A mistake on my part and I am sorry for that. I should have written: "To avoid confusion, assume that the cardinality of the set of natural (I wrote real instead of natural) numbers is X. How could one show that X is the least infinity namely aleph_0?" — MoK

https://math.stackexchange.com/questions/517040/proof-that-aleph-null-is-the-smallest-transfinite-number

I am aware of that. To avoid confusion, assume that the cardinality of the set of real numbers is X. How could one show that X is the least infinity namely aleph_0? — MoK

It's not. Recall the Cantor diagonal argument, which shows that the cardinality of the reals is strictly greater than the cardinality of the naturals.

It's easy to prove that $|\mathbb R | = 2^{\aleph_0}$ and that this is strictly greater than $\aleph_0$.

The question of whether that happens to be equal to $\aleph_1$ or not is called the Continuum hypothesis. It's known to be independent of the standard axioms of math. That is, we can't prove it's true, and we can't prove it's false.

One attempt to resolve the issue is to try to find new, naturalistic axioms that settle the matter one way or another. Others argue that it's not even a well-defined question. The problem has been one of the driving forces in modern set theory ever since Cantor first posed it in 1878.

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

I am looking for proof that the set of natural numbers that each its member is finite has aleph_0 members. — MoK

That's literally the definition of $\aleph_0$.

A set is defined to have cardinality $\aleph_0$ if the set can be placed into bijective correspondence with the natural numbers. Clearly the natural numbers themselves have this property, with the identity map as the bijection.

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

I don't understand this argument. How could aleph_0 be a number and a set at the same time? — MoK

All numbers are sets. In math, everything is a set.

What do you mean by this? Do you mean that the set of natural numbers is the set of aleph_0? aleph_0 is a number. How could you treat it as a set? — MoK

As a set, $\aleph_0 = \mathbb N = \{0, 1, 2, 3, 4, 5, ...\}$

So you are saying that aleph_0 is not a member of the set of natural numbers yet the number of its members is aleph_0. I am however puzzled how all the members of the natural number set are finite yet it has aleph_0 members. — MoK

Let's look at some simpler examples.

1) Consider the set $\{3, 18, 58, 334\}$. That's a set with 4 elements, yet 4 is NOT an element of the set.

2) Consider the set $\{4, 18, 58, 334\}$. That's a set with 4 elements, and 4 IS an element of the set.

So we see that the cardinal number of a set MAY or MAY NOT happen to be an element of the set.

So far so good? Ok now into the infinite realm.

3) The set $\{1, 2, 3, ...\}$ does NOT contain the cardinal $\aleph_0$. I hope you can see that. If you claim otherwise, which element is it? It's not 1, it's not 2, it's not 3, etc.

4) The set $\{1, 2, 3, ..., \aleph_0 \}$ DOES happen to contain $\aleph_0$.

We see that in general, a set MAY or MAY NOT happen to contain its cardinality as a member.

In the case of the natural numbers, they do not contain their cardinality $\aleph_0$.

But there are surely $\aleph_0$ natural numbers, literally by definition. As we've seen from these examples, that fact gives us no information as to whether $\aleph_0$ is a member of the set of natural numbers. We have to look. And in this case, $\aleph_0 \notin \mathbb N$.

But if we tossed $\aleph_0$ into our set to get $\{1, 2, 3, ..., \aleph_0\}$, then we would have a set of cardinality $\aleph_0$ that happens to also have $\aleph_0$ as an element.

It can go either way, as the examples show.

By the way, how do we "toss $\aleph_0$ into the set?" The formal operation is taking a union. That is:

$\aleph_0 \notin \mathbb N$

but

$\aleph_0 \in \mathbb N \cup \{\aleph_0\}$

and BOTH sets have cardinality $\aleph_0$.

There doesn't seem to be anything about the races for the Senate and the House. But isn't it just as important as the Presidency? I have the impression that unless the President and Congress are the same party, the President is pretty much hog-tied. What's happening there? Is it as tight as the Presidency? — Ludwig V

Pretty tight in both houses. In general, the US economy does better when the opposition party controls Congress. The government can get into less mischief that way.

I'll pass on the details of the tower tragedy. Just thought it was interesting that a root cause was environmental do-gooderism, implemented badly or not.

I am still a registered Democrat, but it has been awhile since I have thought of myself as one. — jgill

Same here. It's the Dems who changed, not me.

This is counter-intuitive to me. Consider a function f with the domain D={1,2,...N} where N is a finite positive integer. The domain has N members and N is a member of the domain of f. Could you please explain what happens when N is aleph_0? — MoK

Sure. If you had a set $\{1, 2, 47, \aleph_0\}$, that would be a set with four elements, one of which is $\aleph_0$. Then if you had a function $f$ defined on that set, $f(\aleph_0)$ would make sense.

But $\aleph_0$ is NOT A MEMBER of the set of natural numbers. None of 1, 2, 3, ... are $\aleph_0$. So no function defined on the natural numbers takes a value at $\aleph_0$, since $\aleph_0$ is not a natural number.

if we bring in time and do the series Zenonian as i proposed (one at a time), and with each new slisce changed the color of the new slice, i can ask "what color" the top of the cylander would be. — Gregory

Uh-oh. We had a very lengthy thread about supertask a while back. Best leave that one alone :-)

The reason i brought up Hawking's "no boundary" thesis is that i was thinking maybe geometry and limits are incomplete by themselves and need the 4 spatial dimensions and 1 time dimension in order to make sense of it. — Gregory

Well math is math and physics is physics. Math is a tool for physics but they aren't the same thing. You're making connections that I'm not sure I see. But maybe there's something to it.

The mathematical continuum is perfectly clear. It's the set of standard real numbers. Personally I think it's very unlikely that the real numbers are instantiated in the world.

Yes, we are on the same page and thank you very much for your contribution. I learned a lot of things and refreshed my memory. — MoK

Glad I could be helpful.

If so how does one explain what happens to my mind when you crush my head between two boulders? — Benj96

Nobody knows what happens to the mind (or soul) when we die.

Oh, please! If there had been any do-gooding at all involved, it wouldn't have happened. It was greed and laziness. Complacency, if you like, in that Government trusted the builders to do the right thing. — Ludwig V

It's a general theme of mine that environmental do-gooding generally results in disaster and misery.

Thanks for the opportunity for a good rant. I hope I haven't bored you. — Ludwig V

Your response was interesting. Clearly you're getting more and better info about his tragedy over where you are. I have to depend on my alt-right sources.

If i cut a cake horizontally starting from the halfway point upwards with each slice being half the size of the one immediately below, what would the top of the cake look like? Isn't it indefinite? But you can definitely look at the cake, from all angles, and see that it has definite position in relation to its parts. So how do we reconcile the indefinite with the definite? I think this is what must be asked about the continuum. — Gregory

The continuum is a mathematical abstraction. It has no representation or instantiation (as far as we know) to anything in the physical world.

This is all explained by the mathematical theory of limits. The sequence 1/2, 3/4, 7/8, 15/16, ... is an infinite sequence that has the limit 1. The "top of the cake" is the limit of the sequence. In terms of a cylinder, the very top would be a circular disk of zero thickness, the same as any other horizontal slice. That is, the intersection of a cylinder with a horizontal plane (parallel to the top and bottom of the cylinder) is a circular disk.

It can be confusing to think about cakes, because cakes are made of atoms; and the Planck limits preclude our making fine enough horizontal slices to produce a zero-thickness slice. Cakes are not cylinders, and Math $\neq$ Physics!

Hawking would say that four dimensional Euclidean space, with a time dimension that both 1) acts as space, and 2) is described by imaginary numbers, gives an answer to this question. That is to say, the universe as a whole gives the answer to the continuum. But how do imaginary numbers relate to geometry? — Gregory

There is no evidence that anything in the physical universe is a mathematical continuum. It's possible that there is, but this is a deep open question that (with respect to current science) is more philosophical than scientific.

How imaginary (perhaps you mean complex) numbers related to geometry is a pretty cool subject, but far afield from understanding the nature of limits.

Here's the coolest example I know of how complex numbers relate to the geometry of the plane.

Take a regular n-gon (triangle, square, pentagon, hexagon, heptagon, octagon, etc.)

Place it in the complex plane such that its center corresponds to the origin of the plane, the complex number 0; with one of the vertices as the point (1,0) in the plane, or the complex number 1.

Now the vertices of the n-gon are exactly all complex n-th roots of 1; that is, they are all the complex solutions to the equation $z^n = 1$.

As a concrete example, consider a regular 4-gon, or square, with one vertex at the point 1 in the complex plane. Where are the other vertices? At $i$, $-1$, and $-i$. And these are exactly the four complex numbers whose fourth power is 1.

