Uhh... that's really misrepresenting it. Basically, the US has the same system as the majority of Africa and a couple of failed (middle) Eastern States.

overview universal healthcare in the world

The NHS is set up differently than other European countries. Part of the reason why it's struggling is because conservative governments keep reducing funding increases (it still increases but at a much lower rates) and lowering taxes, making it appear as if it becomes disproportionaly more expensive as part of the budget. — Benkei

I'll stipulate that we disagree on health care policy. This thread's not the place. But if health care is "free," who pays for it? Taxes would go through the roof. And what makes you think the Democratic party could run health care for 300 million people? You already forgot the Obamacare rollout debacle?

These types of posts are so funny. — Maw

I'm glad to see a Bernie supporter maintaining a sense of humor. I'd assume most are in a state of shock and depression. Liz quit today and failed to endorse anyone. It's down to Bernie and Biden. How did the Democrats come to this? I wonder if lifelong centrists who swerved left, like Cory and Kamala, realize that they would have had a better chance staying in the middle. I'd gladly have supported the business-oriented Democrat Cory Booker, but not the bug-eyed Spartacus nonsense that started when he decided to run for president.

I think they were trying to reject Trump. Closing ranks is the way to do that. We'll see if the left side of the party is wise enough to realize that. — frank

Every AOC-backed candidate lost. Cenk Uygur, running to fill Katie Hill's vacates House seat, lost badly. Leftist candidates lost to centrists nationwide. It could mean nothing or it could be that we've reached peak Woke and the voters have had enough. We'll find out.

One interesting take I read is that the Dems might have been wrong to coalesce around Biden. The GOPs failed to coalesce to stop Trump, but Trump won the general election. There's something to be said for that. Getting behind senile and corrupt old Joe will be a disaster.

And likely Bernie will do what he did in 2016: give support to Joe as he did the last time with Hillary.

The Good Loser. Same repeat now with Joe coming soon. — ssu

Yup. Bernie has the heart but not the cojones. Just like he let Hillary off the hook on her email scandal He should have hit her hard on her corruption and carelessness with classified documents. He should hit Joe hard on his corruption. There's a debate coming up soon, we'll see if Bernie wants it or not.

TF proves there is no such set. But meanwhile set theory proves there is that set. The set is the universe for a model of TF. The set itself is not a member of that universe. — GrandMinnow

You're agreeing with me. You simply haven't made your case IMO. If it's the extension of a predicate and it's not a set in TF then it's a proper class. It may well be (and of course is) a set in some more powerful theory such as ZF.

dbl post

Most of the Western world has those policies and has no problems paying for them. There's nothing unrealistic about them. — Benkei

Yeah yeah Denmark population 5 million. Britain's NHS is a total horror show disaster. Canada's too. I'll take the opposite side of the "free health care free college free everything" debate if you please.

Bernie is right behind Joe, with a less than a million vote difference, and we still have about 60% of delegates left to go. — Maw

The fix is already in. The deal's done. My condolences to the Bernie fans on the board. Note also that the mythical youth vote didn't bother to show up. Of course a lot can happen between now and the convention so we shall see. But there's no way to spin Super Tuesday as anything but a shocking defeat for not only Bernie, but also all the AOC-backed Congressional candidates. The Dem voters have rejected ultra leftism.

Like his ideas about which pieces of legislation resulted in harming the most vulnerable members of American society? Like his ideas about what actually caused the tremendous disparity in wealth that we see today? Like his ideas about not continuing to allow money lenders to peddle misleading predatory and/or damaging financial instruments to everyday trusting Americans? Like his ideas about doing what it takes, over the long haul, to cultivate a politics that benefits nearly everyone across the board? — creativesoul

Late night on Tuesday. Biden beat Bernie decisively. Looks like Dem voters aren't buying what Bernie's selling. Biden will be a disaster of course as a candidate but these results are certainly interesting. A big Bernie win was predicted but instead the Dems got Amy and Pete to quit and endorse Bernie and all the Dem voters fell into line. Rarely if ever have the Dems been this organized recently.

Regarding your questions, I am not arguing with Bernie's idealistic beliefs; only his specific policies, their costs, and his highly unrealistic plans to pay for them.

Interesting post with some thought put into it. Thanks. — 0 thru 9

Thank you! Tell that to my many critics ...

But getting to the fine print... Ok, fair enough. You either don’t think Bernie can win, or if the unthinkable happened, it would be like having a former Hippie as president, throwing dollar bills and big doobies (marijuana) out of the Federal Reserve window to his crazed, brainwashed snowflaky fans. With Bob Dylan and the remnants of the Grateful Dead (including some holy relics from Jerry Garcia), Snoop Dogg with a reformed Public Enemy, performing a free concert on the White House lawn in some bizarre combination of Woodstock and the March on Washington. Bernie is inaugurated wearing a tie-dyed shirt, with Noam Chomsky standing next to him. (Ok, maybe that is going too far for a joke, lol. Anyway...) — 0 thru 9

Ok no totally I would love that. It's his Marxism not his hippiedom. I wasn't an official hipped but I certainly identified with them. And we were all against the war. I marched on Washington. I loved the hell out of this paragraph. You reached right into my forgotten past and hit the nail on just about every point. I saw Chomsky speak against the war. I love the crazed hippie president image. I'm imagining Bernie as Fat Freddy from the Fabulous Furry Freak Brothers. I was so into those comics. If Bernie was promising an actual return to the hippie ethos I'd put on my tie-tie and beads and sign up for the whole program.

You are not a “democratic socialist”. That is fine. Yes, obviously Bernie calls himself that. And his opponents do also, but use “scare quotes”, make clucking noises of disapproval, and warn of dire consequences. — 0 thru 9

Even if I concede the poin[ that Bernie is not a Stalin socialist but more like a Mr. Rogers socialist, I would not change a word of what I wrote. Because Bernie has many followers who ARE Stalin socialists or worse. The #CancelCulture out there is like Chairman Mao's cultural revolution, complete with public shaming sessions. struggle sessions, they use to call them. The far left scares the shit out of me lately and they're all way into Bernie.

We have all seen the juicy sound bites. (Though I’m not sure what exactly these terms are supposed to mean anymore. I know in general what they are intended to mean. But words, labels, ideologies, and especially political philosophies have been ever so slowly drained of whatever meaning they once had. It all seems to be advertising, propaganda, and personal branding. Like how Christianity has somehow mutated into an apologist for the war machine. Oh pardon me, “national defense system”. Anyway, please excuse this tangential philosophical point). — 0 thru 9

Bernie has a lengthy track record of making statements in support of the most appalling regimes. I'm taking the man at his word and you want me make excuses for him. I'm not buying that.

But whatever other valid points you make, it is an extreme exaggeration to say Sanders “wants to destroy” the whole system. — 0 thru 9

His plans would double the annual spending of the Federal government. Doing that would blow up the world's economy. Now we could be here all night arguing THAT particular claim, but it's what I believe and you should take that for an indication of where I'm coming from. If someone did have that belief, then it would be rational for them to say that Bernie is going to blow up the country's economy -- even if you disagree with the premise. Fair?

Come on now. Let’s be fair. That is practically calling him a communist, or something worse, but without the directness to do so openly. — 0 thru 9

It was Bloomie who called Bernie a Communist. I'm not calling Bernie names. I'm pointing out that his policies, as outlined on his website, would blow up the US economy and almost certainly take the rest of the world economy with it. That is my considered opinion. I'm not calling anyone names. I like Bernie as a person. Wish he'd hit Hillary harder on the emails in 2016, he could have been president. He would have beaten Trump then. Not now, unless there's a huge economic crash -- which could happen.

Because it is a totally unfounded cheap shot, either implied or explicit. Like calling Sanders “ignorant”. Ok, sure.. Par for the course in an election campaign, “sticks and stones”, etc. Hyperbole and humor. I do it too. (Like this silliness for example. Trump’s new slogan: “Four more years! Let’s Have Another Orangasm!” :snicker: Although come to think of it, Joe Biden has been looking a little orange lately. Is there a tariff-caused shortage of natural-looking makeup for men? Bernie is pale and proud, lol). — 0 thru 9

It is not a cheap shot. It's my informed opinion having glanced at Bernie's plan on his website, and verified in several different places that he will double the annual spending of the federal government. If anyone did that there are certain disastrous economic consequences that would follow. I don't want to spend time arguing this particular point. I'd like you to acknowledge that this is my opinion, and agree to the validity of the argument that IF I believe as I do, THEN it is rational for me to state that Bernie would destroy the world economy virtually overnight.

There are no personal attacks or cheap shots whatever in my comments.

But most observers can see these type of ploys as a desperate attempt to trip up the runner who is 50 yards ahead of everybody in this preliminary track meet. Tackling is not officially allowed in a foot race. Just because it happens and often goes unpunished, doesn’t mean nobody notices or cares. Ok, maybe this is just some sideline forum of internet opinion, mixed with some occasional philosophical insight. But if one wants to stand on their words, they have to have some kind of rational foundation. (Or even a relatively honest emotional one. That is acceptable, if expressed fairly. Emotions are part of who we are, of course). — 0 thru 9

I hope I have outlined my rational foundation with sufficient clarity. It is as follows:

P1: Bernie's own detailed plan on his website would, according to most fair observers, would double the annual spending of the federal government.

P2. That would be a Very Bad Thing; for reasons I don't feel like enumerating because it would amount to spending the time to boil down my thoughts and draft a good response and I don't want to do that at the moment. So I ask you to simply accept that I believe doubling spending is a Very Bad Thing.

C: Therefore electing Bernie would be a Very Bad Thing.

