank" as in financial institutions. — Ludwig V

Yes. "Approach" is a term of art in mathematics. It has a specific technical meaning that is unambiguous. It is not the same as the everyday meaning.

An old word given a new definition. Perhaps. — Ludwig V

Term of art. A lovely legal phrase. Lawyers commonly have to deal with the jargon of whatever discipline a a particular dispute is about.

We can think of this as a FUNCTION that inputs a natural number 1, 2, 3, ... and outputs 1/(2 to the power of n).
— fishfry
That's a very neat definition. I'll remember that. — Ludwig V

There's a class math majors take called Real Analysis, where they teach you all this; and after which you are forever clear in your mind about things that were formerly vague and fuzzy. Sadly nobody but math majors takes this class, leading to so much confusion.

But you can see, surely, how difficult it is to shake off the picture of a machine that sucks in raw materials and spits out finished products. — Ludwig V

Yes, that's a "function machine," a visualization when we teach functions to high schoolers. And of course mathematical functions are routinely applied to real world processes. A vending machine is a function of two variables: put in money and push a particular button, and the appropriate product comes out.

So the picture, or visualization of a function as a process or a machine is perfectly valid. The mathematical abstraction that strips away the process or machine interpretation is for the purpose of clarifying our ideas.

But actually, you are describing timeless relationships between numbers. Or that's what you seem to be saying. — Ludwig V

Yes. The elements of a sequence have a timeless relation to the index set 1, 2, 3, ...

The converse of extensionality is not provided by the law of identity. It is provided by the indiscernibility of identicals. — TonesInDeepFreeze

Oh. In that case @Metaphysician Undercover is right and you are making a point I can't agree with.

I take your point about set equality as expressed in the Wiki page on extensionality, which says, "The axiom given above assumes that equality is a primitive symbol in predicate logic."

If you mean something else, we're back to square one.

I think that this is what the so-called "paradox" of supertasks is all about. What is revealed is that at least one or the other, space or time, or both, must not be continuous. I think that's what Michael has been arguing since the beginning. Tones attempted to hide this behind sophistry by replacing the continuity of the real numbers with the density of the rational numbers. — Metaphysician Undercover

I've bowed out of the supertask discussion, having not typed anything new in weeks. It would be inappropriate for me to comment on anything @Michael said, since he'd then be obliged to reply and we'd be right back in it again. @TonesInDeepFreeze merely made the point that 1/2, 1/4, etc only requires the rational numbers. Perfectly sensible observation.

The real issue is that if one of these, space or time, is not continuous, then it cannot be modeled as one thing. There must be something else, a duality, which provides for the separations, or boundaries. But I don't think anyone has shown evidence of such a duality, so we have no real principles to base a non-continuous ordering system on. — Metaphysician Undercover

I'm fully supertasked out.

I'd say this is similar to Tones' use of "identity" in set theory. We take a word, such as "approach", which clearly does not mean achieving the stated goal, and through practise we allow vagueness (to use Peirce's word), then the meaning becomes twisted, and the use of the word in practise gets reflected back onto the theory. So we have the theory stating one thing, and practise stating something different, then the meaning of the words in the theory get twisted to match the practise. Practise says .999... is equal to 1, so "approach" in the theory then takes on the meaning of "equal". Practise says that two equal sets are identical, so "equal" in the theory takes on the meaning of "identical". These are examples of how theory gets corrupted through practise when the words are not well defined. — Metaphysician Undercover

Let's keep the Infinity theory in that thread. Well I'm not a moderator here so nevermind, do what you like. I prefer not to engage in these thread-hijacking points here. Every discipline has its terms of art, which confuse non-practioners. When a doctor tells you your liver is "unremarkable," that's great news and not an insult.

I'm an antique. Truth for me is associated with proof. — jgill

You are not old as Godel's proof, which was published in 1931. Godel's results are therefore more antique than you. Perhaps you're a logicist at heart. They thought mathematical truth was derivable from logic.

https://en.wikipedia.org/wiki/Logicism

Our relationship was mostly one-sided, with me being the main beneficiary of our conversations. I'm glad to hear that you found some benefit in meeting me as well. This is a great way to conclude our conversation. Cheers! — keystone

Likewise.

