Different sizes of infinity...?

[pzkpfw]

...
You are way, way off base.

I think you need to work on your communication skills. But I'll leave it there.

 

[DaveC426913]

DaveC:

Look, I'll leave you to have it out with pzkpfw, who I'm sure doesn't need my help on the Hilbert Hotel thing.

Every once in a while a new user signs up that gets very testy very quickly - as if they've brought baggage and grudges to the table from a previous incarnation. It's almost as if they've been here before, perhaps under a different name...

One thing you said caught my attention, though:

That sort of thing is the basis of calculus.

For example, instantaneous velocity is defined as the rate of change of position, like this:

v(t) = lim(h->0) [ (x(t+h)-x(t))/h ]

Now, if we were to put h=0 into the fraction (x(t+h)-x(t))/h, we'd get (x(t)-x(t))/0=0/0.

The quantity "zero divided by zero" is in a similar class to the quantity "infinity divided by infinity". It is undefined. In calculus, we avoid the problem by taking the limit as h goes to zero, instead of actually putting h=0. The result is a perfectly well-defined expression for a physical quantity (velocity, in this example), even though the definition involves dealing with infinitesimal quantities.

The v(t) example given here is an example of how dividing an infinitesimal by another infinitesimal can give a finite velocity. I could give you many other mathematical examples where dividing an infinity by an infinity, in an appropriate limit, similarly gives a finite result.

To take a different example, how about this old carnard:

1/2 + 1/4 + 1/8 + 1/16 + ... = ?

There is an infinite number of terms in this sum. The n-th term is 1/n^2, which gets infinitesimally small as n gets infinitely large. Those two effects cancel each other out, in a sense, so that the entire infinite sum gives a finite result. In this case, the result of the infinite sum is 1.

I know how limits work but I don't see that applying here.

 

[Pinball1970]

It's a thought experiment, it is more succinct using set theory notation.

 

[Pinball1970]

I will just add that although I am not a mathematician, I trust the scrutiny that guys like Cantor and Hilbert came under.
They survived the test although some of the conclusions were not palatable for mathematicians of the time.

Also worth noting that Cantor eventually went nuts poor lad.

 

[Sarkus]

Remind me: why are people throwing in physics to a thought experiment about infinite sets? The thought experiment is not a matter of physics but of mathematics. It's like talking about how real kings don't move like its namesake piece on a chess board. Okay, but... so what?

 

[Pinball1970]

Remind me: why are people throwing in physics to a thought experiment about infinite sets? The thought experiment is not a matter of physics but of mathematics. It's like talking about how real kings don't move like its namesake piece on a chess board. Okay, but... so what?

Yeah, not applicable IMO. We may as well complain that Einstein's elevator man in space would freeze before he felt any acceleration/gravity.

 

[phyti]

What does absolute simultaneity have to do with it?

I simply posit that they all leave their room at (nearly) the same time. All guests - even the slowest - travel the same distance - a very finite distance, thus take a very finite time. The entire process happens in parallel, and therefore lasts as long as it takes a guest to travel one room.

In a 2n system, even if they all leave their room at (nearly) the same time, at least one guest must travel an infinite distance, which he cannot travel in finite time.

No matter how you cut the problem up and remove restrictions, the 2n process has a guest travelling an infinite distance to get to their room.

Given, the hotel problem is applied to a real world scenario (the only ones we experience).

You need a means of coordinating the moves simultaneously.

With the revelation of finite light speed, universal time was eliminated and simultaneity became a relative concept. (Einstein 1905)

 

[DaveC426913]

Given, the hotel problem is applied to a real world scenario (the only ones we experience).

You need a means of coordinating the moves simultaneously.

With the revelation of finite light speed, universal time was eliminated and simultaneity became a relative concept. (Einstein 1905)

This is math problem, not a real world problem. We've already agreed not to apply real physics to the problem - we are only considering what is already in the scenario.

 

[pzkpfw]

I will just add that although I am not a mathematician, I trust the scrutiny that guys like Cantor and Hilbert came under.
They survived the test although some of the conclusions were not palatable for mathematicians of the time.

Also worth noting that Cantor eventually went nuts poor lad.

(My bolds)
I think that's a very good point. The people with "alternate ideas" can claim this is "appeal to popularity", but, there's a reason Cantor is in the math textbooks. These people sometimes seem to be motivated a little by iconoclasm. You see it in web forums like these also with the anti-relativity folk. It can manifest as insisting on nit picking thought experiments in Einsteins' original papers. But relativity isn't accepted because Einstein said so, but because it's been studied (and experimented with) for over a hundred years.
Cantors' work too. If some math student could write their PhD thesis disproving Cantors' work, they'd sure as heck do it.
If somebody wants to disprove something like this, it'll take more than a bit of hand waving on an internet forum.