I hope, as I've tried to explain, that if you accept that if I hold P2, then it's valid to conclude C. Even if you disagree about P2. I hope you regard this as an exercise in rationality. In fact you may be confusing me with other people, because I've never disparaged Bernie personally. I really like the crazy old coot. I just don't want him to be president.

Now look... (Just kidding. Don’t you hate when debaters start with that bossy-sounding introduction? It’s like... LOOK... (pregnant pause... either signifying depth of thought, or perhaps an unspoken insult. Such as: LOOK... ya big goofball etc... Almost as bad is someone saying “LISTEN... blah blah...” I’m waiting for the first debater to go all in with “LOOK... LISTEN... and LEARN...” ) Sorry for the rant. — 0 thru 9

Did not follow that para.

Anyway, I am disillusioned (or perhaps “realistic”) about the Democratic Party. — 0 thru 9

It's hard not to be. Over the years they've lost their moral center and what we see today is the end result. At least I hope it's the end, if they get any worse they'll have to be designated a terrorist organization. [That's hyperbole for the purpose of intended humor; not a point on which you need to accuse me of name calling].

(I refuse to say I’m “woke”. Nor am I a “Bernie Bro”. Nor any other kind of “bro”. Buzzwords are as annoying as flies). But I had hopes for the Obama presidency. I thirstily sipped the Kool-aid, but only a little. I thought maybe, somehow he would understand, hoped he would care, figured he would at least try to make some little thing fairer. — 0 thru 9

As a symbol he was revolutionary and much needed. As a president, his greatest trick was to convince people he was a leftist. His foreign policy was Bush's 3rd and 4th term. Instead of apologizing for and prosecuting Bush's torture regime, he institutionalized it. He had many many scandals but gets away by saying he was scandal-free because the medial lets him get away with it. He had his foot on the brake of the economy. Trump's economy would have been Obama's if Obama had a clue.

Great symbol, so-so president. I liked his no-drama approach, we all prefer that over Trump's bombast.

Maybe he was slightly better than the alternative. — 0 thru 9

Oh no. McCain would have blown up the world. I am not on Team McCain, all the people who want to make a saint out a corrupt warmonger. Between Obama and McCain I'd take Obama 100 times out of 100, even in hindsight. All in all Obama was a pretty good president but he had a lot of flaws that his defenders don't admit, and the reaction to his flaws led to Trump.

Maybe the Middle East would have exploded with 4 more years of Neo-Con meddling (and that’s putting it very politely). Maybe not. What do I know? Very little probably. My point is that the Democratic Party (which is neither democratic, nor much of a party) is NOT “liberalism” or even “the left wing” in its entirety. Not even close. The two-party system is an effective monopoly, a good cop/bad cop routine. Two sides of the same coin. They speak for no one except themselves mostly. — 0 thru 9

On that we agree wholeheartedly. I hate partisanship. I believe the worst the Dems say about the GOPs and the worst the GOPs say about the Dems. I hate both parties. They both deserve to die.

In fact that's why we had a Trumpian populist insurgency in 2016 and now a Bernie populist insurgency in 2020. The centrist consensus isn't working and people are starting to notice.

And this week we see the establishment fighting back ... with Joe Biden? This senile and corrupt warmongering, civil-liberties-hating, tool of the banking industry? Do people even know who Joe Biden is? You can be sure Trump will remind us.

If Bernie were any more independent, he’d be on the sidelines with the rest of us. If he were any less independent, he’d be another gravy train rider looking for the path of least resistance. He definitely is NOT Frodo Baggins trying to destroy the evil Ring, nor Luke Skywalker trying to blow up the Death Star. He is not even trying to “level the playing field”... whatever that means. There is no playing field. There is a pyramid and a ladder, with those at the top of it pouring boiling oil on those below. Maybe at one time, the middle-class dreamed that there was room for more at the top of the pyramid, but there never was. Not a pretty picture. At best, Bernie Sanders SEEMS to be TRYING to go in a new direction that is at least a little tiny bit fairer for most people. I’ll take that chance, and hold him to his wager. — 0 thru 9

You're right. Bernie's just an old leftist who's been spouting the same slogans for 40 years and finally the country has become ripe for his message. He's more of a symbol than an individual for his followers. But what a character ... there's something about him, the crazy old uncle with the hair and the hands waving and the delivery and the cadence ... he's got his act down really well. He's actually a very effective speaker. Like I say, I like the guy. Just not his ideas.

The casino has stacked the odds against us, and rigged the slot machines. Even the glittering showgirls are picking our pockets. Now it seems the only way to win... is to leave. — 0 thru 9

If by that you mean that this country's going down, sooner rather than later, there's a good chance. The spending and the stupidity on both sides are out of control, probably past the tipping point.

I think the tl;dr is that you think doubling federal spending is a good idea and I think it will be the end of civilization as we know it. Or perhaps you disagree with that number. That also could explain our difference of opinion. It actually comes down to this point I think.

Read further down in the Wikipedia article, and you will see the axioms for first order PA. There is no predicate 'is a number'. — GrandMinnow

If you mean the part where it lists the axioms only referring to the successor function $S$ and the symbol $\in$, why can't I define

$\mathbb N = \{x : x = 0 \lor \exists y (x = S(y))\}$?

Regarding your notion about improper sets relative to PA as personal visualization, I didn't ignore it - you even quoted me remarking on it. I said I don't opine as to what does or does not make sense in your mind. But I said your notion makes no sense to me. And I would add that I think it does muddle discussion. But I didn't say you shouldn't think it. — GrandMinnow

Yes I think that appeared in your post after I wrote that. But I really don't get why you think the idea is so terribly wrong.

But now I realize that writers often use 'HF' to stand for a class. So my choice to use 'HF' as the abbreviation was not good. From now on, I won't use it to stand for the theory (ZF~Inf)+~Inf. Instead I'll use:

TF = (ZF~Inf)+~Inf — GrandMinnow

I'm still confused on this point, since nobody ever describes ZF as ZF-C + (not-C). I sort of see your point that "ZF" is being noncommittal on the point, but generally from context you can tell when they explicitly mean ZF-C + (not-C). So perhaps people should be more formal about that.

To be more precise, whatever symbol 's' we pick, TF does not support a definition:

s = {x | x is a natural number}

because the theory does not prove that there is a such an object. — GrandMinnow

As above, what of

$\mathbb N = \{x : x = 0 \lor \exists y (x = S(y))\}$?

Surely I can form that class directly from the symbols 0 and S referenced in the PA axioms.

In class theory, it is well understood that a proper class is a class that is not a member of any class. All I'm doing is pointing out that we can also say that in set theory and conclude in set theory that there are no proper classes. It might be annoying, because it's not a very useful series of formulations. But it its technically correct, and I find that it sharpens the picture. Especially it goes against a common misconception that we can define a predicate symbol only to stand for a relation (sets are 1-place relations) that has members. No, we can always define an empty predicate. For example:

dfn: Jx <-> (x is odd and x is even)

is allowable, even if rather pointless. — GrandMinnow

We're agreed on all this but I don't see how it invalidates my point. But I'll concede that if I understood you better you might well be right.

That's not a model of PA. w+2 has no successor.
— fishfry

Was a typo of omission; I meant {w, w+1, w+2 ...} — GrandMinnow

Yes sorry, obvious in retrospect but now I don't remember what point was being made ...

Yes, we can have a predicate 'is a natural number' in TF. And upon an interpretation of the language, it has an extension (a subset of the universe for the model) and that extension is a set, not a proper class. — GrandMinnow

You're saying that my definition of $\mathbb N$ is a set? I confess I don't follow your reasoning but I admit I don't necessarily disagree with it. I just don't follow it. Maybe I haven't the background to follow your argument. I don't see now this could be a set in TF.

Instead, in the absence of the axiom of infinity, we do not have a supporting existence theorem for a definition:

N = {x | x is a natural number} — GrandMinnow

Right, but I gave the correct formula based on the formal axioms you pointed me to in the Wiki piece. We can't form a set with that definition but it's certainly the extension of a predicate. And if it's that, and it's not a set ... that's what I call a proper class. Maybe the people who told me that were speaking more loosely than I realized, that would be your point.

No one is advocating anything like the USSR or Cuba or Venezuela. No one. That's imaginary. — Xtrix

Lot of interesting developments since we last talked! The DNC fix is in. Pete and Amy dropped out and endorsed Biden after speaking with Bloomberg. They're both young and ambitious, the party will reward their loyalty down the line. Steyer's out, he spent $250M for nothing and never made any kind of impression at all. Just the rich guy at the end of the debate stage. Bloomie's events protested over stop-and-frisk. African-Americans turned their backs on him in church yesterday. And Chris Matthews finally got shitcanned. About ten years overdue IMO.


If I were to stipulate that Bernie doesn't want the US to be the USSR, Cuba, Venezuela, and Mao's China rolled up in one; wouldn't you at least agree that this is a credible charge that he will be accused of anyway? His record on this is terrible. He's made many public statements and has many political alliances that argue my side of the proposition and not yours. I get that deep down you think he's a harmless old lefty who means well. Unfortunately he has a lot of friends far more dangerous than that.

I believe that the movement Bernie represents is extremely dangerous, even if Bernie himself is far more kindly than his followers.

Why you keep invoking Cuba or our supermarkets is beyond me. If you can't see that this is sheer stupidity, maybe it's not worth it talking to you. — Xtrix

Deep down you must know I'm right else it wouldn't bother you.