Both are right, and well said. In both PA and Z without infinity (even in Z with the axiom of infinity replaced by the negation of the axiom of infinity), we can define each number natural number, and in Z we can prove the existence of the set of all and only the natural numbers. — TonesInDeepFreeze

I agree with everything you wrote in this post.

There's many things they don't teach in school when looking at what my children have to study. Usually the worst thing is when the writers of school books are too "ambitious" and want to bring in far more to the study than the necessities that ought to be understood. — ssu

I hear awful things about the teaching of math these days and teaching in general, but I have no personal experience. I did try to help a friend's 13 year old with her math homework once and couldn't make heads or tails of it.

I looked at this. Too bad that William Lawvere passed away last year. Actually, there's a more understandable paper of this for those who aren't well informed about category theory. And it's a paper of the same author mentioned in the OP, Noson S. Yanofsky, from 2003 called A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points. Yanofsky has tried to make the paper to be as easy to read as possible and admits that when abstaining from category theory, there might be something missing. However it's a very interesting paper. — ssu

Thanks so much for that reference and to Yanofky's YouTube channel. RIP William Lawvere.

We do have significant disagreement concerning your claim to a proof that "X=X", when X signifies a set, means that X is the same as itself by virtue of the law of identity. You have not provided that proof in any form which I could understand. — Metaphysician Undercover

I ain't draggin' your butt through the Wiki page on extensionality again. That was a very dispiriting experience the last time I did it. However, I'll point you at the relevant sentence.

"The axiom given above assumes that equality is a primitive symbol in predicate logic."

https://en.wikipedia.org/wiki/Axiom_of_extensionality

What is the case, is that "X=X" is an ambiguous and misleading representation of the law of identity. This is because "=" must mean "is the same as", to represent that law, but it could be taken as "is equal to". — Metaphysician Undercover

I don't make a distinction between "same as" and "is equal to." In math they're the same. If you have different meanings for them, it does not bear on anything I know or care about.

Notice that in the axiom of extensionality it is taken to mean "is equal to". Therefore when Tones takes "X=X" to be an indication of the law of identity there is most likely equivocation involved. — Metaphysician Undercover

I don't think so. I don't agree with you.

So, do you recognize, and respect the fact that group theory is separate from, as a theoretical representation of, the objects which are said to be members of a specified "group"? — Metaphysician Undercover

Yes, as ichthyology is different from any particular fish or school of fish or class of fish.

And, I'm sure you understand that just like there is a theoretical representation of the group, there is also a theoretical representation of each member of the group. In set theory therefore, there is a theoretical "set", and also theoretical "elements". — Metaphysician Undercover

I don't know what you mean by "theoretical" elements. The integers form a group under addition. Is the number 5 theoretical? Well it's abstract, as numbers are. What of it?

So when Tones says that a set may consist of concrete objects, this is explicitly false, because the set is the theoretical representation, and the elements of the set are theoretical representations as well. Through such false assertions, Tones misleads people and earns the title of sophist. — Metaphysician Undercover

Is concrete different or the same as physical? 5 is a concrete mathematical object, I guess. "Concrete" is not a term of art in this context, although there's a thing called a concrete category.

You're making up your own definitions of words and arguing with me about them. I am lost.

When Tones speaks about the set "George, Ringo, John, Paul", these names signify an abstract representation of those people, as the members of that set, the names do not signify the concrete individuals. You, Fishfry, have shown me very clearly that you know this. So there is an imaginary "George", "Ringo" etc., which are referred to as members of the set. The imaginary representation is known in classical logic as "the subject". We make predications of the subject, and the subject may or may not be assumed to represent a physical object. Comparison between what is predicated of the subject, and how the object supposedly represented by the subject appears, is how we judge truth, as correspondence. — Metaphysician Undercover

Too deep for me. I take Tones's point that = in set theory derives from = in the underlying logic. I have no problem with that and it was perfectly obvious as soon as he drew my attention to it.

What is important to understand in mathematics, is that the subject need not represent an object at all. It may be purely imaginary, like your example Cinderella. This allows mathematicians to manipulate subjects freely, without concern for any "correspondence" with objects. Beware the sophist though. I believe that when the sophist says that the members of a set may be abstractions, or they may be concrete objects, what is really meant if we get behind the sophistry, is that in some cases the imaginary, abstract "element", may be assumed to have a corresponding concrete object, and sometimes it may not. Notice though, that in all cases, as you've been insisting in discussions with me, the elements of the sets are abstractions, as part of the theory, and never are they the actual physical objects. Failure to uphold this distinction results in an inability to determine truth as correspondence. And that is the effect of Tones' sophistry — Metaphysician Undercover

Ok you should take this up with Tones. You failed to convince me that I am a victim of anyone's sophistry.