And this works for any regular n-gon. The vertices of the regular 17-gon are the seventeen 17th complex roots of 1.

That's one of the coolest things I know. And it is one example of the deep relation between complex numbers and geometry, which is the question you asked. But it's got nothing to do with the definition of limits or the nature of the mathematical continuum, but it's definitely interesting. I only mentioned it since you asked about the relationship of complex numbers to geometry.

Here's some Wikitude on the subject.

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

None of this has ANYTHING to do with relativity or spacetime or the physical world. Physicists use the mathematical continuum to model spacetime, but there is no evidence whatsoever that time or space or spacetime are literally the same as a mathematical continuum. They use the mathematical continuum as an approximation that seems to work, to the limit of our ability to measure the results of our experiments.

Remember: Math $\neq$ Physics!

It seems that there is no operation of infinite division in the real number system. That was something I didn't know. — MoK

There are limits. As an example, consider the sequence 1/2, 1/4, 1/8, 1/16, ...

We all know that the limit of this sequence is 0. You can certainly call this infinite division if you like, as long as you understand what limits are. There are indeed infinitely many elements of the sequence, and you CAN think of this as "infinite division."

You can even think of doing it "all at once" if you like

What it means formally is that the elements of the sequence get (and stay) arbitrarily close to 0.

What it does NOT mean is that there is some kind of magic number that the sequence attains that is a "distance of 0" from 0, but is not 0. That's a faulty intuition.

In fact the formal theory of limits, once one learns it, is the antidote to all our non-rigorous, faulty intuitions about infinite processes.

Does that help, or perhaps refresh your memory? I'm pretty sure that physicists must be exposed to the formal theory of limits at some point.

Oh yeah, I can guess that. We, physicists, work with the infinities all the time. Of course, mathematicians do not agree with how we deal with infinities but strangely physics works. :) — MoK

Yes I understand and agree. But I am a little surprised that you seem to think that a sequence attains some kind of mysterious conclusion that lies at a distance of 0 from its limit, but is distinct from the limit. That's just not right.

Did my mention of limits ring a bell at all? Or raise any issues that we could clarify or focus on?

Because your idea of endless division is perfectly correct. But all that shows is that endlessly halving leads you to the limit of a sequence. But there's no "extra point" in there that's distinct from but at a distance of 0 from the limit.

Let me know if we're on the same page about this.

Not intentionally. If I've upset you, I apologize. — Ludwig V

No worries, I no longer remember.

I did happen to run across something yesterday. The British government put out a big report on the Grenfell disaster.

The Spectator put out a summary blaming the incident on "complacency."

https://www.spectator.co.uk/article/the-vital-lesson-that-must-be-learnt-from-the-grenfell-inquiry/

Spiked-Online noted that the reason the tower burned was that it was wrapped in flammable cladding that had been installed for environmental reasons. In other words the building itself would not have burned but for the cladding that had been wrapped around it as insulation. And now the government is busy removing the flammable cladding from other buildings.

So the loss of life was attributable to liberal do-gooding. Needless to say the official report did not make this point. Thought I'd pass this on.

https://www.spiked-online.com/2024/09/05/why-was-grenfell-covered-in-cladding-climate-targets/

Why does that concern you? — Ludwig V

I'm beyond explaining this. Let's agree to disagree.

Everybody who has power has an opposition. The opposition always thinks that those with power should be "reined in" or crushed. (Actually, if you think about it, that's really a very mild comment compared with what some people say). Most people with power are either "reined in" by the opposition or their own failures. — Ludwig V

Eminently sensible and moderate.

I've no idea whether Musk will be reined or crash and burn. At the moment, it's impossible to tell which it is to be. The sooner the better, as far as I'm concerned. There'll only be another like him afterwards. — Ludwig V

You're trolling me now. I'm kind of done here. I can always tell when I'm at the point when I have nothing else to say without repeating myself.

It depends what you think is anodyne. Compared to the way that some people carry on (without being thrown in jail), it probably is anodyne. But most people's comments are just hot air - unpleasant, but not harmful. Look at the consequences. — Ludwig V

Then why is Starmer throwing pensioners in jail for remarks that are unpleasant but not harmful? But like I say, I'm repeating myself.

There was a famous speech in the 60's by a Conservative politician named Enoch Powell, in which he drew everyone's attention to the flood of immigration into Britain, painted a terrible picture of the abolition of the "British way of life" and announced that there would be "rivers of blood" in the end. Was he reporting? Or was he inciting? I don't know what his motivation was, but I know what happened as a result. It wasn't rivers of blood, but it did involve bloodshed and it was very ugly. — Ludwig V

Yes I remember Enoch Powell. Don't recall the incident you're referencing.

You may have seen the reports of the report released about the fire in Grenfell Tower. Everybody is very shocked and horrified. In a way, so am I. But I have known it was coming ever since the then Government relaxed the building regulations. It was only ever a matter of when and where. It was obvious. It was also always obvious that when it happened most people involved would say it was not their fault, even though it is obvious that they all contributed. No clean hands. — Ludwig V

Now we're into building regulations? Not following. The US infrastructure is likewise decrepit. Gotta fund the wars, you know.

There has never been a golden age when there was no censorship, no authoritarian squelching of opposition. It was ever so, it will always be so. — Ludwig V

Ok. I can't respond with anything I haven't said before. You are justifying evil by saying there's always been evil. Fine.

I'm a somewhat old-fashioned middle-of-the-road liberal and I felt more comfortable 20 or 30 years ago. I grew up in the post-WW2 consensus/settlement. It was never what it seemed to be and it fell apart anyway. (If you want a date, it was the election of Margaret Thatcher in 1979 that did it.) Once that has happened to you, you never, ever buy in to anything else with the same innocent, deluded conviction. — Ludwig V

I'm a big fan of Maggie as you might imagine. Though I wasn't at the time. Both she and Reagan look much better in retrospect. I used to be a liberal too. Something happened over the years.

If you do decide not to continue, that's fair enough. I wouldn't want to (couldn't) detain you if you have better things to do. So long as you aren't leaving for the same reason that you left the Lounge. — Ludwig V

Well ... they're a lost cause over there.

In this case, I just see that I haven't said anything new in quite some time.

Better to let me know when you make your decision, so's I know what's going on. If and when I make the same decision, I will let you know. OK? — Ludwig V

Well ... I guess I'm done. But I've never had a long private convo like this. You could post something on the public area, at least that way we'd get some fresh meat once in a while.

I literally can't think of anything to say that I haven't already. Kamala came out today railing against Elon Musk's freedom of speech. What should I do, say I object and plan to vote against her? Doesn't matter anyway, I vote in California which will go for her by millions of votes. My vote literally doesn't count.

So I guess I'm done. More for lack of anything new to say than any other reason. Appreciate the chat. But I hope we can engage in public where others can jump in. I think it's weird that the moderators buried this chat.

ps -- I haven't anything better to do!! LOL. Am I leaving too soon for your taste? I don't mean to be short. I just haven't got anything else to say. Maybe my concerns about the creeping authoritarianism of the globalists is misplaced. I can sum it up in a cartoon I saw the other day.

This was a reply to the above comment from fishfry who claimed between any two distinct real numbers, there is always another one strictly between. The distance between two points is zero if the number of divisions is strictly infinite so there cannot be a point between two points in this case. — MoK

This is incoherent at best, wrong at worst. I explained this to you at length. But look. You are trying to prove there are two points without a third between them, by claiming there are two points at a distance of zero. Can you see the circularity of your argument?

I explained this to you at length in a post you didn't bother to engage with.

I am a retired physicist — MoK

I believe you. Physicists attempting to do math are often a source of humor and/or horror to mathematicians. But I'm sure you know that :-)

Honestly. I explained this to you at length. What you've worked out for yourself is the equivalence of the two definitions of a dense set: (1) That there is third point between any two; and (2) That there is a sequence of distinct points approaching any given point as a limit. You proved that with your bisection idea. Your idea is essentially correct. Your intuition about what it means is mathematically wrong.

It is very crank (confused, ignorant and fallacious) to conflate the limit of a sequence with an out-of-thin-air claim of an operation of infinite division. — TonesInDeepFreeze

What Tones said.

Hmmm. Care to explain? (I recall having difficulty with filters, ultra filters, etc. in grad school a half century ago. I only encountered them in passing - not in my specialty area) — jgill

I'm not sure what @sime meant by that statement either. But ultrafilters are just a set theory gadget that lets you rigorously construct the hyperreals of nonstandard analysis.