None of this has to do with my comment, that the wealth in the US has been concentrated to the top, especially the 1% (it's actually closer to 1/10 or 1/100 of 1%). — Xtrix

Yes, and inequality is an inevitable byproduct of a system that produces such wealth. In a capitalistic society there will be winners and losers; but on average, and overall, pretty much everyone's far better off than they are under the communist dictatorships of the twentieth century. Every single one of them has been bloodthirsty, brutal, repressive, and ultimately totally unproductive.

Bernie's a socialist. Socialism fails. It has always failed and as it fails it kills a lot of people.

I absolutely agree with you that our current system is pretty screwed up. The inequality has gotten out of hand. This is what everyone recognizes, it's what's given rise to Trump and Bernie. Nobody's interested in mainstream Republicans or mainstream Democrats. This is a crisis for the country.

But if you think socialism (or "democratic socialism" if you prefer) is the answer, then I humbly beg to differ.

Stop arguing against imaginary opponents. — Xtrix

I'm responding to the points you're making.

Outside of your information bubble, they don't exist. If all you know how to do is respond to straw men and imaginary opponents, that's OK. Just let me know so I don't waste my time trying to explain anything. — Xtrix

You are not required to respond to anything I write. If you did happen to make a coherent point, I'd acknowledge it. I see you defending a system and way of thinking totally discredited in the twentieth century. As I said I do agree that our system is a greatly distorted, corrupt, and unsustainable version of what it's supposed to be. We disagree on the remedies. Socialism can never be the answer because it's a flat out failed ideology that causes misery and horror wherever it's implemented.

Yes, we all agree the economy has worked very well for them, and they continue to prosper. The system that's been in place has been a state-capitalist system, rigged for the wealthy who can lobby for legislation, subsidies, contracts, tax breaks, and bailouts from the government (our tax money). Bernie does indeed want to destroy that. I agree with him. — Xtrix

I'd like to blow up that system too. It's part of the neoliberal consensus. The global elite suck the wealth of the world upwards from the middle classes. "Davos man." There is much merit in the socialist critique. Marx predicted most of what we know of as late stage capitalism. He saw the inevitable disaster that capitalism must become.

It's not the socialist theorizing I object to. It's the authoritarianism of leftists that I oppose. With Bernie the cure would be much worse than the disease.

Who said shutting down commerce? Try reading again what I wrote. Bernie wants to destroy a rigged system that distributes the wealth of this country to the top 1/10th of 1%, and I agree with that. I think such a system which produces such enormous inequality should be dismantled or at least heavily corrected. This is the exact opposite of what you're saying -- it's in FAVOR of the working and middle classes. It has nothing to do with "shutting down the American economy." Nothing. Nor did I ever say that. Nor has Bernie said that. It's a ridiculous statement that, once again, exists only in your imagination. — Xtrix

The goals may be noble but the policies would be economic disasters. The Green new deal, the radical environmentalism, the opening of borders (you know Bernie used to be a sensible immigration restrictionist just a few years ago. He knows immigration's bad for workers); the confiscatory taxation of the successful, as if you think all the wealth of the country would still be there if we abolished billionaires, whatever that means.

It's very easy to tax wealth. All we need is the political will, which Bernie has. The working and middle classes will not pay for it, the wealthiest Americans and the corporate sector, however, will. — Xtrix

You have not run the numbers. If you stripped every billionaire in the country down to the clothes on their back, you could run the current federal budget plus Bernie's new programs for a year or two at best. After that you'd have to come for the hundred millionaires, then the ten millionaires, then the millionaires, and finally the high school teacher married to the firefighter making $150 or $200k together.

If you sit down and run the numbers: WHERE does the money come from, HOW MUCH, money, and how long does that run the newly doubled federal budget -- you see you will run out of the "rich" pretty quickly and soon be be into the middle class.

Run the numbers.

Because it's the agenda of Donald Trump. It's every policy that's come out of the Trump administration: deregulation, privatization, corporate tax cuts, etc. — Xtrix

Tax cuts stimulate the economy. Corporations don't pay taxes, they collect them. Now I disagree with Trump in that he cut taxes but then allowed the budget to blow up. Same mistake Bush 43 made. I disagree terribly with the government's profligate spending combined with tax cuts, borrowing, and printing. It's not going to end well. But I'm sure you've noticed that there's no interest in Washington for getting the trillion dollar deficits under control. You know that every nickel the government spends is authorized by the House and personally signed off on by Nancy Pelosi. The financial trainwreck in Washington is bipartisan.

That's not true. Neoliberalism has little to do with wars. It has far more to do with increasing the military budget (to line the pockets of defense contractors), which Trump has done. — Xtrix

Hillary and Obama's foreign policy was a disaster. Trump is getting us out of Afghanistan. Of course we lost, but he has some sort of face-saving deal and there's a chance he could get the troops the hell out of there. Hillary would have had us in ten more wars. The neoliberals are part of the neocon war agenda. That's why Hillary, Schumer, DiFi, and all the other "liberal" Democratic senators signed on to the Iraq war. You give us our social programs and you can have your wars. That in a nutshell is the unholy neocon/neolib alliance that's destroying this country; that both Trump and Bernie oppose.

He has done nothing on trade except re-named NAFTA and started a stupid trade war with China which changed literally nothing. — Xtrix

Trump stood up to Xi. China has over a million Uyghurs in concentration camps. In the end we're all going to have to stand up to China. One could ague that Trump standing up to Xi is why Hong Kong has whatever autonomy it has left. Cozying up to China is one of the biggest pieces of the neoliberal project. Trump sees the future in this regard with far more clarity than many.

His proposal of building a wall will go down as one of the stupidest ideas in history. — Xtrix

As I've indicated elsewhere on this site I'm a longtime follower of US-Mexico politics and I've lived in Mexico. I strongly oppose Trump's wall. But I also opposed the equally cynical Secure Fence Act of 2006, which Hillary and Biden the other power Dems voted for. And it was Obama who built the kid cages in 2014. What I object to is liberals who attack Trump for implementing essentially the same policies Dems have been implementing since the Bill Clinton administration. The militarization of the border did not start with Mr. Trump you know. It's the other side of the sanctuary cities and drivers' licenses for the undocumented. Total hypocrisy. The same people who support DACA built the cages. You want to complain about Trump's wall but not take responsibility for the Democrats' complicity for the screwed up border policies.

As for war -- yes, he wants to stay out of war. — Xtrix

That's good, right? We have a point of agreement.

You're confused. Sorry for the accuracy. Try to stop arguing against your imagination. — Xtrix

Yeah yeah whatever. The fix is in against your guy Bernie, what do you think about that? Tomorrow's Super Tuesday, we'll know a lot more by the end of the day.

True. And being accurate about what's really happening in the current administration and about Bernie's actual policies is all the more important. I highly recommend making an effort to do so. — Xtrix

Doubling the spending of the federal government while making unrealistic estimates of where the revenue will come from strikes me as a recipe for disaster.

PA doesn't have sets, and second, even if it did that would not be a valid specification of a set since it violates the axiom schema of specification.
— fishfry

Hard to discuss a counterfactual here. — GrandMinnow

That made me chuckle.

So let's turn to TF.

It's not a matter of being consistent with the axiom schema of specification.

Instead, in the absence of the axiom of infinity, we do not have a supporting existence theorem for a definition:

N = {x | x is a natural number} — GrandMinnow

Yes ok I get that you assert that.

But when Wikipedia says that the first axiom of PA is, "0 is a natural number," are you asserting that "is a number" is something other than a predicate?

My chain of reasoning is:

P1: I take Wiki's account of PA as reasonably accurate (whereas I suspect that perhaps you don't);

P2: Wiki says that the first axiom of PA is "0 is a natural number";

C: If "is a number" is not a predicate, what the heck is it? In which case N = { x : x is a natural number} is the extension, or "predicate satisfier" as I think of it, of the predicate "is a natural number." It's not a set. What is a predicate satisfier that is not a set? It's a proper class.

Can you please tell me where you disagree?

It goes back to Frege. A predicate defines a class. Of course he thought that a predicate defined a set, which was falsified by Russell. But a predicate still defines a class.

This is how I understand what's going on.

No. I was giving practical advice to not overlook that when we read natural language renderings of formulas, then we can't expect that how we naturally take such locutions in English is preserved with every interpretation (model) for the formal language. — GrandMinnow

Yes ok. Agreed.

Dems had an interesting day. On the one hand, party stalwarts like Pelosi and others signaled some level of acceptance of the Bernie freight train bearing down on them.

Nancy Pelosi says she would be comfortable with Bernie Sanders winning the Democratic presidential nomination
.

At the same time, the NYT published a story in which they interviewed 93 superdelegates, who were astonishingly open and up front about how they intend to shaft Bernie.

Democratic Leaders Willing to Risk Party Damage to Stop Bernie Sanders.

Also, the powerful and influential Rep. Clyburn of South Carolina endorsed Biden, who immediately jumped up in the polls. If Biden shows he can hold the African-American vote in South Carolina Saturday, he may do better than expected three days later on super Tuesday ... in which case the corrupt and senile old coot may make it to the nomination. Bloomie's a bust, time for them to run Biden up the flagpole again as the great centrist hope.

Stay tuned.

ZF-inf says: There is no x such that all natural numbers are a member of x.
— fishfry

That is not correct. — GrandMinnow

Ok. In general terms, I noticed you didn't engage with my point that "N is a proper class in PA" is a personal visualization that I find helpful; even though it is not literally true and, according to your thinking, is so flagrantly false that I shouldn't even think it.

If you didn't engage with this point I assume you accept it and are just explaining some of the technical points I brought up, which I appreciate.

