To infinity and beyond.

[Walt Zingmatilder]

Just to point out Vlad's edited his post that you are replying to here, after your reply, to provide what he sees as a contradiction.

Thanks

Let me post it again
However, some “absurdities” arise when groups start leaving. If a finite group of 5 checks out, the hotel still has an infinite number of rooms filled. But consider what happens when two different infinite groups leave the hotel. Having the infinite group in all rooms greater than 5 check out leaves only 5 rooms filled. Alternatively, when the infinite group of all even numbers checks out, the hotel has an infinite number of odd rooms filled. In equation form (paralleling the addition equations above), this gives:

a – b = c   Just like addition above
∞ – a = ∞   OK so far
∞ – ∞ = 5   First infinite group leaving
∞ – ∞ = ∞   Second infinite group leaving
The last two equations are contradictory! [/quote]So infinities produce contradictions. Can this be exemplified in physics?

You can’t subtract one value from another value and get two different results. Thus, many people have used this contradiction to argue that actual infinities cannot exist in the physical world. However, mathematicians recognized this dilemma and solved the issue by noting that subtraction is not a well-defined operation for infinites. Lest this strike you as defining the problem away

It does.

[Gordon]

Moderator

Vlad

Perhaps you could make clear, since the OP is yours, where the relevance is to Theism of Atheism in your OP and in the subsequent discussion, since the thread is currently on the Theism and Atheism Board.

I'd have to say, the content seems more appropriate to Philosophy.

Please advise.

[Walt Zingmatilder]

Moderator

Vlad

Perhaps you could make clear, since the OP is yours, where the relevance is to Theism of Atheism in your OP and in the subsequent discussion, since the thread is currently on the Theism and Atheism Board.

I'd have to say, the content seems more appropriate to Philosophy.

Please advise.

I’m happy wherever it is thanks.

[Stranger]

From your edited post (as NS noted):

From the link
However, some “absurdities” arise when groups start leaving. If a finite group of 5 checks out, the hotel still has an infinite number of rooms filled. But consider what happens when two different infinite groups leave the hotel. Having the infinite group in all rooms greater than 5 check out leaves only 5 rooms filled. Alternatively, when the infinite group of all even numbers checks out, the hotel has an infinite number of odd rooms filled. In equation form (paralleling the addition equations above), this gives:

a – b = c   Just like addition above
∞ – a = ∞   OK so far
∞ – ∞ = 5   First infinite group leaving
∞ – ∞ = ∞   Second infinite group leaving
The last two equations are contradictory! You can’t subtract one value from another value and get two different results. Thus, many people have used this contradiction to argue that actual infinities cannot exist in the physical world. However, mathematicians recognized this dilemma and solved the issue by noting that subtraction is not a well-defined operation for infinites. Lest this strike you as defining the problem away, you encounter a similar solution for a much more familiar mathematical idea. Let me illustrate with an interesting proof:”. How does noting that subtraction is not well defined for infinities help out in the argument that actual infinities could exist?

Actually, there are two answers to this. The first is given in the article that explains that zero is undefined under the operation of division because it leads to similar contradictions. This doesn't stop real zeros from appearing in nature. So the whole contradiction, using mathematically invalid operations, means physically impossible 'argument' goes straight out of the window.

This alone should answer your claim.

However, we could get more technical and point out that we can make it work for the Hilbert Hotel example if we use explicit sets. Infinite cardinal numbers are defined by the cardinality (loosely 'size') of sets.

The Hilbert Hotel has rooms numbered 1, 2, 3, 4,...., and the cardinality (loosely 'number') of rooms is the cardinality of the set of natural numbers: ℕ = {1,2,3,4,...}, denoted by |ℕ| = aleph-0 (ℵ₀). Using set difference as 'subtraction', in the first subtraction, we are 'subtracting' all the numbers greater than five, so

{1,2,3,4,...}\{6,7,8,...} = {1,2,3,4,5} and |{1,2,3,4,5}| = 5.