That's not quite fair. I do agree that free speech is a Good Thing. So I am bothered by Putin and Xi Jinping. But I don't think that criminals should be allowed free access to their victims — Ludwig V

Recent developments in the West are very concerning. Robert Reich, Clinton's Secretary of Labor, just called for "reining in" Elon Musk.

https://www.theguardian.com/commentisfree/article/2024/aug/30/elon-musk-wealth-power

Famous law professor Erwin Chemerinsky just published a book calling for dumping the U.S. Constitution.

https://www.theguardian.com/books/article/2024/sep/01/erwin-chemerinsky-no-democracy-lasts-forever

There are many other examples. You talk about Putin and Xi but you don't seem concerned about the creeping -- actually now galloping -- authoritarianism and censorship in the west. I'm very concerned; you much less so. So I don't think my point was unfair. For a Brit to ignore these issues lately I find very strange. They're putting people in jail in your country for very anodyne online comments.

The song but not the singer. I don't disapprove of some enivironmentalists, but I do get bored with them. — Ludwig V

With you there.

The truly depressing thing is that the poor are screwed by climate change and by the attempts to reduce it. — Ludwig V

My very point. Environmentalism is elite virtue signaling.

Well, not to go on about it, I can accept that there is some work around trans people to be done. But the recent publicity has been provoked by some thoroughly objectionable trans people (and some "trans" people). My partner has some acquaintance in those circles and tells me that many trans people just want a quiet life and are horrified by them. — Ludwig V

Well yes, of course. It's always the extremists who make the news.

But doctors are doing double mastectomies on perfectly healthy 12 year old girls. That's something tht needs to be pushed back on.

The really basic question is why there is no decent candidate on either side. All the people who might have make a good shot at an impossible job seem to have taken a back seat. — Ludwig V

Longstanding problem. Bush-Kerry. Trump-Clinton. Trump-Biden. Trump-Harris. etc.

I'm kind of running out of steam on this site. Might need to wrap this up soon.

There is, indeed. It may not be perfect, but some arrangement like that is all there is. — Ludwig V

You are agreeing that free speech is a virtue then. Yet you don't seem too bothered by the globalist war on free speech.

Don't be ridiculous. — Ludwig V

It's a great heresy to be against the environmentalists these days. But of course IMO one can be against the environmentalists yet for the environment. That would be me.

It was always obvious that dealing with climate change would be a mess, and that it might well be ineffective. We can probably organize some response after the event. There will be some mitigation, but nothing less that world-wide panic will trigger serious attempts at mitigation and that won't happen until serious climate change has kicked in. As usual, the poorer countries will suffer most, and much of their population will leave, looking for somewhere safer to live. There'll be a lot of trouble. — Ludwig V

The poor countries suffer from radical environmentalism. When you raise the cost of energy, the limousine liberals aren't affected. The poor are. And the third world suffers the most.

Fair enough. We can achieve things. It's just that it takes a disproportionate amount of shouting and shoving to make things happen. It helps when people can see the effects themselves. (see above) — Ludwig V

The effects are virtue signaling among the first world elite; and terrible suffering in the third world, out of sight. This is my point. I oppose the environmentalists.

Yes. Temperate. So too hot and too cold are both problems and climate change will cause more of both. But the temperate north and south of the world will be less badly affected than the equator and tropics - apart from the effects of sea level rise and the increase in extreme weather events. — Ludwig V

I don't know how we got here but environmentalism isn't one of my favorite conversational topics. I know what I think and I don't bother to talk about it much.

No. I looked at the wikipedia article. It seems quite plausible. But I'm very difficult to convert. I'm going to be reading "Techofeudalism" soon, in a futile attempt to keep up to date. — Ludwig V

I shall read the Wiki page :-)

I don't know about you, of course, but I was liberal when liberals were a minority and thought to be insane. Then things starting going our way. Splendid - until I realized that younger generations would want to push everything further. I've gone some way with them, but not all the way. Much of what they are pushing for now seems to be dubious, at best. They don't remember what it was like to be what it is to be an oppressed minority, so they feel no need to compromise and make room for different views. But hey! no-one listens to doubts and compromises any more. — Ludwig V

Right. But most longtime liberals haven't noticed. They've gone from gay rights (good) to transing the kids (bad) without missing a beat.

Now that Biden has gone, the context has changed. He looks different in a different context. I think you'll find that the right wing will get some of what it wants - not all. That's what's happened to liberalism. Life has to go on and forces compromises. Remember, liberals are as fearful as conservatives. — Ludwig V

Liberals are stupid and mean these days.

Did you see the Kamala "interview?" If the Democrats get away with this the country is doomed. Not just policy-wise. But that Americans would have validated the four year Biden swindle, propping up a senile candidate who campaigned from his basement; and then swapping in the historically unpopular Harris, hiding her from the press while her fans swooned. It's very bad if they get away with this. And honestly, not too much better if Trump wins. He's past his prime for sure.

Not in this case: — schopenhauer1

My very first thought :-)

Yes. I do worry about that argument. But since Stalin was on the left and Hitler on the right, it seems like there's no safety anywhere. — Ludwig V

There's safety in free speech and a limited, Constitutional republic. Me and Thomas Jefferson against the world.

Any more than there is against the possibility of all-out nuclear war (or indeed against the reality of climate change) These things are hard to predict. — Ludwig V

Sigh. I probably shouldn't reduce your esteem for me any more than I already have, but I'm not much of a climate fanatic, either. The question is whether we should wreck our economy and throw billions into poverty to effect a hypothetical fraction of a percent change in the average global temperature, which is ridiculously hard to measure anyway.

The air and water are a lot cleaner than in the 1970s, so I'm all for the environment. I love the environment. Just not the radical environmentalists.

Besides, the Obamas own beach front property in two states (Massachusetts and Hawaii), so clearly they're not too concerned with the rise of the oceans. Besides, warmer temps are GOOD for life and colder temps are BAD for life. So a lot of what passes for environmentalism these days is ass backward.

The world is stumbling into nuclear war. US foreign policy is a bloody disaster.

Yes. I expressed myself badly. Perhaps I was in a bad temper. My point was that most people are sore losers and it's very hard to tell when a protest like that is valid. — Ludwig V

The lawfare against Trump is wholly illegitimate and many liberal legal minds have so opined.

I am a disillusioned liberal. Still a registered Democrat. I'm just horrified by what's become of my former fellow liberals and Democrats. Some of them see it and most of them don't.

I'm afraid the Telegraph has been tracking my viewing of its articles. There's a limit on free views of them and I've hit it. But I do know that there was a case like that and there was a lot of reporting of it. I don't pretend to know the rights and wrongs. — Ludwig V

I don't either. We'll all find out how this plays out in the next few years. No question that the liberal governments of the West have decided to throw open their borders to hordes of people who don't share their traditional values.

You know Christopher Lasch's book. The Revolt of the Elites? The idea (I haven't read the book -- I no longer read books, only Wiki pages and articles in scurrilous right wing rags) is that rather than the people revolting against the elites, these days the elites are revolting against the people. Hard to argue with that thesis, we see it all around us.

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

I rather think you have a bad day. I'm sorry about that. — Ludwig V

LOL. For a while I was trying to engage over in the political threads in the Lounge, but it's just a bunch of mindless checkbox liberals throwing insults. So I gave up. I am a little burnt out on the standard liberal talking points against Trump. Heck I don't even like him much, he's old and tired and bitter. But he's all we've got against the continuation of what's been going on.

Apologies for getting triggered :-)

Some of the public are quite likely not happy. Others are more bothered by the rioting and are perfectly happy. Starmer has read the mood perfectly. — Ludwig V

"If you object to stabbing six year old girls to death, you just might be a right winger."

Yes. I won't use it again. And I'm all ready to slap down anyone who tries to.[/erquote]

LOL. I think I overreacted.
— Ludwig V

I don't think he cares much what I think, — Ludwig V

Not today. Today, he's putting people in jail who express ideas you don't express. So you let me know when an authoritarian regime has ever known when to stop. As he was consolidating power, Stalin killed his most fervent supporters. Hitler did the same.

and anyway, I don't think he's listening. — Ludwig V

What makes you think that? All digital communicates get stored. Nobody looks at them till your friend's friend's friend's friend whom the government doesn't like, steps out of line. Then they roll up the whole chain. Like I say. Find me an authoritarian regime that ever knew when to stop.

But you never know. Everything leaks in the end. But I do choose carefully about who I raise it with. — Ludwig V

Doesn't matter. Some friend of a friend might say something the government doesn't like. Your argument here is, "Who cares if someone else goes to jail for saying something the government doesn't like. They won't do that to me." History has not been kind to that argument.