ZF-Inf is ZF but without the axiom of infinity. (The '-' here means 'without'; it doesn't mean 'the negation of'.)

(ZF-Inf)+~Inf is ZF but with the axiom of infinity replaced by the negation of the axiom of infinity. — GrandMinnow

Right, perfectly well understood. But why don't they do the same thing with ZF-infinity +(not-infinity)? Nobody every does that. Rather, ZF-infinity means ZF plus the negation of the axiom of infinity by default. Why is that?

No, the finite ordinals are a proper subset of the set hereditarily finite sets. For example, {0 2} is an hereditarily finite set but it's not an ordinal. — GrandMinnow

Oh ok it's all the finite sets. I think I knew that then forgot. All good.

Is there something else special about them?
— fishfry

They may be of interest for many reasons, but for starters, they are the usual universe for a model of "finite set theory" = (ZF-Inf)+~Inf = HF. — GrandMinnow

Right.

so interpretation is a technical term that I think I don't know. I know what it means to interpret an axiomatic theory, ie assigning meaning to the symbols or at least assigning elements of some model.
— fishfry

This is a different sense of 'interpretation' (but closely related). Simplifying here: We interpret a theory T into a theory T' by defining the symbols of T in the language of T' so that every every theorem of T is a theorem of T' plus the added definitions. And we say the theories are equivalent when there is such an interpretation from T into T' and vice versa. (This deserves a sharper statement, but it's too many technical details for a post.) — GrandMinnow

I didn't follow all that but it's ok, I'll check out the def one of these days. Especially since bi-interpretability came up the other day. PA is bi-interpretable with ZF-infinity + (not-infinity) but apparently I'm not getting the subtleties of that statement.

How are HF and N different?
— fishfry

HF is a theory. N is a set. — GrandMinnow

Aiiiyyyy now I'm confused. You just explained to me that HF are all the finite sets in ZF. So if I'm understanding you, HF is the inductive definition of the collection of all finite sets in ZF; and that collection of finite sets is a model of HF. But it's wrong to also call that HF?

You're saying [in HF] I can't form the predicate that I think characterizes N.
— fishfry

Defining a predicate symbol is not a problem. But there is no definition of a constant symbol (such as 'N') such that N = {y | y is a natural number}, since HF does not prove that there exists an object that has as members all the natural numbers. — GrandMinnow

So your argument comes down to saying that since N is not a definable symbol in PA, I can't say "N is a proper class" because I have no idea what N is. Is that right?

In any language, in any theory, we can define whatever predicate symbols we want. It's only function symbols (including constant symbols, where a constant symbol is a 0-place function symbol) that require the supporting existence and uniqueness theorem

I can't form the collection of all numbers because I haven't got the language to do that.
— fishfry

In HF, you have the language, but you don't have the existence theorem ExAy(y is a natural number -> y e x). — GrandMinnow

Didn't track that sorry. But I'll agree that N's not a definable symbol.

in any model of ZF-inf there is not a set of all natural numbers
— fishfry

No, the sentence ExAy(y is a natural number -> y e x) is a theorem, but that doesn't preclude what the members of the universe for the model may be.

For every infinite cardinality, there is a model of ZF-Inf with a universe of that cardinality. And that universe can have as members any sets whatsoever. Same for (ZF-Inf)+~Inf. Same for PA. — GrandMinnow

Yes ok ... not sure where this is going ...

For example, we can have a model of PA whose universe is {w, w+1, w+2} and each of those members of the universe is infinite. — GrandMinnow

That's not a model of PA. w+2 has no successor. What am I missing here?

But wait, (ZF-Inf)+~Inf has a theorem ~Ex Ix [where 'I' is a defined 1-place predicate symble we are read in English as "is infinite"], so how can the universe of a model have a member that is infinite? Well, because for such a model, the predicate symbol 'e' is interpreted not as the ordinary membership relation but rather as some other "bizarre" relation and so also my 'I' be interpreted differently from "is infinite". When we talk about models in general, we can't presume that any given model of a theory "captures" the way we ordinarily "read off" the theorems of the theory. If we want to narrow the discussion to only models that adhere to the way we "read off' the theorems, then we should confine to talking about standard models. — GrandMinnow

"Read off"? Is that a technical term? Lost. I do understand that set theorists prefer models in which $\in$ is set membership.

in ZF there is no such definition or thing as a proper class.
— fishfry

In ZF, we may define:

x is a proper class <-> Ey y e x & ~Ez x e z

And we may prove:

~Ex x is a proper class. — GrandMinnow

That's very interesting. Makes perfect sense. I've never seen this done. They always say that ZF doesn't talk about proper classes so whenever we say proper class, we are being informal. In fact I've heard that so many times that it's probably why I do use "proper class" informally. But your definition makes perfect sense. I do wonder why I haven't seen it.

When the Peano axioms say, "O is a number,"
— fishfry

First order PA doesn't have a primitive 'is a natural number'. — GrandMinnow

Grrrrrr. The first thing it says is that "0 is a number." Is that not true? You know, everything you say directly contradicts Wiki. Is that what you meant earlier when you talked about Peano's original formulation? Is what everyone thinks of as PA not what you mean by PA?

Here's Wikipedia on the subject:

The Peano axioms define the arithmetical properties of natural numbers, usually represented as a set N or $\mathbb N$. The non-logical symbols for the axioms consist of a constant symbol 0 and a unary function symbol S.

The first axiom states that the constant 0 is a natural number:

0 is a natural number.

Now if by PA we mean what is described by Wikipedia, then I am frankly right and you are wrong. So it must be that by PA you mean something else. Is that the source of our misunderstanding? I will stipulate that they shouldn't have used the word set, but rather collection. Or (ahem) proper class!

Peano's historical own formulation should not be conflated with first order PA. — GrandMinnow

Ah this is what I meant a moment ago. You don't think PA is what is described by Wiki, and I and probably millions of other people are confused about this. Is that what you are saying?

the set of all sets is a proper class
— fishfry

There is not a set of all sets, not even in class theory. There is the class of all sets, and it is a proper class. — GrandMinnow

I'm terribly sorry, of course I know better. See how you have me confused!! I meant that the collection of all sets is a proper class. Just misspoke myself.

And I explained why referring to proper classes in discussion about set theory can be understood as an informal rendering for an actual formal notion in the background, but that is lacking here in saying N is a proper class in discussion about PA. — GrandMinnow

It's still a good conceptual metaphor IF not literally true. But again, according to Wiki your point is not so clear. So you must be objecting to the common Wiki understanding of PA.

And N is a set, which is not needing exceptions in view of the fact that in PA there can be no definition N = {x | x is a natural number}. — GrandMinnow

There is no SET such as that because first, PA doesn't have sets, and second, even if it did that would not be a valid specification of a set since it violates the axiom schema of specification.

But "is a natural number" must be a predicate, what else can it be? It's a thing that classifies all the objects in the universe into "yes it's one of these" and "no it's not." So the number 3 is a number and a tunafish sandwich is not a number.

Therefore I can form the COLLECTION, or "predicate satisfier," or as it's officially called the extension of the predicate, N = {x | x is a natural number}. N is a class and it's not a set. So it's a proper class.

If one wishes to say "N is a proper class with respect to PA" but not formulate the exact mathematical meaning of "with respect to" or even to a clearly articulate an intuitive/heuristic notion that is still consistent with the ordinary mathematical result that N is a set, and hopefully has value as a metaphor rather than confusing the subject with impressionistic use of terms, then, of course, I cannot opine whether or not in one's own mind it somehow makes sense nevertheless. But I do say, and have explained, that it makes no sense to me. — GrandMinnow

But it makes perfect sense to me! And you haven't explained what "is a number" could be, if not a predicate. And for any predicate whatever you can always form its extension consisting of all and only those things that satisfy it. Which, if it's a set, is a set; and if it isn't, is a proper class.

tl;dr: By PA do you mean something other than what's called PA on Wikipedia? That's the only way your post makes sense.

impressionistic use of terms — GrandMinnow

You say that like it's a bad thing! LOL.

What wealth? You mean the wealth of the 1%? — Xtrix

Comrade Bernie is that you? Celebrating the wise and just rule of Fidel Castro who taught the peasants to read while he appropriated their land and imprisoned them for wrongthink?

Well ok then. Been to the supermarket lately? Seen the bounteous harvest in the produce department, the shelves full of all kinds of wondrous goods, the meat and fish sections filled with good stuff to eat? Maybe you'd prefer the stores in Venezuela or the Soviet Union or the aforementioned Cuba.

This conversation is frankly beneath me. Please go vote for Bernie. You know one theory I've heard is that the establishment Dems have this problem on their hands, their radical leftists in the AOC/Bernie wing of the party. One strategy is to let Bernie win the nomination then get slaughtered in the general election. Then in 2024 the Hillary wing of the party can take back control. Trump won't have a strong successor, least of all a dead fish like Pence. The Dems are a lock in 2024 with a centrist candidate and their left wing having been fully discredited.

But please, tomorrow as you go through your day, look around at the abundance around you. The bustling commerce, the well-stocked store shelves. Ask yourself if you'd rather live here or in Bernie's Cuba.

LOL. I can't believe you actually said that. Are you joking? You have no idea of the actual, literal wealth of the US -- spread throughout society, though certainly terribly unequal -- relative to the rest of the world?