I'll return to the schoolkids example briefly to tell you why I didn't like it. — Metaphysician Undercover

Jeez man you already told me you don't like it, so I stopped using it. To me it's a good illustration of how a collection may be ordered in many different ways.

Using that example made it unclear whether "schoolkids" referred to assumed actual physical objects, or imaginary representations. That's why "real-world analogies" are difficult and misleading. The names, "George", "Paul", etc., appear to refer to real-world physical objects, and Tones even claims that they do, but within the theory, they do not, they are simply theoretical objects. If we maintain the principle that the supposed "schoolkids" are simply imaginary, then they have no inherent order unless one is stipulated as part of the rules for creating the imaginary scenario. Set theory ensures that the elements have no inherent order, but this also ensures that the elements are imaginary. — Metaphysician Undercover

Sets are mathematical abstractions. I don't know what you mean by "imaginary," which is a term of art in math referring to complex numbers with real part 0.

This is wrong, and where Tones mislead you in sophistry. A set is not identical to itself by the law of identity. The set has multiple contradictory orderings, and this implies violation of the law of identity. — Metaphysician Undercover

No you are as wrong as can be about that.

We allow that "a thing", a physical object has contradictory properties with the principle of temporal extension. At one time the thing has a property contradictory to what it has at another time, by virtue of what is known as "change", and this requires time. But set theory has no such principle of temporality, and the set simply has multiple (contradictory) orderings. — Metaphysician Undercover

You have been misunderstanding this point for years, and I surely have nothing new to say on the topic.

As I said, the reference was to the identity of indiscernibles, not the law of identity. You recognize that these two are different. The proof was not by way of the law of identity. If you still believe it was, show me the proof, and I will point out where it is inconsistent with the law of identity. — Metaphysician Undercover

I'd walk you through the Wiki page on the axiom of extensionality, but this also is something I've done for years with you, to little productive effect.

We agree on this very well. The principle we need to adhere to, is that this is always an "abstraction game". If we start using names like "Ringo" etc., where it appears like the named elements of the set are concrete objects, then we invite ambiguity and equivocation. And if we assert that the elements are concrete objects, like Tones did, this is blatantly incorrect. — Metaphysician Undercover

You must be misrepresenting what Tones said, since he made his point with me; and after that, I found several clear references supporting his point.

The three fundamental laws of logic, identity, noncontradiction, and excluded middle, are inextricably tied together. Therefore one cannot discuss identity without expecting some reference to the other two. There has been some philosophical discussion as to which comes first, or is most basic. Aristotle seemed to believe that noncontradiction is the most basic, and identity was developed to support noncontradiction. — Metaphysician Undercover

Aristotle thought bowling balls fall down because they're "like the earth," and fire goes up because it's "like the air." But never mind that. You are swimming in murky logical waters that have absolutely nothing to do with anything I'm saying.

What C.S. Peirce noticed, is that if we allow abstract objects to have "identity" like physical objects do, as Tones seems to be insisting on, then necessarily the validity of the other two laws is compromised. Instead of denying identity to abstract objects, as I do in the Aristotelian tradition of a crusade against sophistry, Peirce sets up a structure outlining the conditions under which noncontradiction, and excluded middle ought to be violated. — Metaphysician Undercover

I'm ill-equipped to argue Peirce and Aristotle with you. I don't think your points bear on set theory.

You are missing the point. The law of identity refers explicitly to things, "a thing is the same as itself". A "set" is explicitly a group of things. Therefore when you say X = X, and X is a set, rather than a thing, then "=" does not signify identity by the law of identity. — Metaphysician Undercover

A set is not a "thing?" A set is a thing in set theory. It's an abstract thing, to be sure. But it's still a thing.