I can see your point. The problem is that whether you cheer on the rioters depends on whether you agree with them. You and I don't have to be impartial, so that's ok. Law enforcement does. But it's nigh on impossible, but I think most of them do try. — Ludwig V

I don't want to keep discussing this. Floyd versus J6 is just as blatant an example as you can find. Two billion dollars in property damage and twenty dead; versus a few old ladies wandering aimlessly around the Capitol building. Many MANY completely nonviolent J6 protesters have been in solitary confinement for three years. This is an outrage; and a bigger outrage is that it's not generally recognized as such.

I do think it is hilarious to hear Trump bleating on about how all the prosecutions against him are political. — Ludwig V

I absolutely and without reservation share his bleats. Even liberal legal scholars have been outraged by the New York 34-felony case. It's a legal travesty, the kind of thing you see in banana republics.

I don't know whether or how much they are influenced by political considerations. — Ludwig V

100%. None of those cases would ever have been brought if Trump weren't Trump.

The thing is, he wants to make all prosecutions political, by appointing people who agree with him politically to, for example, the Supreme Court and throughout the legal system. — Ludwig V

Bullshit. You're just spouting leftist propaganda. It's not worth my time to have these arguments.

What matters is whether he is guilty or not - the fair trial. — Ludwig V

Stop. Please. Just stop.

He does the same thing about elections. If he likes the result, he accepts it. If he doesn't he decides that the ballot was rigged. His losing the election is not evidence that the ballot was rigged. He's not the only one, but he's the most prominent one. — Ludwig V

Man I've been hearing this leftist claptrap since 2016. Enough already. I don't begrudge you your beliefs. I do choose not to engage with them.

I agree with all of that. The liberals focus too much on the individuals and the hard-liners too much on the numbers. There's a real need to balance and consensus. — Ludwig V

Ok whatever.

Where would we be without rebels against authority? But choose your issues. — Ludwig V

I am. Today, these ain't them.

You do like the contentious topics. Yes, some people are very trigger-happy. I find "Let's agree to disagree" followed by ignoring them works quite well.
I've seen a bit of Quora (and Reddit). They look a bit too much like snake-pits for me. — Ludwig V

I like the math and computer sections of Reddit. Quora is a pale shadow of its former self.

Don't we all? But sometimes there is a deeper issue - the arrogance of the opinion or its wilful blindness, for example, rather than its content. — Ludwig V

I would say you have much willful blindness about the Democrats' corruption of the justice system to go after Trump. But then I'd be arguing this tedious subject again.

The first day is the hardest. The hard thing is to disagree nicely - especially with sensitive people. But if you can, you might actually persuade the other side to move a bit. — Ludwig V

I have never persuaded anyone of anything in decades online :-)

I saw this today.

https://www.telegraph.co.uk/news/2024/08/26/violent-offenders-increasingly-let-off-with-apology/?ICID=continue_without_subscribing_reg_first

It's about how the Brits let stabbers go if they apologize to their victims. Meanwhile, old ladies who say the wrong thing online go right to prison.

Maybe it's all lies. How would I know, right?

Therefore, we have — MoK

$\frac{\aleph_1}{2^{\aleph_1}}$?

The burden is on you to justify this notation. Perhaps you can begin with your theory of cardinal division. You'll have a hard time making sense of it. Trust me, if there were any mathematical theory that justifies this notation, I'd have heard of it. I haven't and there isn't.

BUT! Your underlying idea of continually subdividing an interval is correct. It just doesn't show what you think it does.

If as @jgill notes you are simply proving that you can find a sequences approaching $a$ and $b$ as limits, that just amounts to a restatement of the fact that the real numbers are dense: that between any two, there's a third. An equivalent condition is that we can find a sequence approaching as a limit any given real number.

And the equivalence is shown by continually taking the midpoint, as you did. But your notation is fanciful and undefined. Your basic idea is correct, but it does not show that there are two reals without a third between them. On the contrary, it shows that if between any two reals there's a third between them, then we can find a sequence approaching as a limit any given real.

Give this a read.

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

ps -- How did you get to $\aleph_1$ as the limit of 1, 2, 3, 4, ...? Did you mean perhaps $\aleph_0$?

In summary, your idea is correct, even if your notation isn't. But your idea only shows that the two definitions of a dense set coincide. If you can always find the midpoint between two points, then you can always find sequences converging to each of the two points.

Disgusted of Tunbridge Wells. — unenlightened

"On the whole, I wish I'd stayed in Tunbridge Wells."

-- Claude Rains character at the end of Lawrence of Arabia.

Couldn't find a shorter clip but it's at 4:18 here.

https://www.youtube.com/watch?v=pZdLM2ENld8

There are still plenty of places in the country where it is legal to kill a fetus. In any case, it was a shaky legal precedent, not a right. Now everyone can go about it the right way. — NOS4A2

I'll take the other side of that for sake of discussion.

I have heard over the years that Roe was bad law. Even some liberal, pro-choice legal scholars made that argument.

But as a moderate pro-choicer (safe, legal, and rare as Bill Clinton put it), I say that Roe was working. It kept abortion off the ballot. It's analogous to Obergefell. Before Obergefell, gay marriage was an issue in every election. Now, whether you support or oppose gay marriage, it's the law of the land. You can blog your opinion, but it's never on the ballot. It never affects an election.

In the same way, Roe kept abortion off the ballot. It may have been bad law in the eyes of legal scholars, and it upset the pro-lifers, but politically it was working.

I say that if the so-called conservative justices were secretly working for the Democrats, things couldn't have turned out worse than they are now. Abortion kept the 2022 red wave from happening. It's an issue in 2024. It's Kamala's strongest issue. I've seen her give pro-abortion speeches and she is really, really good at it. She has her heart in the issue and she has her talking points straight.

Dobbs has been an electoral gift to Democrats and it is going to continue forever. It's worth a few points in every election from national to local and it's going to be till Congress does something about it, and they never will.

And it brings out the worst in the pro-life forces. This idea of arresting women who cross state lines is completely insane. I've "crossed state lines" from California to Nevada to visit gambling casinos. Nobody ever objects to that, even though laws against gambling used to be rooted in moral arguments.

Dobbs has unleashed the worst impulses on the right. It's just a disaster for the GOP.

Point taken. If Government and Corporations are collaborating, normal people don't stand a chance.[/quote]

Mark Zuckerberg was in the news today, sending a letter to Congress admitting that he was pressured by the government to help cover up the Hunter Biden laptop, which probably swung the 2020 election to Biden. He said he regretted being part of that cover up. Too little too late but better than nothing.

The Internet, which we all naively thought would be a tool of our liberation, instead turns out to be the instrument of our enslavement. China's social credit system on steroids, coming to a gulag near you.

Oh well you'd probably just say I "resent" that boot stomping on my face, forever. I should just get with the program and love Big Brother. I have no other choice anyway.

There was a landmark case in the US about this. The difference is that platforms (internet, phone, slowmail and, I think, couriers) are not responsible for the content of what they carry, only for delivering it. — Ludwig V

Right. Section 230.

https://en.wikipedia.org/wiki/Section_230

But Government can intercept and read them. Newspapers and publishers in general (broadcasters as well) do have responsibility for the material they publish; I think the difference is that they have editorial control over it, i.e. pick and choose what they publish. The point about platforms is that they don't pick and choose. The internet providers won the case, and have been dodging the small print about Government access ever since. — Ludwig V

Right. But it's tricky. Nobody, not even freedom-loving and rule-resenting me, thinks online platforms should be allowed to carry criminal material.

You know the reason I'm a little triggered by you saying I resent rules is because it's true. I've always been this way, always a rebel against authority.

Nigel is indeed very likeable when you first meet him. When you get to know him better - not that I know him, but I have followed him and had him pushed in my face for quite a while - you may well decide that he is a sleaze-bag. I doubt if he seriously cares about anyone but himself. — Ludwig V

I admit to being taken in by his superficial charm. Plus the UK is getting pretty stabby lately and the public is not happy when the only people going to prison are the ones calling attention to it.
I see no checks on his power at the moment.

I'm very mindful of that.[/quote

They say the Telegram case is a warning to Elon Musk, that he's next. The powers that be don't like free speech.
— Ludwig V

That would be worrying. But people setting up a meeting with the intention of rioting - those I worry less about. — Ludwig V

I'm not talking about people actually inciting riots. There are old ladies being tossed in jail for much much less.

And again -- in the US, the ruling class cheered on the Floyd riots and threw the J6'ers in solitary. So it's two-tier policing again.