Yes, we all agree the economy has worked very well for them, and they continue to prosper. The system that's been in place has been a state-capitalist system, rigged for the wealthy who can lobby for legislation, subsidies, contracts, tax breaks, and bailouts from the government (our tax money). Bernie does indeed want to destroy that. I agree with him. — Xtrix

All those people driving to and from work on the freeway, you want to shut down all that commerce. How many would starve under your plan? Are you insane? You seriously want to shut down the US economy? If you did that, ONLY the 1% would survive. They already have their bunkers. The rest of us working stiffs would be crushed in a depression that would make the 1930's look like the good old days.

I would grow out of this fear of "socialism" and try learning something about what Bernie's proposals really are and whether they make sense. — Xtrix

I glanced at his detailed plans on his website. I don't believe his numbers because he hasn't factored in the adjustments people will make in response to his taxes. He'll pass laws, the wealthy will find ways around them. and the middle class will pay. The middle class MUST pay for such enormous spending programs because the rich have lawyers and the poor have no money. This is very basic.

I'd vote for Bloomberg/Clinton over Trump.
— Xtrix — Xtrix

I'm on the opposite side of that proposition. But I did realize that if by some miracle Pence won the 2024 GOP nomination, I'd vote for whoever the Dems run. Pence would be just awful, he's a dim bulb and his brand of social conservatism is of no use to me, I oppose it. That's where I'd draw the line. Trump is a once in a lifetime historical figure. I don't see anyone following his act. Which adds weight to the lose-with-Bernie and win in 2024 theory of the centrist Dems.

Bloomberg and Clinton are exactly why the public wants Trump and Bernie. You cling to the neoliberal consensus perhaps because you don't know how truly evil it's become. Didn't the Iraq war teach you anything?
— fishfry

Given the context, it was very easy to see that I don't like either, but was demonstrating how "low" I would go just to get Trump out of office. How is that hard to understand? — Xtrix

I do understand. I have a different opinion. But you did get me to realize that I'd vote for Bloomie against Pence. Or pretty much any other Republican on the current scene. I don't like many or even any of the GOPs.

As for "neoliberal consensus"...do you even know what that is? — Xtrix

Yes. Hollowing out our industrial base and outsourcing it to China. Endless wars, not just wars but stupid wars. Expensive stupid wars. Open borders for cheap labor, further destroying the working class. The globalist project that took over in the 1980's and really got going in the 90's.

Because it's the agenda of Donald Trump. It's every policy that's come out of the Trump administration: deregulation, privatization, corporate tax cuts, etc. — Xtrix

No, I disagree. Trumps policies on trade and immigration go directly against neoliberalism. He hasn't started any new wars and he's trying to get us out of the ones we're in. Of course he's been rolled by the likes of Bolton and other warmongers. It's damned hard to fight the establishment alone. But his big overarching politics are directly opposed to the neoliberal consensus of the past thirty years.

So you either don't know what you're talking about, or voted in favor of neoliberalism. — Xtrix

Trade, immigration, war. Trump's solidly opposed to the neoliberal consensus and he's achieved quite a lot in that direction. He stood up to China at a time when the Bloombergs of the world want to sell what's left of this country to China. For my part I'm against cybertotalitarian surveillance states and I stand with the million Uyghurs in concentration camps. Trump is standing up to Xi and that is every bit as historic as Nixon's visit to China. You are missing the big picture and you are wrong on the facts.

I assume you're just confused, though, because the word "liberal" is in it. — Xtrix

Anyone who sleepwalks through their American life and doesn't see the incredible material abundance all around them is not one to talk about others being confused.

Excuse me as I laugh myself out of this dialogue. — Xtrix

That's cool, I'm politicked out for a while. I saw that Democratic debate last night and I'm still rolling on the floor laughing. It must be awful to be a committed Democrat right now. But these are good days for the Bernie brigade. I like crazy Bernie personally. He could have won in 2016. He beat himself when he said nobody wanted to hear about Hillary's emails. If he truly had the stones to be president he would have gone after her hard on her corruption. He'd be president today.

2020, I don't think that's going to happen. But that's what they said about Trump in 2016 and Bernie's 2020 campaign is weirdly parallel. Not being taken seriously then the whole party panicking to stop him and the moderates unwilling to get out of each other's way. The parallels are eerie. Anything could happen.

Latinos and African-Americans came out for Bernie Sanders, a 68 year old Jewish guy
— fishfry

ahem, 78. The difference might matter. — Wayfarer

Yes thanks. I wonder why I didn't catch my typo. I think I somehow can't believe a 78 year old who had a heart attack a few months ago is the likely nominee and is surrounded by screaming youngsters like a rock star.

I think he has a chance to win Trump. I hope that finally the Dems can pick a good candidate, not a bad candidate like Hillary. — ssu

I believe Bernie could have beaten Trump in 2016. I don't think he can beat him in 2020 unless there is a humongous economic collapse. And there is currently a seriously nonzero probability exactly that. The Fed's been blowing bubbles of digital money into the system since the last financial crisis, which in effect never actually went away. It just got papered over, literally. When the bill comes due it will be a crash the likes of which the world has never seen. That's the theory, anyway. Maybe coronavirus has already triggered it. If it happens, Bernie can win.

If the economy stays good, Trump wins.

If Bernie wins Trump, I think he will be like Lopez Obrador. Mexico hasn’t gone the way of Venezuela, even if the President is a leftist. And likely won’t the US either, even if the GOP will portray a Sanders ”regime” putting the US on the road to Venezuela like socialism. — ssu

You know that's a very interesting point. Lopez Obrador has been called the Bernie Sanders of Mexico. And you are exactly right, people call him a socialist but he's actually a pragmatic populist. However I don't believe Bernie is the same! Bernie's a dangerous true believer IMO. I think a Bernie presidency would be a disaster on a historic scale.

If you are saying that the DNC won't be able to screw him because it would be too obvious, I respectfully stand by my cynicism. But I am definitely impressed by the post-Nevada vibe in the country. Latinos and African-Americans came out for Bernie Sanders, a 68 year old Jewish guy from a virtually all-white state. It's something to behold. It's what this country's all about.
— fishfry

Needless to say I agree, except with the cynicism. I'm more optimistic in that case...or maybe more "hopeful." Time will tell. — Xtrix

I'm a glass half empty type. But this is the DNC we're talking about. The courting of the superdelegates is well under way. And there's no law that says Bloomberg can't just buy them.

Dick Morris thinks that this is exactly the plan. So if I'm cynical about the lengths the Dem establishment will go to in order to stop Bernie ... I'm not alone.
— fishfry

Now here I really disagree. This is wild speculation and I see no evidence for it. It's true that Bloomberg is throwing a lot of money around, but that it's part of a conspiracy to elect Hillary Clinton? Come on. — Xtrix

Wild speculation has its place. It lets us explore the boundary between the possible and the unlikely. Dick Morris has known the Clintons a long time.

Not one you'd vote for? Given the alternative and the importance of this election? — Xtrix

I'm not a socialist.Not even a democratic socialist. The US got its wealth through a system Bernie wants to destroy. He has no understanding of the economy at all. It would be insane for him to be president. The alternative is Trump. I have come to see Trump as a highly flawed but historic figure. In the past three years he's shown the world how corrupt the media and the Democratic party are. You think that's only right wing spin. I used to be a left winger. It's spin I believe because I watched it happen and I think for myself. I stand with Trump, warts and all. As opposed to what's become of the Dems. And Bernie? No no no no no. Unbelievable that an ignorant guy like that could be in charge of the country.

That's mind boggling. — Xtrix

A lot of liberals just don't get it. I used to be a liberal. I'm off the reservation. Just how it is. I'm not alone. A lot of former liberals are in shock at what's become of our former side. So yeah, I'm mind boggled too.

I'd vote for Bloomberg/Clinton over Trump. — Xtrix

Bloomberg and Clinton are exactly why the public wants Trump and Bernie. You cling to the neoliberal consensus perhaps because you don't know how truly evil it's become. Didn't the Iraq war teach you anything?

One believes in climate change, the other doesn't. That's enough of a reason right there. — Xtrix

The US under Trump led the world in reducing CO2 emissions last year. So even on the facts you're wrong. Trump is not anti-environment. That's just something Rachel told you.

https://www.theblaze.com/news/us-led-world-in-reducing-co2-emissions

I'm not advanced. But I do have a methodical understanding of some basics. — GrandMinnow

Thanks. You wrote a really interesting post and I have a lot of questions and comments. Before going into them I have to say that my stance remains unaltered: which is:

Regardless of whether it's technically accurate, it can sometimes be a useful visualization of proper classes.

So it doesn't matter if you're right on the technical part. If I myself find it a helpful visualization, that is my right. We are all entitled to our visualizations.

So at best you could possibly argue that even though I have the right to my own private visualization, I should not mention it aloud in polite company. I'll take that under advisement.

As an analogy, suppose an engineer takes calculus class and spend the rest of his professional life believing that dx is an infinitesimal. We could of course correct him mathematically, but it's a harmless belief for an engineer or physicist and a highly effective conceptual aid. Even many mathematicians who technically know better privately think of dx as an infinitesimal when analyzing a calculus problem. I hope you take my analogy as on point.


Also I did find at least one professional set theorist who was willing to make an admittedly offhand remark in agreement with my thesis. And I agree that it's not even clear what he meant. But he referenced the idea. If my intuition matches professor Caicedo's, I consider my point to have been made to my own satisfaction.

All that said, you wrote a lot of really interesting things so I'll try to comment.

(1) There is a difference between ZF-Inf and (ZF-Inf)+~Inf. I'll call the later 'HF' (the theory of hereditarily finite sets). — GrandMinnow

Oh yes right away you have actually identified a couple of points of confusion or ignorance in my mind.

In my reading the last few days I do keep seeing the notation ZF-infinity +(not infinity) and this is puzzling me greatly.