Right, this is the point. "Time", or temporal extension allows that a thing may have contradictory properties, at a different time, yet maintain its identity as the same thing, all the while. This is fundamental to the law of identity. Without time (as in mathematics), the multiple orderings of a set, which Tones referred to, are simply contradictory properties. That is a good example of the issue Peirce was looking at. — Metaphysician Undercover

No, orderings are not "contradictory properties." Technically, an order on a set is another set, namely the set of pairs (x,y) for which we mean to denote that x < y in the ordering. The ordering is distinctly and noticeably separate from the set it applies to.

Fine, but can you respect the fact that "equal" does not imply "identical", despite the sophistical tricks that Tones is so adept at. — Metaphysician Undercover

That distinction has no meaning or relevance in my understanding of the world. "equal" and "the same as" are entirely synonymous. I do take your point that 2 + 2 = 4 does not mean that they are the same as strings. We've been over this many times, as have many philosophers. The morning star and the evening star are the "same object," (which turns out to be a planet and not even a star) but they have different senses. What of it, this is not news to anyone.

No, that's simply wrong. A particular apple is a physical object. A set is an abstraction. An instance of an apple is a physical object. Your supposed "instance" of a set is an abstraction, a concept. The two are not analogous, and I argue that this is a faulty, deceptive use of "instance". — Metaphysician Undercover

Would you agree that "number" is a general abstraction and that 5 is a partcular instance of number? Isn't that the most commonplace observation ever?

An instance is an example, and understanding of concepts or abstractions by example does not work that way. — Metaphysician Undercover

5 is a terrific example of a number. One of the best I know.

Assume the concept "colour" for example. If I present you with the concept "red", this does not provide you with an instance of the concept "colour". — Metaphysician Undercover

It doesn't? Red is not an instance of the concept of color? How do you figure that?

An instance of the concept "colour" would be the idea of colour which you have in your mind, or the idea of colour which I have in my mind, expressed through the means of definition. — Metaphysician Undercover

Incoherent. Red is a color. Red is an instance of the concept of color.

Each of those would provide you with an example of the concept of "colour", an instance of that concept. The concept "red" does not provide you with an example of the concept of "colour". — Metaphysician Undercover

What??

Nor does a specific "set" provide you with an example or instance of the concept "set". — Metaphysician Undercover

Of course it does.

What you are saying in this case is completely mixed up and confused. — Metaphysician Undercover

Same back atcha.

Michigan is in play in large part because Biden is the nominee and pissing off Arabs with what he's doing in Gaza. That's why I think Whitmer is an ideal candidate since she can take any stance on Israel and win Michigan easily but apart from his age, Biden's foreign policy is a big drag on the ticket. — Mr Bee

Good point, Whitmer is popular in Michigan and can withstand the Palestinian-supporting component of the left. Makes sense. Still tricky to leapfrog Kamala. Do you mean Whitmer for veep or prez? Kamala has a constituency within the party.

I've heard there's also a "good conscience" rule the DNC can add for delegates to not vote for Biden, but right now the Dems are trying to convince grandpa that he is perhaps not the best driver in the world and relinquish his car keys voluntarily. Next week will probably see the dam breaking. — Mr Bee

Oh I see I hadn't heard that. Internet says that "DNC rules encourage but don't specifically require delegates to vote for the candidate they're pledged to support. Instead, the rules say, “All delegates to the National Convention pledged to a presidential candidate shall in all good conscience reflect the sentiments of those who elected them ..."

That does seem like an out. I thought the delegates were firmly bound, but evidently not.

My take on this situation is that the Dems are in denial when they say, "Dems are trying to convince grandpa that he is perhaps not the best driver in the world ..." In fact Joe has stated that he's in it to win it, and he has Jill and now Hunter on his side. And he's President of the United States. There's a lot of power in that. The Dems are going to have to force Joe out. And I don't think they'll be up for it. The unseemliness of the Dems trying to destroy their own president.

My take -- my out-on-a-limb prediction -- is that in the end, the Dems will not persuade him to drop out. They will either need to impeach him, or invoke the 25th Amendment. And I predict the Democrats will not have the stones to do that. And besides, if they do move to impeach or invoke 25A, the Republicans will oppose them! Imagine the hilarity that would ensue. The GOP would love to run against Biden. Without GOP support the Dems can't get rid of Joe.

I think the Dems made their bed last year when they decided not to have an open, competitive primary. They are stuck with Biden until Jill says so. And she didn't come this far to give up now.