Happy medium is exactly right - but also the problem. You do know, don't you, that illiterate people can also make a contribution? Not sure that reducing welfare for everyone in order to discourage immigrants would play very well in politics. — Ludwig V

Milton Friedman said you can't have open borders and a welfare state. That's the point I'm making.

In fact in the abstract, I'm an open-borders type. I say let everyone go where they like, but don't give anyone handouts. Then the productive people would gravitated to to the most free-market jurisdictions.

But of course that's not practical, because when people show up you can't just let them starve in the streets. So my solution is purely theoretical and idealistic. In real life, I'm just glad I'm not a big-city mayor, I'd have no idea what to do.

IS have claimed responsibility for events that they had no hand in. On the grounds that anyone who does something they approve of is a supporter. I'm not sure where that issue has got to now. — Ludwig V

Yes true. But the stabber was an Islamic refugee. And the German people are unhappy, hence their own anti-immigrant movement.

You know I like immigrants. If the government would impose some order on the system, it wouldn't be creating a right wing backlash. I don't like racist hooligans. But we have to try to grapple with the government policies that they are reacting too.

Yes, indeed. It's not a popular theology, but the ancient Greeks believed it and the Vikings had a special god, Loki, for mischief. They reckoned that one of the primary functions of human beings is to provide amusement for the gods. Not a bad idea. Conventional heaven seems rather boring. — Ludwig V

Yes definitely. God has a sense of humor.

I'm not surprised. But once you have conceded that, it's just a question of what and where. Not that it's an easy question. — Ludwig V

Pretty vile speech is affirmed over and over again by the US Supreme Court. It's a principle not often supported any more even in the US.

Well, I was never talking about the law as such. I didn't know about the Supreme Court. My intention was to use a cliche as a quick way of making a point. — Ludwig V

Yes sorry hope I didn't overreact. I did happen to read about the fire in a crowded theater example, and it turns out it was never against the law, and it was only kind of a sidebar issue to some legal case that's long since been overturned anyway. So everyone uses the example incorrectly.

What you can't do is incite everyone to murder the theater manager. That's a direct incitement to violence.

there are scenarios in which intentionally lying about a fire in a crowded theater and causing a stampede might lead to a disorderly conduct citation or similar charge. — Ludwig V

Yes I'm sure they'd throw the book at someone for doing that. So maybe it is illegal after all. I have no idea.

This was more what I was gesturing at, but more as a moral criticism that a matter of legal action.
Still others have categorized hate speech in a similar way.
I do have a problem about restricting that. Freedom of speech includes the right to give offence. — Ludwig V

Yes right. Just don't let Two-teir Keir hear you say that :-)

And believe me, with Harris and the Dems a pretty good chance to get elected, free speech will be over in the US soon enough.

And I agree with that. It's not contradictory. The reconciliation is that it seems only natural that if someone insults and abuses me, I would want to deck them, but that would be to lose the argument, so instead I would try to make them shut up. In a democracy, if that's the will of the people, I won't object. — Ludwig V

I'm thinking of online mostly. I'm on Quora a lot arguing about the JFK assassination, and people just get vile about the most trivial differences of opinion. And sometimes I do the same thing. I'm trying to be nicer and more civil online. Been at it for about 24 hours now :-)

Yes, then there's nothing much to further discuss. I did some work on Godel coding for compression algorithms with countable denumerable alphabets, such as color encoding like RGB for satellite TV to deliver true 4,8,12K video. It was a fun task that led me to believe that every denumerable ordered task can be sped up or optimized by actually archiving the read to the CPU with already post-processed information, and thus labeling it as if a "brick" to every further task to be done on similar logic. Eventually, with so many bricks, you could compile the task on the CPU, to just be read out to the memory. To process the information wouldn't be anything too far-fetched; but, the archive file might be quite big to cache. The optimization might be quite profound in my mind. — Shawn

I'm not an expert on this topic. As I understand it, memoization is just used to store results of calculations and things like that. I could be wrong because my knowledge is from some years ago.

Your idea seems much more extensive. I would not want my little comment to end this interesting discussion! Your concept seems to go further.

Is this something that is done already on hardware, or only on software to this day? — Shawn

I know it as a software technique, but I'm not up to speed on the state of the art.

... have a copy on a hard drive ... — Shawn

Are you perhaps talking about memoization?

In computing, memoization or memoisation is an optimization technique used primarily to speed up computer programs by storing the results of expensive function calls to pure functions and returning the cached result when the same inputs occur again. — Wiki

and

A memoized function "remembers" the results corresponding to some set of specific inputs. Subsequent calls with remembered inputs return the remembered result rather than recalculating it, thus eliminating the primary cost of a call with given parameters from all but the first call made to the function with those parameters. — Wiki

The question is whether Telegram is facilitating free speech (good) or facilitating criminal activities (bad). I think that if he couldn't help the bad people taking advantage of Telegram. But he could at least try to prevent them or at least help police and prosecutors nail them. — Ludwig V

Comes down to what responsibility platforms have. Being litigated all over the world at the moment.

Not really, though politics played a big part. Prosecutions in Athens were only brought by private citizens; there was no such thing as Government legal action. It was a very different world. The real problem that many of his followers were right wing. But there's no evidence that he agreed with them and some evidence that he believed in the Athenian constitution, which the right wing opposed. — Ludwig V

I don't really know much about it. I heard he got a bad deal. Still, corrupting the youth. That's the kind of charges agains Telegram and other social media companies. "Disinformation." Who decides what that is?

— Ludwig V

Ok!

I find it hard to believe that he didn't realize he was liable to arrest if he went to France. What were his people doing? It looks as if he and they just assumed that because he was OK in the USA, he must be OK in France. That's the kind of attitude that seriously annoys the rest of the world. — Ludwig V

Nobody can figure out why he landed his private plane in France. Perhaps he expected to get arrested and wants the legal fight. Personally I spent a night in jail once and did not like it. I wouldn't go to jail to prove a point.

True. But fascism does. — Ludwig V

Oh no, that's the point. Mussolini defined fascism as the merger of state and corporate power. That's exactly what happened when the US government pressured the social media companies to censor and suppress speech. That's exactly what fascism is. No jackboots. Just the state and corporate power crushing the freedom of the individual. It's rampant these days. Very dangerous. Because it comes dressed as benevolence. "We just want to keep you safe from misinformation." Soft fascism if you like.

I don't know about that case. I agree it looks bad. But on the principle, the difference between murder and manslaughter is your intention i.e. what is in your thoughts. — Ludwig V

We talking about the Telegram guy? Brand new story, he just got arrested lately. One account said "... he’s now jailed, and facing 20 years for the heinous crime of “allowing people to speak privately to one another in a manner the EU cannot readily surveil.”

I find that concerning. We'll see how this plays out.

Fair enough. I don't expect us to agree about much. — Ludwig V

But we sure can yak!

I'm quite happy to understand what you think and find out what we agree about. After that, agreement to disagree is fine, and certainly much better than exchanging abuse. — Ludwig V

I don't like online abuse. Or like that great Rolling Stones line ... "I went down to the demonstration, to get my fair share of abuse." Love that line.

You seem to resent any restrictions on free speech. — Ludwig V

Resent? Not sure what you mean. I support the First amendment. One of the best things about the US. I believe freedom of expression is one of the most basic and vital of all human rights. It's under attack all over the so-called liberal west. I find that troubling. I see no resentment there. I see the defining political issue of our time. The freedom of the individual to say what's on their mind.

The classic question here is whether you have no objection to someone shouting "fire" in a crowded theatre or stadium when they know darn well that there is no fire. (Thus causing mass panic and distress, injury and death) Nobody doesn't limit free speech. The only question is what limitations are appropriate.[/quote]

I hope you know, and as a professional philosopher you should know, that this is a bad example, was never a principle of law, and isn't about what you think it is. Even Wikipedia has a decent writeup.

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

Another good writeup:

https://www.theatlantic.com/ideas/archive/2022/01/shouting-fire-crowded-theater-speech-regulation/621151/

You are wrong to use that example. It's totally weak and incorrect argument. It does not mean what people think it means. It's not illegal. It was never illegal. The legal ruling in which it appeared has long been overturned.

In the US, direct incitement to violence or unlawful action is illegal. Just about anything else, no matter how vile, is legal. Of course that is under attack these days.

That's what I call the honey-pot effect. That's a thorny problem too. — Ludwig V

Thorny enough that Brits are rioting. Americans haven't gotten to that point yet. America's a big place, you can drop in ten million foreigners and the disturbances will still be local.