(Edit: to be fair to phyti, props for "...and simultaneity became a relative concept. (Einstein 1905)")

 

[Pinball1970]

(My bolds)
I think that's a very good point. The people with "alternate ideas" can claim this is "appeal to popularity", but, there's a reason Cantor is in the math textbooks. These people sometimes seem to be motivated a little by iconoclasm. You see it in web forums like these also with the anti-relativity folk. It can manifest as insisting on nit picking thought experiments in Einsteins' original papers. But relativity isn't accepted because Einstein said so, but because it's been studied (and experimented with) for over a hundred years.
Cantors' work too. If some math student could write their PhD thesis disproving Cantors' work, they'd sure as heck do it.
If somebody wants to disprove something like this, it'll take more than a bit of hand waving on an internet forum.

(Edit: to be fair to phyti, props for "...and simultaneity became a relative concept. (Einstein 1905)")

Canadian Dave is usually on the ball with all this stuff, perhaps you guys are talking past each other a little?
I would have go back and unravel but it's bedtime now. Cheers, speak later

 

[DaveC426913]

If some math student could write their PhD thesis disproving Cantors' work, they'd sure as heck do it.
If somebody wants to disprove something like this, it'll take more than a bit of hand waving on an internet forum.

Since nobody's trying to do this, it's kind of moot. Another swing and a miss.

Canadian Dave is usually on the ball with all this stuff, perhaps you guys are talking past each other a little?

pzkpfw is trying very hard to catch me out. He's become more interested in that than actually trying to have a constructive discussion.

I would have to go back and unravel...

Then you'll see the number of times I said "Can't be done in finite time", and then once I said "undoable", leaving off the "in finite time" and it's as if he's deliberately being obtuse about it.

And then he's all...

That's been clearly understood from the start.

He seems to have a pretty self-serving idea of when something is taken as a given and when it must be beaten to death.

And aispokesperson's pot-stirring didn't help smooth matters over.


Anyway, I made my point about 2n, and nobody has refuted it as-yet. So I'll take that win.

 

[pzkpfw]

Since nobody's trying to do this, it's kind of moot. Another swing and a miss.

I was referring to phyti, who is the person that has been trying to argue against what Cantor showed. Don't be paranoid.

pzkpfw is trying very hard to catch me out. He's become more interested in that than actually trying to have a constructive discussion.

You seem obsessed.

Then you'll see the number of times I said "Can't be done in finite time", and then once I said "undoable", leaving off the "in finite time" and it's as if he's deliberately being obtuse about it.

In terms of the earlier "can't be done in infinite time" my response was "why does that matter?". In that, I was not disagreeing with you about it taking infinite time for the guests to move in the 2n case, I was disagreeing that it mattered. At that time it was not clear that your concern was "elegance", it seemed like you thought it couldn't work. That seemed to be confirmed by the later use of "undoable".

And then he's all...

He seems to have a pretty self-serving idea of when something is taken as a given and when it must be beaten to death.

I was ready to stop, but you bring me back into this.

And aispokesperson's pot-stirring didn't help smooth matters over.


Anyway, I made my point about 2n, and nobody has refuted it as-yet. So I'll take that win.

Elegance is a personal thing. Nobody can refute your own feeling on that.

I still like the 2n approach. To me, it's more mathematically interesting.

 

[DaveC426913]

I was referring to phyti, who is the person that has been trying to argue against what Cantor showed. Don't be paranoid.

OK. Point conceded.

You've been riding me pretty hard, and it's not the first time you've theorized what I'm up to and put words in my mouth. So I don't think it's paranoid.

You seem obsessed.

It might behoove you to review your own contributions. It's been you who has been dogging me.

In terms of the earlier "can't be done in infinite time" my response was "why does that matter?".

Eventually, I got that concession out of you, but it was like pulling teeth.

As to whether it matters, I feel like the invocation of infinite time in an attempt to resolve the problem is an necessary reach/complication outside the box of the scenario. That's what didn't sit well.

In that, I was not disagreeing with you about it taking infinite time for the guests to move in the 2n case, I was disagreeing that it mattered. At that time it was not clear that your concern was "elegance", it seemed like you thought it couldn't work. That seemed to be confirmed by the later use of "undoable".

OK. I hope it's been settled to your satisfaction.