Nothing I can do about it, I’m not even an elector (although my son lives in the US and is a dual citizen.) I’m still holding out hope that Biden will see reason — Wayfarer

This is my thesis again. The Dems are hoping Joe will quit. But Joe has said he's not quitting, and he and Jill and Hunter are circling the wagons. In the end the Dems are going to have to act; by impeaching him, 25A-ing him, or deliberately incapacitating him.

(and rather uncharitably wishing he’d have a mild stroke which would take the matter out of his hands.) — Wayfarer

Can you see the irony, dare I say depravity, of hoping fo such a thing? In a candidate you supported five minutes before the debate?

I'm struck by the viciousness of the Dem and left response. All those who had Joe's back five minutes ago, and are now stabbing him in the back. And why is the response so emotionally intense? Because these are all of the people who didn't say anything a year ago, when they could have called for open and competitive Democratic primaries. They didn't say anything in 2020, when Biden was doing badly in the polls and the DNC did the Clyburn deal to install Biden. Along with Kamala, who'd dropped out of the 2020 race in 2019, polling in single-digits in her own home state. She got taken apart by Tulsi Gabbard in a debate, and never recovered. The media are pumping her up this week, but her negatives aren't going away.

Now the bill's come due, and the Dems are hoping Biden strokes out soon. You're not the only one. Perhaps that stroke won't come along by itself, ya know? Slip Joe a little something in his bowl of ice cream. That's what the Dems have come to.

Is that extreme? Just look at what you wrote. You are not the only Dem thinking that way. But a year ago when the DNC decided not to have real primaries, you said nothing. This is a fiasco of the Dems' own making.

But if he stays the candidate, I’m now convinced that Trump will win, — Wayfarer

That's been clear a long time, and even from before the general public found out about his tragic age-related cognitive decline. Biden's policies are unpopular. You can't fix that with a younger candidate. Of course I'll stipulate that you disagree with me on policy, and I'm not here to argue that. Many voters are not happy with how things have been in the Biden administration and swapping in a younger candidate with the same policies is not going to change that many votes.

and that it will be an unqualified disaster for America and the rest of the world (but that’s not something I’m going to debate outside the Trump thread, of which I’m steering clear.) — Wayfarer

I understand your feelings about that. No need to discuss the respective merits of the candidates. The scandal is what's interesting. This Biden mess is going to be the biggest political scandal in my lifetime, bigger than Watergate. Just swapping in a new candidate is not going to solve the Dems' problems, It's raise a whole host of new ones, starting with fundamental democratic legitimacy. Will voters stand for yet another last-minute DNC back-room deal?

Yes. I was saying in a complicated way, that a long post is not, for me, a bad thing. — Ludwig V

Oh I see. I prefer shorter posts so I don't get lost in the quoting!

That's a useful tactic. I shall use it in future. — Ludwig V

Yes, I just highlight the whole post and say Quote.

He did indeed. It was very common back in the day. It was disapproved of by many, but not treated as unacceptable. I don't think anyone can really understand how horrible it is unless they've actually experienced it. — Ludwig V

"Back then" wasn't that long ago, this scandal's just a few years old IIRC. I didn't follow the particulars. I'm sure it's just as common today. Or maybe not as 1950's, say. Thinks have changed since then. Still. The male-female thing, those are very deep energies being played with. You are not going to stamp it out with rules. You can change the form in which the scenarios are played out. What Eric Berne called the games. Games People Play, remember that?

Exactly. There's a lot of refinement needed. But that's the basic idea. What those objects are is defined entirely by their use in mathematics. — Ludwig V

There's another funny thing that goes on. Sometimes you make a definition, and it DOESN'T bring a mathematical object into existence. For example we write down the axioms for a group. But we have no idea if there are any groups, or if the axioms are perhaps vacuous. So the next step is to exhibit some groups, like the additive structure on the integers. This is a very common pattern in math: Make a definition, then show that there's something that satisfies the definition!

I was just being pedantic. It was a thing in the era before Descartes &c. But I understood that the distinction was "potential" and "actual". Nonetheless, the idea of a "completed" infinity catches something important. — Ludwig V

Right, potential/actual versus potential/completed. I've heard them both. Since they don't come up in math I kind of use them interchangeably. But they probably have more specific technical meanings or contexts I don't know about. But for me, I just regard the axiom of infinity as the sharp boundary between the two concepts. Induction on the one hand, versus a "completed loop."