What if you disagree with the existing laws about immigration? People who have a problem with immigration want restrictive laws as well. Most people expect some level of control. The really thorny argument is how much control should there be. (At one point, the law in the UK did not allow any immigration at all. It didn't work very well.) — Ludwig V

The Biden-Harris administration had an open border. They tightened it up this election year when it became a political problem.

No nation can have an open border when the people coming in are by and large illiterate peasants with few work skills and massive social needs.

I may be exaggerating about the police state, but how would you feel about employers having to get government approval before offering anyone a job? — Ludwig V

Counterintuitively, I'm a libertarian on that. I believe in the free exchange of labor. But Mexico is not exporting brain surgeons. They're exporting illiterate peasants. You have much the same problem in your country, along with a certain degree of anti-western religious feeling. Not by all, but by some. You did hear about that stabbing in Germany. At the "diversity" festival no less. God is a joker.

Or hospitals having to check your status with the government before treating or even examining you? Or hotels, landlords and restaurants contacting the police before letting you have a room? Schools asking permission before they take on your child? Have a look at what China is doing on the surveillance front. — Ludwig V

No, I disagree with all of that. I don't claim to have the answers. I'm pro-freedom. If you abolished the welfare state I'd be for open borders. Some happy medium. Fewer social services in order to discourage people from showing up who can't support themselves. With that proviso, I'd let everyone in who can make a contribution. That's actually my belief, not that I'm certain it would work.

Whose line is it over? Yours? — Ludwig V

Not yours? People thrown in prison for tweets the government doesn't like?

But you are not living here and you are not a citizen. The job of the UK government in the UK is to keep in line those who are way over the UK lines (by law). That's what they are doing. — Ludwig V

Not that benign from what I hear. We'll have to see how it plays out. You know the censors never stop with the people YOU don't like. When they came for the trade unionists I said nothing, etc.

There's a paradox. In the UK, there is practically no coverage at all of what they are doing at the moment. They are invisible. — Ludwig V

Is that right? It's all Meghan and Harry all the time over here. Probably because the New York Post is all over it. Another scurrilous right wing gossip rag I read every day.

She does seem to have got the Democrates back in contention. She seems to have worked out that joy and confidence are more attractive than fear. It's a brilliant move against Trump. — Ludwig V

For sure she's a big upgrade over senescent Biden. She can stay up past 4pm and whip up a crowd. We'll see how long she can get by without ever having a press conference or an interview. She can read a teleprompter very well, but she's often a disaster when speaking extemporaneously.

I'm also wondering if his age is catching up with him, and whether it will create difficulties for him when it comes to voting. That would be ironic. There's a rather old-fashioned phrase in English English "hoist with his own petard" it means roughly "blown up by his own grenade". Very satisfying. — Ludwig V

Trump is old and seems tired and out of it. No question. He's 78, nobody should be running for president at that age. And whether you think the lawfare and impeachments and Russiagate have been justified or not, he's been under enormous stress for eight years. Most humans would have long since been broken.

But yes he is the old and feeble one now.

Well it will help if, in the mean time, we do not treat as terrorists people who are not terrorists. Islamic terrorists are a tiny minority of Islamic people. The vast majority of them disapprove of them. Other Islamic people have suffered from them as well, you know. — Ludwig V

I agree. Then again there's that German stabber. Islamic terrorists have taken credit. People don't ike that kind of thing and it only takes 1% to ruin it for the rest. Not very fair to the 99% of hard working, loyal, peaceful Muslims in Europe.

I'm sure he will, and if he doesn't, there are plenty of his supporters and officials who will sit on his head. — Ludwig V

I see no checks on his power at the moment.

On the contrary, I'm seriously worried that the whole world is moving to the right. The dictators (Russia, China, Iran, North Korea and all the small fry) think things are going their way. They recently had a global conference to swop tactics and sympathy - somewhere in S. America, I think. The UK, I believe, was represented by Nigel Farage! Talk about the emerging global government. It's quite likely to be a right-wing government. — Ludwig V

I like Nigel. He's fighting the emerging globalist government, as is Trump. The globalists talk like leftists and rule like ... well, fascists. Without jackboots.

Not what I said. So either you can't read or your memory is a sieve. — Benkei

My apologies either way.

I define G2 as an interval between two immediate points with no point between them (what I call an abrupt change in OP). I am interested in understanding whether there are gaps of type G2 in the set of real number given the definition of the set of real number which can be found here. — MoK

There are provably no such G2 gaps in the real numbers. Between any two distinct real numbers, there is always another one strictly between them.

The real numbers are dense. That means that between any two distinct real numbers, there is a third one between them. For example between 5 and 7 we find 6. Between .001 and .002 we find .0015. And so forth.

In fact there's a formula to find the exact midpoint between two distinct reals. If $x$ and $y$ are distinct reals, then $\frac{x+y}{2}$ is halfway between them.

Also note that even the hyperreal numbers of the nonstandard reals, which have actual infinitesimals, are also dense. Between any two hyperreals there's another one distinct from those two.

There are no "adjacent points" in the real numbers. When you think of the real numbers, don't think of a string of bowling balls. Think of infinitely stretchy maple syrup or taffy. You can stretch and stretch but there are always more points.

infinitesimal_1 = 0.0...01 — MoK

This notation is meaningless. In decimal notation, each digit position to the right of the decimal point corresponds to a negative natural number power of 10. That is, starting from the decimal point and going right, the position values are 1/10, 1/100, 1/1000, and so forth.

There is not an "infinitieth" position. There is just no such thing in the notation. And that's not how the hyperreals work.

To emphasize this point, note that in even in the hyperreals, you can always divide by 2. So if .000...1 was a hyperreal infinitesimal, what is .000...1/2? You have no notation for that. In fact, .000...1 is meaningless in the reals and meaningless in the hyperreals.

And what if we divide by 10? You'd need an extra 0 in there somewhere ... but you can't add another 0 if you already have Aleph-0 of them. Make sense?

Forget .000...1. No such notation. Meaningless in every context, real or hyperreal.

Preferably, no sex (subject to self-discipline) — Tarskian

Self-discipline ... I don't even wanna ask.

ps -- This just came across the wire. The head of Telegram was just arrested in France, for a "lack of moderation" on the platform. Europe is cracking down on free speech. I think that's very bad. You are not so unhappy. As a philosopher how can you support crackdowns on free speech? Didn't Socrates run afoul of the Starmer types? "Corrupting the youth." Exactly the kind of vague charge the Eurocrats are using to suppress free expression.

Pardon the scurrilous right wing site link, it's factual info that I'm sure is replicated elsewhere.

https://www.breitbart.com/tech/2024/08/24/justice-europe-style-telegram-ceo-pavel-durov-arrested-due-to-lack-of-moderation-on-platform/

ps -- I'll stipulate that he's charged with all kinds of awful things. The French must think they have a case. Interesting to keep an eye on this one. Another article:

https://nypost.com/2024/08/24/world-news/telegram-founder-pavel-durov-arrested-at-paris-airport-report/

"Law enforcement believe that Telegram’s lack of moderation and the tools it offers, such as cryptocurrencies, make it complicit in global drug trafficking, pedophilia and fraud. "

Yeah the prosecutor's press conferences always make people seem awful. We'll see what they can actually bring to trial. I don't know the guy, not defending anything he may or may not have actually done. If he's enabling illegal activities, that's different than if he's only enabling free speech. We'll have to wait and see.

Evidently he screwed up by landing in France.

... and a central pillar of his platform being that he's anti-war. — Baden

I'm old enough to remember when liberals considered that a virtue.

And there weren't any new wars while Trump was president. The world didn't blow up till Iran and Russia saw Biden's weakness. Some of us out here credit Trump and blame Biden for that. But don't worry about the defense contractors. Kam shouted out her strong support for the continuation of the wars. Yippee.

Politically it will probably work out for her — Mr Bee

Oh yes I quite agree. Price controls are popular. It's not till a ways down the road that the shortages and lines (queues for my British cousins) begin.

An old article has been making the rounds. (pdf link)

Forty Centuries of Wage and Price Controls

Price controls have been failing for 4000 years. But yes you are absolutely right. They are very popular. Nixon's wage and price controls were popular till they failed.

sure buddy. Keep telling yourself that. I suppose a high level of delusion is necessary to be a Trump supporter. — Benkei

You know as I remember it -- I didn't go back to check -- you and I were going back and forth about whether it was a "coup." And you were claiming Biden stepped down willingly. At some point I didn't feel like arguing about it any more. Especially since neither of us are privy to what actually happened. Nancy and Barack and George Clooney didn't share their innermost thoughts with me; and I assume not with you either. So we're both guessing. The evidence support the proposition that Biden was forced out is pretty strong. The other day Pelosi said, "I did what I had to do." Another data point for my opinion.