In the second, we are 'subtracting' all the even numbers, so

{1,2,3,4,...}\{2,4,6,...} = {1,3,5,...} and |{1,3,5,...}| = ℵ₀ = |{1,2,3,4,...}|.

I'll take questions.   

[jeremyp]

:-*I think the point is he admits to contradictions where there is more than one conclusion possible.

No. He does not.

The contradictions only arise if you have an incorrect understanding of the maths involved.

All you have today is provide a physical example of such a situation.

Why? As I said, the maths does not show a contradiction. The contradictions do not exist ands the article does a reasonable job of explaining why in layman's terms.

Similarly you are proposing subtraction as not well defined for infinities. You need to show this in the physical realm to justify showing that infinities are real.

No I don't.

[jeremyp]

a – b = c   Just like addition above
∞ – a = ∞   OK so far

This is meaningless. Subtraction is undefined for infinities.

∞ – ∞ = 5   First infinite group leaving
∞ – ∞ = ∞   Second infinite group leaving

These are meaningless. Subtraction is not defined for infinities.

Did you look at the other example using division by zero?

If a = b then x / (a - b) is meaningless. Division is not defined for zero

And yet, the concept of zero certainly exists.

[jeremyp]

I have provided where he establishes a contradiction or apparent contradiction.

He does not establish a contradiction. Subtraction is not defined for infinities.

[jeremyp]

However, we could get more technical and point out that we can make it work for the Hilbert Hotel example if we use explicit sets. Infinite cardinal numbers are defined by the cardinality (loosely 'size') of sets.

The Hilbert Hotel has rooms numbered 1, 2, 3, 4,...., and the cardinality (loosely 'number') of rooms is the cardinality of the set of natural numbers: ℕ = {1,2,3,4,...}, denoted by |ℕ| = aleph-0 (ℵ₀). Using set difference as 'subtraction', in the first subtraction, we are 'subtracting' all the numbers greater than five, so

{1,2,3,4,...}\{6,7,8,...} = {1,2,3,4,5} and |{1,2,3,4,5}| = 5.

In the second, we are 'subtracting' all the even numbers, so

{1,2,3,4,...}\{2,4,6,...} = {1,3,5,...} and |{1,3,5,...}| = ℵ₀ = |{1,2,3,4,...}|.

I'll take questions.   

This is actually a much better answer than "subtraction is undefined for infinities". In fact it shows that Vlad's argument relies on obfuscating the fact that infinities are not numbers in the normal sense and that addition and subtraction are not the traditional operations we would like to believe.

[Walt Zingmatilder]

From your edited post (as NS noted):

Actually, there are two answers to this. The first is given in the article that explains that zero is undefined under the operation of division because it leads to similar contradictions. This doesn't stop real zeros from appearing in nature.

Given that the statement “There me be no such thing as nothing, what real zeros are you talking about?Or to put it another way...”show me a physical zero”

So the whole contradiction, using mathematically invalid operations, means physically impossible 'argument' goes straight out of the window.

Anybody?

This alone should answer your claim.

However, we could get more technical and point out that we can make it work for the Hilbert Hotel example if we use explicit sets. Infinite cardinal numbers are defined by the cardinality (loosely 'size') of sets.

The Hilbert Hotel has rooms numbered 1, 2, 3, 4,...., and the cardinality (loosely 'number') of rooms is the cardinality of the set of natural numbers: ℕ = {1,2,3,4,...}, denoted by |ℕ| = aleph-0 (ℵ₀). Using set difference as 'subtraction', in the first subtraction, we are 'subtracting' all the numbers greater than five, so

{1,2,3,4,...}\{6,7,8,...} = {1,2,3,4,5} and |{1,2,3,4,5}| = 5.

In the second, we are 'subtracting' all the even numbers, so

{1,2,3,4,...}\{2,4,6,...} = {1,3,5,...} and |{1,3,5,...}| = ℵ₀ = |{1,2,3,4,...}|.
h
I'll take questions.   

I am not denying it works in maths. Can you show me it in concreto or in nature?

[Walt Zingmatilder]

No. He does not.

The contradictions only arise if you have an incorrect understanding of the maths involved.