For example in discussions of the axiom of choice we see ZF or ZFC. I've never seen anyone say, "ZF + not-AC". The not-AC is emplied when you say ZF. And in a given model of set theory, there either is or isn't a choice function on every collection of nontempty sets. It has to be one way or the other, we don't have to say it twice.

But when it comes to infinity, lately I keep reading ZF-infinity + not-infinity and I have no idea why they're doing that! I'd think that likewise, in any model of ZF-infinity, there either is or isn't an infinite inductive set containing the empty set and its successors.

So why do we need to say "+ not-infinity?"

The language of HF is the language of ZF (i.e. the language of set theory). — GrandMinnow

Right, the hereditarily finite sets. Why are they considered interesting? I just thought they were the usual finite von Neumann ordinals, ie the natural numbers. Is there something else special about them?

PA and HF can be interpreted in each other. — GrandMinnow

Ok so interpretation is a technical term that I think I don't know. I know what it means to interpret an axiomatic theory, ie assigning meaning to the symbols or at least assigning elements of some model. But I'm not sure about thie bi-interpretability business.

The usual universe for HF that we have in mind is the set of hereditarily finite sets. And of course N is also a universe for HF. — GrandMinnow

Ah. How are HF and N different?

(2) Most textbooks take 'is a set' as informally primitive, but we can be precise in the language of set theory:

x is an urelement <-> (~ x=0 & ~Ey yex)

x is a class <-> ~ x is an urelement

x is a set <-> (x is a class & Ey xey)

x is a proper class <-> (x is a class & ~ x is a set)

In set theory, we can prove:

Ax x is a set (though, as mentioned, most textbooks don't bother with something so basic). — GrandMinnow

Ok, very interesting. Haven't seen those defined formally before. I learned the "informally primitive" way.

(3) The language of class theory (such as Bernays style class theory, which I'll call 'BC') has a primitive predicate 'is a set' (or a many-sorted language is used, which is essentially the same as using a primitive 1-place predicate), so in BC 'is a set' is not defined but instead certain axioms are relativized to sets.

In BC we prove:

Ex x is a proper class — GrandMinnow

Ok I never actually looked at BC but I've heard of it. Also I believe Morse-Kelley has classes.

(4) I explained why "N is a proper class in PA" [or whatever paraphrase] is, on its face, not coherent. — GrandMinnow

I have seen no such explanation; or if I have, I haven't understood it. A proper class is the extension of a predicate that's not a set. That's how I understand the term. I haven't thought about this interms of BC or Morse-Kelley so maybe you're right, I don't know.

But I allowed that one is welcome to adduce some particular mathematical statement instead. And I explained why it would not be a correct statement in set theory (and I would add, not even in BC). So maybe we turn to HF. — GrandMinnow

I admit I haven't followed the argument but that's probably my fault.

Since HF is in the language of set theory, in HF we can define any predicate of set theory, and we can define any operation of set theory for which we can prove existence and uniqueness in HF. — GrandMinnow

Ok.

HF proves ~ExAy(y is a natural number -> y e x). So there is no definition of 'N' (in the sense of the set of natural numbers) in HF. — GrandMinnow

Oh I think I see where you're going. You're saying I can't form the predicate that I think characterizes N. Am I on the right track at all?

So, while HF can have predicates 'is a natural number', 'is a set', and 'is a proper class', still HF can't have the definition N = the set of natural numbers. — GrandMinnow

In other words, if I'm understanding, then I can't form the collection of all numbers because I haven't got the language to do that. So my argument fails.

NOTE: At least the TECHNICAL argument fails; but my intuition still likes it!!

As far as I can tell, the best we could do in NF is this theorem:

If Ax(Ay(y is a natural number -> yex) -> x is a proper class). But that holds vacuously, since there we have ~ExAy(y is a natural number -> yex).

So, as far as I can tell, we are still thwarted from making sense of "N is a proper class in PA" or even "N is a proper class in HF". — GrandMinnow

I don't understand the details but perhaps this is a good argument. In which case I'm technically wrong that N is a proper class but it's still a useful intuition; and if not for everyone, at least it is for me. You can't enjoin me from thinking my thoughts, misguided though they may be.

And in set theory (and even in BC, if I'm not mistaken) the universe for a model is a set, not a proper class. — GrandMinnow

Yes, models are supposed to be sets. That's my understanding.

(5) Caicedo says, "in ZF without the axiom of infinity [...] you cannot prove that w is a set, but you can prove that as a (perhaps proper) class, it satisfies both first and second order PA."

I don't know why he says 'perhaps' there. And without more explanation, I don't understand what he's saying. — GrandMinnow

Yes. Perhaps he is speaking INFORMALLY because he has a similar intuition as mine. Perhaps he even has a context in mind where the statement could be made rigorous. Perhaps not. But we DO know that at least one professional set theoriest thinks it's something you can say in this context. That means a lot to me.

I do understand that, in ZF-Inf, there is not a proof that there is a set of which all natural numbers are a member (that's another way of affirming the independence of the axiom of infinity). — GrandMinnow

Right. Because if their were, it would witness the axiom of infinity! But I take this differently, not in terms of proofs. I imagine that in any model of ZF-inf there is not a set of all natural numbers. Because if there were it would prove the axiom of infinity, and therebey contradict ZF-inf. Am I wrong about this?

But when he says "you can prove", does he mean prove in ZF-Inf? Proof of satisfaction with models takes place in set theory, not in ZF-Inf nor in HF. And in set theory, universes of satisfaction are sets, not proper classes. — GrandMinnow

I'm afraid I have no idea what his point may have been; only that he used the magic words proper class.

What is understandable to say is:

ZF-Inf does not prove there is an x such that all natural numbers are a member of x. — GrandMinnow

Now I do not understand this. ZF-inf says: There is no x such that all natural numbers are a member of x. You said we can't prove there isn't; I'm saing that there isn't. This seems subtly different.

HF proves there is no x such that all natural numbers are a member of x. — GrandMinnow

Ok the distinction between ZF-inf and HF is totally lost on me. If I understood it perhaps I'd be enlightened.

PA and HF are mutually interpretable. — GrandMinnow

Ok. But not quite the same in some way I can't grasp.

The set of natural numbers N is a universe for a model of ZF-Inf or of HF. — GrandMinnow

Yes I believe that.

But saying "in Pa (or in HF), N is a proper class" makes no sense. — GrandMinnow

a) If you say so; and

b) But can you forbid me from thinking it? What if I promise not to tell anyone else to think it? But taken informally, it's a good visualization of what is meant by proper classes.

(6)
absent the axiom of infinity, w (or N) is a proper class.
— fishfry

No, absent Inf, it is not a theorem that N is a proper class. — GrandMinnow

Of course it's not a theorem, in ZF there is no such definition or thing as a proper class. I thought I made it clear that I understand this point. So it should be obvious that I'm speaking vaguely and metaphorically and not literally.

But in fact we're at this same "theorem" impass. It's true that there's no theorem. But morally, N is a proper class!! This is the nub of our disagreement.

Indeed, absent Inf, there is not even possible a definition N = the set of natural numbers. — GrandMinnow

Yeah yeah. You're technically right and morally wrong. When the Peano axioms say, "O is a number," what exactly is "is a number" if not a legal predicate?

Rather, absent Inf, there is not a proof that there exists an x such that all natural numbers are in x, and there is not a proof that there is no x such that all the natural numbers are in x. In other words, "there is an x such that all natural numbers are in x" is independent of ZF-Inf. However, (ZF-Inf)+~Inf does prove "there is no x such at all natural numbers are in x", but still, it does not say anything about such a thing (which does not exist anyway in NF) being a proper class or not. — GrandMinnow

I'm losing my train of concentration here, don't think I got much out of this para. My fault.

(7)
Yes, you can't define the ordinals in PA because you can't get to the first transfinite ordinal ω by successors.
— fishfry

In HF, we can define the predicate 'is an ordinal' and for any finite ordinal, we can define a constant for it. But, as you mention, we can't define a set that has all the finite ordinals as members. — GrandMinnow

Ok we found something we agree on!

But even in set theory, there are specific ordinals that don't have a definition (there are more than countably many ordinals, but only countably many definitions we could form). — GrandMinnow

Ok. I concede all your points even though there are some I don't understand, and among those I understand, some I disagree with.

But what of it? It's obvious that since there are no proper classes in ZF, when I speak of proper classes in ZF I mean metaphorically. You can go into any set theory book and they'll tell you that the set of all sets is a proper class, even as they say it's not technically so because we're working in a set theory that doesn't have proper classes!

Ok my head is officially confused. There is much I don't understand about these matters. But I retain my private intuition that N is morally a proper class with respect to PA; and that if I were to share my intuition in print, I would be helping more people than I'd be hurting.

Ok for that!! More words than this subject is worth. Not a hill I need to die on. I'll consider myself suitably chastized for my belief that N is a proper class with respect to PA. As Galileo whispered as they forced him to recant his belief that the earth moves around the sun: And yet it moves.

That's fair, but all of that is minor compared to '16. Sanders was a relatively unknown candidate at the beginning, came out of nowhere, and so they didn't quite know how to handle him. They thought they could just sweep him aside without much backlash. They were obviously wrong. — Xtrix

I'm impressed by the Bernie-mania right now. Suddenly everyone's realized all at once that his candidacy is for real. None of the other candidates inspire much enthusiasm. Bloomie's face-plant and Bernie's huge win in Nevada have snapped everything into focus. The only viable candidate is Bernie and he commands an army of true believers. His campaign is being compared to Trump's in 2016.