I don't think you have much in the way of a supportable point based on the widely-reported pressure that was brought to bear on Biden. And I don't think it's that important a hill to die on. So I withdrew from the conversation. This seems to make you unhappy. I regret that.

Keep telling yourself that. I suppose a high level of delusion is necessary to be a Trump supporter. — Benkei

No doubt. But I came by it honestly, as a lifelong liberal Democrat and currently a disillusioned one. One of the seven to ten million Americans who voted for Obama and then Trump. You don't want to engage with us and that's sad.

The Biden-Harris admin was a disaster and now Kamala is actually running against her own administration. A Martian watching the Dem convention would have had no idea that Kamala has been running the country (in the stead of the non compos demented Biden the past three and a half years. She actually said she's going to fix the border. She's been in charge of the border all this time. I just read that 70% of the voters don't know any of her past policy positions. She may yet get away with it.

Are you saying that US law should apply in the UK? How is that not imperialism? — Ludwig V

The emerging globalist government is cracking down on free speech. You and I are not on the same side of this issue. Perhaps we can agree to disagree. I'll go with the First amendment to the US Constitution. I'm burnt out on this topic, my apologies.

They don't give details (no doubt for fear of being accused of spreading the words more widely), so I can't sort out what's going on. Anti-Muslim is a problem. Anti-establishment is not. It's interesting that the headlines all mention "anti-establishment" and don't mention "anti-muslim". That does puzzle me. — Ludwig V

Your government is way over the line these days. But like I say, I have my hands full fighting off the censors in the US. Hoping for the best for our British cousins. I hear Starmer is letting hardened criminals out to make room for the posters of mean tweets.

Perhaps I am. My parents fought WW2. So I think I have a real understanding of what full fascism is. Believe me, this isn't it. — Ludwig V

Well authoritarianism doesn't always look like jackboots.

Perhaps we just have different ideas about free speech. You have yours. I have mine. Why is that a problem? I don't think anyone thinks there should be no restrictions at all. Even the US has libel laws, doesn't it? — Ludwig V

Americans have extremely wide latitude for free expression. For the moment, anyway.

Sadly, from my point of view, US citizens have been conditioned to hate and fear sensible controls to minimize the harm that some people will inflict on them by exploiting their freedoms - not only in free speech, but also in the matter of gun control. There may be detriments to control, but there are detriments to unlimited freedom. It's a choice. Nothing is pure benefit. — Ludwig V

Well your side is going to soon crush my side. I have no doubt that bad days are ahead. You might call them good days. No unapproved thoughts.

Oh yeah there was that woman arrested for silently praying. That case got dismissed. But still ... arrested for what is in your thoughts?

https://www.bbc.com/news/articles/c4gze361j7xo

I think we should drop this. You know the kind of scurrilous literature I read. Since we talked last I've got 20 articles about the repression of speech in England. I won't bore you with them.

And have they been conditioned as well? Or just making a different choice from you? — Ludwig V

Actively trying to destroy free speech. I say that's bad.

And have they been conditioned as well? Or just making a different choice from you? — Ludwig V

Bad choices.

Jonathan Turley just wrote a book about all this.

https://www.amazon.com/Indispensable-Right-Free-Speech-Rage/dp/1668047047

Perhaps. So long as you are aware. The problem is that many people aren't as concerned about immigration as you are. So, to enforce immigration restrictions, you would need a police state. Indeed, I rather think that you would not be happy about that. — Ludwig V

Not so. Trump's Remain in Mexico policy was keeping a lid on the problem. You don't need a police state to simply defend your own border and enforce the laws already on the books.

I believe it was Milton Friedman who said you can't have both open borders and a welfare state. That's the mistake the US government is making.

By the way, why are you so keen on freedom of speech and so much against freedom of movement? — Ludwig V

I can live with open borders as long as nobody gets government services. But that's not workable, because people get sick and need health care. Kids need education. It's a thorny problem.

But your question is analogous to asking, "Since you're against bank robbery, why are you against bank withdrawals." I'm fine with legal immigration.

I suppose that works. But they are actually very boring people. — Ludwig V

I don't spend much time following the Royals, but they're in the news and hard to miss. Meghan and Harry and all that. England's gift to the US.

It's true. Kam has managed to revive the Democrats, and now it's more of an actual race. I did wonder, in all the fuss about Biden, whether the issue might come back to bite Trump. — Ludwig V

So far Kam still hasn't announced any actual policy stances, nor sat for an interview or press conference. She might get away with it. Trump looks tired and out of it these days.

There's not that many of them. There will be fewer in the third generation. — Ludwig V

One can only hope.

Ever since that business started off, I've been astonished how Israel has mismanaged the propaganda war. They started off with the moral high ground and have surrendered it almost completely. — Ludwig V

Agree. Even their friends are upset with them now. It's a tough situation. And very volatile if Iran and Israel go to war.

Sometimes I agree with the mainstream (that's less pejorative than "establishment"), but not always. No, you would not be subject to arrest in this country on the basis of anything you have said to me. — Ludwig V

I hope your buddy Starmer is as open-minded :-)

Maybe when she sits for an actual interview — Mr Bee

Haha. Good one Mr. Bee. When do you think that will happen? It's her official campaign strategy to never say an unscripted word. We all know what happens when Kam goes off script.

Price controls. Of all the hare-brained schemes. Even WaPo and CNN are against the idea. Price controls inevitably create shortages and bread lines. Nixon's price controls failed. Price controls always fail. They constrain supply and increase demand. Exactly the opposite of what you want.

Well 100,000 Antifa goons and Hamas-loving maniacs are planning to exercise their free speech rights in Chicago. The store owners are boarding up the windows just in case. "The whole world's watching" as they chanted in '68.

The one is the extremist of the rich and evangelicals, the other the extremist of the poor and identity politics. — Eros1982

The Democrats are now the party of the rich. The GOP are now the party of the working class. Exact opposite of how it used to be.

and she's still wrong for the reasons and arguments I gave you that you never engaged. And you're doing the same spiel again, by offering someone else's opinion devoid of the context of my arguments and pretend it's some kind of rebuttal. Learn how a discussion works! — Benkei

I did not engage with your arguments for the same reason I don't engage with flat earthers.

It's perfectly well known that Biden was pressured and shoved out. It's not rationally possible to argue the contrary. I get that you're sincere, but so are the flat earthers. Since we talked a week ago, stories have come out about Biden's seething resentment at the way he was treated. Here's one such.

Biden admits he was pushed out of presidential race, name-drops Pelosi in first interview since exit


It's simply not possible to look at the facts -- Biden was in it all the way on Saturday, then they announce he's got covid, then on Sunday a letter comes out under a forged mechanical signature without any of the standard official notices of withdrawal as required by the FEC.

You just can't spin this any other way than hardball political pressure from the Dem insiders. I absolutely do not understand how you can even pretend otherwise. To be honest I don't recall you making any rational arguments.

I did think it was amusing that MoDo spent her Sunday column the day before the Dem convention to make the point that it was a coup.

But if you want to think Joe Biden woke up on Sunday morning and dictated and signed the letter of his own free will ... you are entitled to your opinion. It is just too silly and unsupportable an opinion to be worth much in the way of discussion.

You're right. I was confused. But it is quite simple. If you break British law in Britain and go home, Britain can sue in US courts for extradition, take you to back Britain and try you. If you break US law in the US and go home, US can sue in British courts for extradition, take you to back to the US and try you. Seems fair enough to me. Most countries in the West have the same arrangement - by treaty, i.e. international law. — Ludwig V

A British official threatened to extradite Americans whose free speech offended him. There is no conceivable way you can spin this. It's disgraceful.

Info or Incitement? — Ludwig V

In Britain a guy was arrested for "anti-establishment rhetoric." If that doesn't bother you, I won't further argue the point.

https://www.allsides.com/news/2024-08-14-1315/general-news-bbc-court-hears-man-arrested-anti-establishment-rhetoric

There's an interesting question about people who are US citizens in the US posting something to Britain that is within US law but banned in Britain. There's a suggestion that they can be extradited, but I find it very hard to believe. — Ludwig V

It's hard to believe they could actually do it; but a British official did threaten it. The British government has gone full fascist. I'm sorry you can't see it. Maybe you're too close.

There's a new law in Britain that if you re-post an illegal post by someone else, you are also guilty of incitement. I agree that's pushing it a bit, but if someone is inciting violence and you join in the incitement, I think there's a case for it - if you can prove it. After all, if you help someone committing a theft, you are also breaking the law. No? — Ludwig V

Anti-establishment rhetoric. As an American accustomed to the robust protections of the First Amendment, I'm appalled. You don't seem very keen on free speech as I understand the term.