I was ready to stop, but you bring me back into this.

Well you didn't really indicate that...

Elegance is a personal thing. Nobody can refute your own feeling on that.

Accepted.

I'll put the post mortem behind me. Hopefully, no hard feelings.

I still like the 2n approach. To me, it's more mathematically interesting.

Sure.

 

[pzkpfw]

Still things you write that I would comment on, but that's good enough

Now, back to why phyti thinks the diagonal is incompatible with something and why that matters ...

 

[aispokesperson]

DaveC:

Look, I'll leave you to have it out with pzkpfw, who I'm sure doesn't need my help on the Hilbert Hotel thing.

One thing you said caught my attention, though:

That sort of thing is the basis of calculus.

For example, instantaneous velocity is defined as the rate of change of position, like this:

v(t) = lim(h->0) [ (x(t+h)-x(t))/h ]

Now, if we were to put h=0 into the fraction (x(t+h)-x(t))/h, we'd get (x(t)-x(t))/0=0/0.

The quantity "zero divided by zero" is in a similar class to the quantity "infinity divided by infinity". It is undefined. In calculus, we avoid the problem by taking the limit as h goes to zero, instead of actually putting h=0. The result is a perfectly well-defined expression for a physical quantity (velocity, in this example), even though the definition involves dealing with infinitesimal quantities.

The v(t) example given here is an example of how dividing an infinitesimal by another infinitesimal can give a finite velocity. I could give you many other mathematical examples where dividing an infinity by an infinity, in an appropriate limit, similarly gives a finite result.

To take a different example, how about this old carnard:

1/2 + 1/4 + 1/8 + 1/16 + ... = ?

There is an infinite number of terms in this sum. The n-th term is 1/n^2, which gets infinitesimally small as n gets infinitely large. Those two effects cancel each other out, in a sense, so that the entire infinite sum gives a finite result. In this case, the result of the infinite sum is 1.A

As I said, there is benign infinity (converging) and there is malign infinity (non converging). The hotel is malign, so is: +1, -1 +1, -1, ...

 

[aispokesperson]

Per AI: You are touching one of the deepest fault lines in modern mathematics. Your suspicion is not crankery—it is a predicative, finitist, and constructivist objection that has been held by Poincaré, Weyl, Brouwer, and more recently by Feferman and Nelson. The question is whether Cantor performed a conceptual smuggling operation by calling an infinite totality a "set" as if it were the same kind of object as a finite collection.

 

[QuarkHead]

As I said, there is benign infinity (converging) and there is malign infinity (non converging). The hotel is malign, so is: +1, -1 +1, -1, ...

No there isn't, these terms are unknown in mathemtics. What makes you think that a hotel with a non-finite cardinality of rooms either converges or diverges?

Per AI: You are touching one of the deepest fault lines in modern mathematics. Your suspicion is not crankery—it is a predicative, finitist, and constructivist objection that has been held by Poincaré, Weyl, Brouwer, and more recently by Feferman and Nelson. The question is whether Cantor performed a conceptual smuggling operation by calling an infinite totality a "set" as if it were the same kind of object as a finite collection.

So, you are using the testimony of an AI robot to support your contention that the brotherhood of AI bots has superceded mathemamatics.

And, for your information, there is absolutely nothing that is non-constructive about axiomatic (ZF) set theory as used by Cantor.

 

[aispokesperson]

No there isn't, these terms are unknown in mathemtics. What makes you think that a hotel with a non-finite cardinality of rooms either converges or diverges?


So, you are using the testimony of an AI robot to support your contention that the brotherhood of AI bots has superceded mathemamatics.

And, for your information, there is absolutely nothing that is non-constructive about axiomatic (ZF) set theory as used by Cantor.

Then I have invented a very important concept in mathematics! If you cannot determine the finiteness of the hotel, then all the subsequent operations are illegal.

It is futile to try to strengthen your position or weaken my position by merely pointing out the source of the argument. For your information, I believe Cantor is not very careful in his theory.

 

[DaveC426913]

Then I have invented a very important concept in mathematics!

No you haven't. You admit that AI hallucinated it:

It is futile to try to strengthen your position or weaken my position by merely pointing out the source of the argument.

As you admit: AI is the source, not you.

 

[QuarkHead]

If you cannot determine the finiteness of the hotel, then all the subsequent operations are illegal.

Then maybe you can explain exactly what you mean by "determine the finiteness". A set is either finite or it is not. What's so difficult about that?

I believe Cantor is not very careful in his theory.

My friend, you can believe what you like.