That's a very helpful metaphor. — Ludwig V

Shoulders of road. Bounds. Glad that helped.

If I am understanding you, you think time is somehow sneakily inherent in math even though I deny it.
Have I got that right?
— fishfry
Yes.
I cannot fathom what you might mean.
— fishfry
Nor can I. That's the problem. — Ludwig V

I quoted that entire exchange. I can't fathom your meaning. But you say you can't either.

Time is a concept in physics. You can see that, right? Math is outside of time. It doesn't describe or talk about time, though it can be used by physicists to model time. And then again, we have no evidence that the mathematical real numbers are even a decent model for time. The real numbers are continuous, but nobody knows if time is.

If you can give an example of what you are thinking, that might be helpful.

Why is this a problem? The traditional view is that mathematics, as timeless, cannot change. Our knowledge of it can, but not the subject matter. (Strictly that rules out creating any mathematical objects as well, but let's skate over that.) — Ludwig V

Ok. You seem to be agreeing with me.

"A sequence does not approach its limit in time" makes no sense. — Ludwig V

And now you're not agreeing. The word "approach" is colloquial. It is not intended to evoke images of panthers stalking their prey, or arriving at your destination in a car. Not at all. It's just the word we use for the limiting process. But 1/2 + 1/4 + 1/8 + ... IS 0; it does not "become" or "approach" zero. It's this language ambiguity that is the source of so much online confusion about the subject. See any .999... = 1 debate. You'll hear that .999... "approaches but does not reach" 1. But the sum is exactly 1 nonetheless, by virtue of the definitions. The math is designed to make it work out.

I may be about to solve my own problem. That doesn't mean that raising it with you is not helpful. — Ludwig V

I'd like to know what the problem is, regardless. If for no other reason than to make sure you're understanding it correctly!

We have to accept that a sequence approaching its limit is not like a train approaching a station. — Ludwig V

Correct. It's a shame we use the word "approach," because many are confused by that.

The train is approaching in space and time. But you can't ask what time the sequence left its origin and when it will arrive at its limit. — Ludwig V

Right. Trains are physical objects. Numbers in a sequence are mathematical abstractions. They don't live in the physical world.

You can call the sequence approaching its limit a metaphor or an extended use. The train approaching the station is the "core" or "paradigm" or "literal" use. The sequence approaching its limit is a different context, which, on the case of it, makes no sense. So we call this use is extended or metaphorical.
We can explain the metaphor by drawing a graph or writing down some numbers and pointing out that the different between n and the limit is less than the difference between n+1 and the limit is less and that the difference between n and n-1 is greater.
And so on. — Ludwig V

I think it's just a confusing use. When mathematicians use the word approach, in their minds they already have the full context of the theory of limits. So they are not confused. But non-mathematicians hear the word and associate it with is everyday context. No harm is done, till these misunderstanders show up to .999... = 1 threads online. Then we get problems.

Yes. I realize this is border country. Godel seems to live there too. — Ludwig V

With supertasks? I don't think so.

Maybe a Josh Shapiro — Mikie

I imagine Shapiro would be a top pick for VP, at least if she were smart. — Mr Bee

Not a good year for Shapiros in the Dem party as long as Michigan's in play.

it's a wild card, things could shift very quickly.) — Wayfarer

What do people in this thread plan to do about Biden? The biggest wildcard is that he's dug in. He is on record as saying, "No one is pushing me out of the race." He's made this perfectly clear. And Jill is fierce. You can talk Kamala and Gavin and Gretchen and Michelle all you like, but Biden's not budging.

Are Democrats ready to either impeach him or invoke the 25th Amendment? If not, how are you going to dislodge him?

Here is what ChatGpt has to say — jgill

Et tu? ChatGPT doesn't know anything about mathematical philosophy. It just statistically autocompletes strings it's been fed.

Then the crank, in his usual manner of self-serving sophistry, misconstrues fishfry. fishfry didn't contradict that the law of identity is different from the identity of indiscernibles. — TonesInDeepFreeze

cc: @Metaphysician Undercover

You guys get a room! LOL. FWIW you convinced me that X = X for sets follows from the definition of = in the underlying predicate logic. How the = of logic relates to the law of identity, I have no idea.