There's a big push in the UK and Europe to get the internet under control. You may not be aware of how much the big internet companies are resented over here. They have a very poor reputation. One has to give them credit for taking the issues seriously, but they don't take effective action. They plead free speech, but no-one believes that. It's about the bottom line and that's not acceptable. — Ludwig V

Free speech is under attack everywhere. That's why it's so vitally important to defend it, and to push back on these awful statements and policies of the Starmer regime. I'm sure Europeans have been conditioned to hate and fear free speech, free expression, and free thinking. That's to their own ultimate detriment. Lot of people in the States want the government control the Internet too.

I'm not sure who you trust on this. But Reuters have a pretty good reputation.
Reuters on deaths on US-Mexico border
Certainly, people die in the Channel regularly. BBC on migrant deaths in the Channel
I don't know how many, if any, are illiterate. Why does it matter? — Ludwig V

You're making an obscure and convoluted point. I'm fully aware of the dangers to illegal immigrants. But most just walk across (in the US) and are welcomed by an administration that refuses to enforce its own laws.

Yes, that's true. The UK does have protection for free speech. Just not as much as in the US. People resent they way the the US internet companies impose your law on us.
However, I really don't care at all what Prince Harry's views are; he has no special knowledge or authority that I'm aware of. I can't understand why people in the US get so excited about our royal family. They are an embarrassment in a supposedly democratic country. — Ludwig V

I have a theory about why the Americans love the British Royals. We get to enjoy all the pomp, the circumstance, and the salacious scandals. And we don't have to pay for it!

Starmer is at least less of a joke than the other lot. Rishi Sunak was better his immediate predecessors, but was undermined by his own party. I have the impression that Trump is still likely to win. — Ludwig V

Kam's got the media on her side and a newly energized Democratic party. Trump is old, seems confused and out of sorts lately, and IMO may be suffering a touch of age-related dementia himself. The election could go either way.

Hopefully, by that time, there will be more home-grown imams and fewer radicals imported from back home. — Ludwig V

The second-generation native born Muslims seem to manage to get themselves radicalized anyway.

There are already a good many of them (home-grown imams) - they just don't get the news coverage. Plus, generations born and brought up here are, on the whole, often atheists or moderates. I think they will settle down. If the other immigrant communities are anything to go by, there'll be a lot of inter-marriage with the general population, anyway. — Ludwig V

You are a glass half full guy! I am not so sanguine.

By the way, 100,000 Hamas-loving maniacs are going to riot at the Democratic convention in Chicago this week. Should be something for the world to see.


Sorry, I wasn't clear. No-one is suppressing my views. Fortunately, I'm pretty much mainstream. I've tried to clarify what I was trying to say and failed, so I'll have to let it go.[/quote]

Of course. You have the establishment view. I often take the anti-establishment view. In your country I'd be subject to arrest.

So are many other Western countries, including Britain, not to mention Japan and Korea. There's a lot of argument about the reasons. Most plausible explanation is that that a modern capitalist economy makes it too hard to bring up children. Either you live in poverty with children or you work to make the money for a decent life without children. Not to mention the gloomy outlook for the West. That also is one of the reasons why Britain actually needs immigrants and allows many in, legally.

The USA is not doing well but is not in collapse - yet. — Ludwig V

"There's a lot of ruin in a country."

You have a disinterest because you were wrong and are unwilling to admit it. That's called not being able to have a conversation. — Benkei

Maureen Dowd, the queen of the liberal chattering class, sets you straight on this point.

Her opinion column in tomorrow's Sunday New York Times on the eve of the Democratic convention:

The Dems Are Delighted. But a Coup Is Still a Coup

Top Democrats are bristling with resentments even as they are about to try to put on a united front at the United Center in the Windy City.

A coterie of powerful Democrats maneuvered behind the scenes to push an incumbent president out of the race.

It wasn’t exactly “Julius Caesar” in Rehoboth Beach. But it was a tectonic shift and, of course, there were going to be serious reverberations. Even though it was the right thing to do, because Joe Biden was not going to be able to campaign, much less serve as president for another four years, in a fully vital way, it was a jaw-dropping putsch. — MoDo

Not every day that Maureen Dowd makes my point for me.

Islam is a missionary religion. It seeks to become the universal religion. The idea of the theocratic Caliphate is an aim that some fundamentalists are committed to. That's true. It's just that I don't think they will succeed. Sadly, they can do a lot of damage while they are trying. — Ludwig V

Well if Islam seeks to become a universal religion, what happens to your nation when there are enough of them to make a political difference? It's no hypothetical.

Christianity has the same ambitions. They are not terrorists, of course. Nonetheless, while I respect their right to campaign for their views, I object strongly to their desire to impose their views on me and suppress mine. — Ludwig V

Who is suppressing your views?

Oh, come on. I think that Islamic fundamentalism is not an existentialist threat to the West. That doesn't mean that terror bombings are ok with me — Ludwig V

We shall see.

I agree with you that they are complicated. The desire to suppress IS and similar groups is perfectly reasonable. But the means employed against Uighurs are grossly disproportionate. — Ludwig V

I agree.

You're missing the problem. People who are willing to die to get in to UK or US are very hard to stop. Public opinion won't support extreme measures (which would probably not work anyway) — Ludwig V

Nobody's "willing to die," they're just walking across an open border in the US. When Trump had his Remain in Mexico policy, the problem was greatly reduced. These aren't armed hordes "willing to die," what are you talking about? These are illiterate peasants walking across an open border that can be closed if the leadership wills it.

Strictly speaking, they are not terrorists. But both of them operate in secret in the UK and elsewhere. — Ludwig V

Ok. China has its own problems though. I hear they're in demographic collapse.

Sorry, I think you are a bit confused. He can arrest and deport (i.e. send back home) US citizens who misbehave. — Ludwig V

He explicitly threatened non-Brits in their home countries. I am not confused about this, it has been extremely widely reported.

"London’s Metropolitan Police chief warned that officials will not only be cracking down on British citizens for commentary on the riots in the UK, but on American citizens as well.

“We will throw the full force of the law at people. And whether you’re in this country committing crimes on the streets or committing crimes from further afield online, we will come after you,” Metropolitan Police Commissioner Sir Mark Rowley told Sky News."

https://nypost.com/2024/08/10/media/uk-police-commissioner-threatens-to-extradite-jail-us-citizens-over-online-posts-well-come-after-you/

The UK also has free speech, but bans incitement to riot. — Ludwig V

Some 60 year old was just jailed for a Facebook post. I can't argue with you about British politics, you being there and all, but we seem to have very different information. You can be jailed for just reposting info about riots, not inciting them.

That seems perfectly reasonable to me. They are lucky that he doesn't apply UK law and throw them in jail.
I haven't seen anything about violent Muslim rioting recently - not in the UK, anyway. Obviously, if no Muslims are rioting, he can't throw them in jail. — Ludwig V

Well I've said my piece on this.

Well, you know best about what's going in the USA. In the UK, the Government has been trying to prevent immigration across the Channel for decades. You would think it was easy enough. But they've failed. — Ludwig V

By incompetence or design? Either way, the people seem unhappy about it.

People are who prepared to die to get here are very difficult to stop. — Ludwig V

Who is prepared to die? Impoverished peasants streaming across the US southern border?

Who employs the cheap labour? When those people are not prepared to employ them, the incentive will disappear. That's what I meant about lack of public support. People are happy to make a fuss, but not willing to pay a bit more for labour. You can't have it both ways. — Ludwig V

Agree on that. Lots of people benefit from broken immigration systems on both sides of the pond.

You're begging the question. The courts think that those people are rioting, and that's not free speech, it's violence. As for people's true feelings, you seem to trust the Telegraph.
Daily Telegraph Southport Counter-demonstrations — Ludwig V

The British courts don't have the US First Amendment, which provides legal protection for the most appalling expressions of ideas. I read that Prince Harry has called the First Amendment "bonkers." The US has very strong protections for speech not found in most other democratic nations.

Yes, but that was just one aspect of their failure to deliver any public services at all. Health, Education, Justice, Defence, not to mention the housing crisis - the list is endless. Obsessed by in-fighting and tax reduction, failed to do their job. — Ludwig V

And now you've got Starmer. Good luck! I should talk, right? We're about to have Queen Kamala.

I agree it is supported by some of the facts. But surely the police are not supposed to throw people under buses - arrest and fair trial? — Ludwig V

During the summer of Floyd that's exactly what they did. There was another case of the three Georgia guys thrown in prison for decades for the accidental death of a known burglar.