If you are saying that the DNC won't be able to screw him because it would be too obvious, I respectfully stand by my cynicism. But I am definitely impressed by the post-Nevada vibe in the country. Latinos and African-Americans came out for Bernie Sanders, a 68 year old Jewish guy from a virtually all-white state. It's something to behold. It's what this country's all about.

It's four years later and almost everyone knows what happened. You have Trump tweeting about it at this point. And Sanders is now the clear frontrunner, so there's no excuse of "Well Hillary won fair and square, the so-called Revolution didn't show up!" and so forth. It's very different -- this time, the DNC is aware that everyone is watching closely and will be livid if there are any shenanigans. The media is slightly better at covering it as well this time around, as they can't ignore Sanders' numbers. They aren't stupid, they must see this. — Xtrix

And yet I read a story just today ... Dick Morris, who's always been a conservative strategist but worked in the Bill Clinton administration and has seen the Clintons close up and personal, says that Bloomberg is just a fakeout. That explains why he doesn't even bother to pretend to be campaigning for president. He doesn't do interviews or campaign events, doesn't bother to prep for his debate, just throws fabulous sums of money at ad buys. All he wants to do is get enough delegates to keep Bernie from getting to Milwaukee with a majority. Once that happens and the convention is hopelessly deadlocked (because the superdelegate party insiders will get behind Bloomie but he still won't get near a majority either); then WHO can possibly come in and be the one person to unify the party?

That's right: As I call her, She Who Must Not Be Indicted: Hillary Rodham Clinton.

Dick Morris thinks that this is exactly the plan. So if I'm cynical about the lengths the Dem establishment will go to in order to stop Bernie ... I'm not alone.

https://www.breitbart.com/clips/2020/02/23/dick-morris-hillary-clinton-will-get-the-nomination-in-a-brokered-convention/

Bottom line: Even as Bernie surges into a national phenomenon ... the plotting continues.

You could be right, in the end. But I both think and hope that you're wrong. — Xtrix

I saw a striking photo of Bernie surrounded by his crazed and adoring young fans. And you know what I thought as I saw their faces? How devastated and crushed and angry and heartbroken they're all going to be when the DNC steals the nomination from Bernie.

I will say this is great entertainment. Suddenly there's excitement on the Dem side. Liz destroying Bloomie so that Bernie can surge. Could that be part of a plan too? Maybe she's hoping to be his Veep. It would be a great ticket. Not one I'd vote for, but it would be a hell of an interesting election.

I agree with most to your post, but I wasn't being facetious: if you know how the process works, what evidence is there that suggests this is most likely to happen? I realize the DNC doesn't want Bernie, but Bernie will end up with most of the delegates in the end. I have a hard time believing that the DNC is stupid enough, given the delegate numbers, to simply hand it over to Bloomberg. That's a disaster.

You could be right, but I need more. Bloomberg plotting against Sanders we knew from the beginning. — Xtrix

The question you ask is, what is the evidence that I think makes the DNC screwing Bernie the most likely outcome. Well, the same people did the same thing to him in 2016. And they changed the rules to let Bloomie in the debate, while Tulsi, who has grassroots support, remains shut out.

The very existence of Bloomberg as a credible candidate is proof that the fix is in. You say you don't think they'd be stupid enough to be so blatant in their corruption. I say the Dems are long past that point. Bloomie had a tough night at the debate but he's never been a flashy debate performer. Many Dems are fully ready to abandon every one of their so-called principles to beat Trump and they know America's not ready for Bernie.

This all seems clear to me. I of course agree that I could be wrong. We'll all find out together, popcorn at the ready.

ps -- I'm pretty impressed by Bernie's strong showing in Nevada tonight. If his movement picks up steam, he could win. That's certainly the vibe tonight. Nobody believes in any of the other candidates.

Here's a nice analysis. It concludes, "The race is now Bernie's to lose." Maybe you're right.

Sanders eviscerates the conventional wisdom about why he can't win

The Atlantic article, which I did happen to read early today, despite offering a simplistic overview, unequivocally concludes that the analogy is not valid (despite the clickbait headline). Did you actually read the article? — Maw

Let's agree to disagree here. Nobody knows if Bernie is McGovern. We'll find out in Milwaukee.

it's a vapid analogy — Maw

Take it up with the Atlantic. Or the LA Times, George McGovern is a cautionary tale for Sanders supporters.

Or if you prefer the other side of the proposition, 2020 is NOT 1972, and Bernie Sanders is NOT George McGovern.

Take your pick.

This was nearly 50 years ago, under vastly different conditions and with a set of voters who are now mostly dead. Not at all analogous. — Maw

Tell it to the DNC insiders plotting to stab Bernie in the back. 1972 is very much in the minds of the Dem insiders. I'm not making this stuff up. It's in the news. I can assure you that the professionals who run the campaigns are acutely aware of history. And today's politics comes directly out of that era.

Here's the Atlantic.

Bernie Sanders Is George McGovern
The similarities between 2020 and 1972 are too astonishing to ignore. But there’s one big difference.

https://www.theatlantic.com/ideas/archive/2020/02/bernie-sanders-george-mcgovern/606883/

What happened in 1972? — Pfhorrest

Dems violently split between their centrists and leftists, just like today. In 1968 the centrists won and the Dems nominated centrist Hubert Humphrey, who had even refused to come out against the Vietnam war. The leftists were marching against the war every day.

Humphrey lost to Richard Nixon in what was at that time the closest election history.

By 1972 the leftists took back the party and nominated George McGovern. Nixon got reelected in a historic landslide, winning 49 out of the 50 states.

Hillary is Humphrey, the centrist beating back the challenge from Bernie in 2016; and Bernie, if he wins, would be McGovern. The pattern is that after the centrists beat back the leftists but then lose the general election, they're discredited and the leftists take over. Hence AOC and the sharp leftward lurch of the Democrats.

The Democratic party powers that be are damned if they are going to let 1972 happen all over again with Bernie. No other candidate can win. That leaves Bloomie as the great centrist hope. That's why his disastrous debate performance was such a shock. But Bloomie's still plotting and I would not count him out. He had the same stiff, wooden, cold demeanor when he won three elections as Mayor of NYC. He's rich and has powerful friends in high places.

Once you get to the second round and the superdelegates take over, Bernie is certain to be screwed.
— fishfry

Explain why you think this is true. I don't see it. — Xtrix

It was reported today that Bloomie's already conspiring with the superdelegates.

There's a fight to the death between the centrist neoliberals -- the Hillary wing of the party -- and the radical leftists. I'm sure you know this.

If Bernie shows up in Milwaukee with a plurality but not a majority of the votes, then the superdelegates will have their way. The superdelegates are party insiders, status quo types. They remember 1972. They are not letting Bernie win the nomination. Blooomie, flawed as he is, is the only status quo candidate who can beat Trump. Biden's not even in the race anymore, nobody bothered to attack him at the debate the other night. It was sad to watch. Today he got confused and said his late son Beau was the Attorney General under Obama.

When I went looking for a link to that last bit the only one I found was on Breitbart.

https://www.breitbart.com/politics/2020/02/21/joe-biden-falsely-claims-son-beau-was-us-attorney-general/

So people who don't read Breitbart and who get their news from MSNBC or the New York Times, have no idea that Joe Biden isn't even on this planet anymore, let alone in the race for the Democratic presidential nomination. I hope you see how news reporting works these days.

The story about Bloomie plotting with the superdelegates was reported yesterday by Politico, a center-left outlet.

Bloomberg quietly plotting brokered convention strategy

https://www.politico.com/news/2020/02/20/bloomberg-brokered-convention-strategy-116407

Couching that in terms of proper classes is off. — GrandMinnow

Your use of the terminology is impressive, and I'm not able to determine whether you have advanced knowledge or not; since if you do, my own level of knowledge of these matters would not be sufficient to let me make that determination.

Nevertheless you are wrong on this particular point.

I didn't make up the claim that absent the axiom of infinity, $\omega$ (or $\mathbb N$) is a proper class. I read it somewhere a while back in a reference I can no longer find.

I did find this discussion supporting my claim. It's in a Stackexchange thread called The purpose of the Axiom of Infinity.

I will quote from the checked reply by Andrés E. Caicedo. Professor Caicedo is a well-known professional set theorist. I linked his home page so that you can determine for yourself his stature within the set theory community. (Click on the link Notes and Papers). He says:

"In particular, the axiom of infinity goes well beyond the Peano axioms (and not simply in terms of consistency strength or expressive power). The Peano axioms are provable in ZF without the axiom of infinity. In this theory, you cannot prove that $\omega$ is a set, but you can prove that as a (perhaps proper) class, it satisfies both first and second order PA."

Perhaps you can take a look at his full reply, which is of interest beyond our conversation and worth reading in general, and put it into context for me. If it means something other than what it plainly seems to say, then I'll concede the point. Else you'll need to.

ps -- In the same Stackexchange thread see Asaf Karagila's comment:

"In a model of ZFC with the negation of the axiom of infinity instead, the natural numbers are just the ordinals of the universe."

In other words, in ZFC minus infinity, the natural numbers are the proper class of ordinals. You may object to my terminology (and Professor Caicedo's), but if the extension of a predicate is not a set, what is it? It's a proper class, right?

I don't know where there would be an actual mathematical formulation of "N is a proper class as far as PA is concerned." — GrandMinnow

If this is in response to something I said, I never claimed such a thing. I use proper class as an informal description of a collection that's not a set. In PA the extension of the unary predicate "n is a natural number" is the collection we call $\mathbb N$. However $\mathbb N$ is not a set in PA. So I say $\mathbb N$ is a proper class. It's the extension of a predicate that isn't a set.

I can see you objecting that PA doesn't talk about sets. But given the objects of PA, namely the natural numbers 0, 1, 2, 3, ... we can certainly define unions, intersections, pairs, powersets, and so forth; and do finite set theory perfectly well. In other words PA is a model of ZF-infinity. In that case we would find that $\mathbb N$ is not a set.

My usage was informal but useful in the sense that when we talk about proper classes like the class of all sets, that's hard to imagine; but $\mathbb N$ is easy to imagine. And if we throw out the axiom of infinity, $\mathbb N$ is a proper class. Unofficially, of course. But $\mathbb N$ is the extension of the predicate "n is a natural number" and it's not a set, so it certainly qualifies. Informally.

Indeed, every consistent theory has a model whose universe is a set. — GrandMinnow

Yes. Have you a demonstration in PA of the consistency of PA? Of course not, since that would violate Gödel's first incompleteness theorem. The only way to get a model of PA is to wave your hands and say the magic words, "Axiom of infinity!" And now you have a model of PA, along with a consistency proof for PA. But without the axiom of infinity you haven't got a model of PA nor a proof of PA's consistency.

I know you know this so I must be missing something. And you know our friends the ultrafinitists, who doubt the consistency of PA. Sounds crazy but that doesn't make them wrong.

It seems to me you have to consider all three of these in comparing uncountable to countable and all the other comparisons. Are we sure there are even only these 3 ways of assessing infinity? — Gregory

Oh there are lots of ways. There's the subset relation, we can say that the set of odd naturals is smaller than the set of naturals because the odds are a proper subset of the naturals. This way of thinking is incompatible with bijections, but you can use it as a definition if you like. Bijection gives you a more interesting theory so that's why it's so common.

There's natural density, that's the idea that the density of the even numbers in the naturals must be 1/2, because the limit of the number of evens in the first n naturals goes to 1/2 as n goes to infinity.

There are probably other ways.

On the subject of religion, will two people in heaven have less eternity than three? — Gregory

Is this for me? Can you please Quote a bit of my text by selecting it and hitting the Quote button that appears? That way I get notified rather than having to keep coming back to the thread. Thanks.

I just said that mathematical infinity has nothing to do with heaven or eternity. Or people for that matter. Or religion. So if this post was for me, I surely don't have any idea.

I feel like bijection is invalid. With the natural numbers, you have to step every odd numbers back in order to biject them and who knows what that does to the infinity on the other side. — Gregory

In 1638 Galileo noted that you can put the whole numbers 0, 1, 2, 3, 4, ... into 1-1 correspondence with the perfect squares 0, 1, 3, 9, 16, ... You can see that this is true, right? Without overthinking it. You can line up the whole numbers and line up the square numbers and connect them with lines such that every whole corresponds to a unique square and vice versa. Without trying to figure it out or overthink it, you can see this is true, right?

I'm probably not smart enough to understand the coming response,.but I like this subject. — Gregory

We're all on that path. I'm not smart enough to understand all the math I wish I knew but I like to read about it anyway. Remember Wiles spent seven years of evenings after his regular day job as a math professor, working on Fermat's last theorem. Seven years of confusion and hard work and struggling to learn all the areas of math he needed in order to figure the thing out. Math is beyond everyone that way. You have to work at it. As Euclid said when the King asked him for an easy way to understand Euclid's great book The Elements: "Sire, there is no royal road to geometry."

Liking it's a good place to be.

Fishfry, if the set is infinite, it's like saying there are the same infinity of points in the pineal gland as in the whole body. That's how it appears to me. I just know infinity from Hegel. He says the finite is the infinite thrown from itself. For him the infinite must be one, not four or whatever — Gregory

I'm afraid I don't know Hegel. I've heard he's difficult to read. I'm not much for the classical philosophers, my limitation. That said, I think it's better, when studying mathematical infinity, to put aside prior notions of philosophical conceptions of infinity.

Mathematical infinity starts from the counting numbers 0, 1, 2, 3, ... that we all have an intuition of as being unending. Given that, there are interesting things we can say. But none of this is intended to resolve any philosophical issues regarding being or the world or heavy things like that. It's only math.

If you can take the math on its own terms, the study of mathematical infinity is interesting and beautiful. Those are the criteria for what mathematicians care about.

It's true that Cantor himself believed that after Aleph-0, Aleph-1, and so forth, was an "absolute infinity" that he called God. Today, Cantor's religious beliefs are not much remembered except historically.

All in all, when approaching this material it's better to put aside all philosophical preconceptions. Mathematical infinity is not attempting to resolve any philosophical issues about the world or God or metaphysics.

In particular, there's no reason to believe the world is made up of dimensionless mathematical points in the same way the real number line or Euclidean space are. The question of how many points are in your pineal gland is meaningless. The pineal gland is made up of organic molecules, or atoms, or electrons, or quarks, or quantum probability waves; depending on the level of discourse. But nothing in the body, or in the world, is made up of mathematical points. Mathematical points are purely conceptual entities, like justice; or fictional entities like chess pieces.

How much math must one know to understand this Catorian proof? — Gregory

Virtually none. There are proofs all over the Internet.

The diagonal argument is the one people usually see. But I think the proof of Cantor's theorem is simpler and more beautiful. It shows by way of a short and simple argument that there can never be a bijection between a set and its powerset.

So the powerset of the natural numbers must be uncountable. It's not difficult to show that there is a bijection between the powerset of the naturals and the set of real numbers (think binary strings) so this shows that the reals are uncountable without the confusion usually generated by the diagonal argument.

In other words, if he ends up with 1400 delegates, it's not as if second place will have the remaining 1591 or whatever it is. The rest will either vote according to who their the candidate who dropped out endorsed or can vote however they'd like at the convention -- but the point is, the distance will be sufficiently large, and this in itself will almost force the DNC's hand to give it to Sanders. — Xtrix

Once you get to the second round and the superdelegates take over, Bernie is certain to be screwed. The only question is whether the Bernie bros will burn down the convention center or the entire city of Milwaukee. (/jk Bernie fans).

The odds numbers don't line up with the whole numbers (you say), but you say they are equal infinities. — Gregory

We DEFINE them to have equal cardinality because there is at least one way of matching them up bijectively.

You can prove "uncountable" infinities don't line up with the whole numbers either, but maybe they are equal as well. Until you prove that "uncountable" cannot be lined up with the wholes you haven't proven Cantor right. — Gregory

Again, this follows by definition. If a countable set is defined as one that can be bijected to the natural numbers, then BY DEFINITION an uncountable set is one that can't. So if a set is uncountable, by definition it can not be bijected with the naturals.

That doesn't in and of itself prove that there ARE any uncountable sets; only that if there were one, it could not be bijected to the naturals. And of course Cantor proved that the reals are one such uncountable set.

And what is your point? — Zelebg

A simulation of a thing isn't the same as the thing.

↪fishfry
Does simulated gravity attract nearby bowling balls?

Does actual gravity attract simulated bowling balls? — Zelebg

No. You just made my point.

Should we consider a simulated cell to be alive or not? I say it is alive in relation to its simulated environment, but if it could interact with the actual world then it would also be alive as any other actual living cell. — Zelebg

Does simulated gravity attract nearby bowling balls?

Bi-interpretability looks like an interesting subject, but unfortunately the Wikipedia page does not elaborate PA versus ZF-infinity as an example. — alcontali

Yes, it seems to be more subtle than mere logical equivalence.

Well, you did use the term "complete" in the sense of induction-complete. I clearly used it in the same way, and then you suddenly backtrack to claiming that induction-complete would be "no such term of art in set theory". — alcontali

You are entirely correct. I said, "The axiom of infinity allows us to take the "output of the completed induction," if you think of it that way ..." Guilty as charged. I have apparently been strenuously arguing against my own terminology and blaming it on you. My bad. Thanks for pointing it out.

So is materializing the same as completing?
— fishfry

It is used as a term for induction-completing (A term you actually introduced by yourself yourself). — alcontali

Ok ... materialization means making a set inductively complete. So if we have a set that contains x, it contains the successor of x and the successor of the successor of x, etc., for all possible applications of finitely many occurrences of the successor operation. A set is materialized if it's closed under taking successors.

In this case I believe you're talking about a limit ordinal. A limit ordinal is not the successor of any ordinal; but is rather the "completion," or closure under the successor operation, of its elements. So $0$ is the only finite limit ordinal; and $\omega, 2 \omega, 3 \omega, \dots, \omega^2, \dots$ are all limit ordinals. They're constructed by taking the upward limit of a collection of ordinals.

But why "materialized?" That seems to load the concept with some kind of metaphysical significance. I could live better with "inductively completed," now that I think about it.

In fact, I never used the term induction-complete or induction-completed before. I only used it because you used it first. I tend to use the term "materialized" instead of induction-completed. Furthermore, it is probably better not to further overload the term 'complete' with additional meanings. — alcontali

Guilty as charged. But I don't love the word materialized either. I think the idea of limit ordinals captures the concept.

...because if there is an all-knowing, all-seeing and all-powerful being, then the answer to every philosophical question becomes "Because God Says". — Banno

If science is true, the same argument can be made.

Moreover, you cannot discount the power of religion in human affairs. Wars have been fought, empires toppled in the name of religion. Religion deserves a place in philosophy for that reason.

When a discussion degenerates in "who is smarter than whom", "who knows more than whom", i.e. the typical, ridiculous conversations in the academia, in which they engage because they simply have nothing else to show for, then I tend to back out. — alcontali

Not my intention.