Courtiers reply? But the problem isn’t with it being mathematically coherent or logically coherent but you demonstrating it physically, continual preening over solving the maths does not equate with showing these things in nature. Where does something having something taken away but leavingq the same amount being left occur in nature?

Why? As I said, the maths does not show a contradiction. The contradictions do not exist ands the article does a reasonable job of explaining why in layman's terms.

The author calls them contradictions exclamation mark though because there are two answers both different. Where does he remove the difference immediately launching as he does into the properties of Zero?

[Stranger]

Given that the statement “There me be no such thing as nothing, what real zeros are you talking about?

A box full of sand, and nothing else, contains exactly zero zebras.

I am not denying it works in maths. Can you show me it in concreto or in nature?

Your argument was that physical infinities were impossible because of contradictions. There are no such contradictions, and the one you gave as an example (from mathematics) actually doesn't contain any contradictions, when we examine it more closely,

If you're now going to run away and simply say that there can't be actual infinities because we cannot directly observe them, that's a whole new realm of stupid.

Are you ready to deny quantum mechanics because the quantum state requires i = √(-1) to describe it, and its development in time, and you can't actually observe either the quantum state or i?

[Walt Zingmatilder]

A box full of sand, and nothing else, contains exactly zero zebras.

Show me a zero zebra or infinite zebras for that matter.

Your argument was that physical infinities were impossible because of contradictions. There are no such contradictions, and the one you gave as an example (from mathematics) actually doesn't contain any contradictions, when we examine it more closely,

If you're now going to run away and simply say that there can't be actual infinities because we cannot directly observe them, that's a whole new realm of stupid.

Are you ready to deny quantum mechanics because the quantum state requires i = √(-1) to describe it, and its development in time, and you can't actually observe either the quantum state or i?

No I am pointing out that doing the working out he came up with two answers for the same operation involving infinities. Where is that demonstrated in nature. He seemed to be equating infinity with what is the largest amount of numbers with zero. That intuitively seems a bad comparison but the chief problem here is evidencing either.

[Walt Zingmatilder]

If you're now going to run away and simply say that there can't be actual infinities because we cannot directly observe them, that's a whole new realm of stupid.

The argument is rather there is no indirect evidence for them, therefore we can deduce they don’t exist and are not actual.

I think your evidential standard for actual infinity is lower for infinities than necessity.

[Stranger]

The argument is rather there is no indirect evidence for them, therefore we can deduce they don’t exist and are not actual.

Both untrue and logically absurd. Oh, and by the same 'argument' we can deduce that no God exists.

I think your evidential standard for actual infinity is lower for infinities than necessity.

You have yet to show that a 'necessary entity' even makes logical sense.

[Nearly Sane]

Given that the statement “There me be no such thing as nothing, what real zeros are you talking about?Or to put it another way...”show me a physical zero”  Anybody?I am not denying it works in maths. Can you show me it in concreto or in nature?

Just to point out that if you take that position, it means all the times you've asked why is there something rather than nothing, you've just said are incoherent questions 

[Stranger]

Missed this nugget of nonsense....

Show me a zero zebra...

How many zebras are there in the room with you right now?

No I am pointing out that doing the working out he came up with two answers for the same operation involving infinities.

Only by using incorrect 'working out'. If you use incorrect maths you'll get incorrect answers. If you use zero incorrectly, then you can prove that any number is equal to any other number.

What's more, I also showed that you got the same two answers by using the correct set theory approach to the problems stated, without any hint of contraction because the two situations are different when stated in those terms. Infinities are not really numbers in the same way as finite numbers are. It was trying to use them as such that produced the different answers to the 'same' questions, because the questions only looked the same if you simplistically used the infinities in the same way as ordinary numbers.

In fact, strictly speaking, all numbers are derived from set theory in modern mathematics, but we are so used to normal arithmetic, which works well for finite numbers, that it doesn't matter for most problems. Similarly, the concept of cardinality is not used for normal numbers because 'size' or 'quantity' work just fine for finite numbers. Also, with infinities, there are very significant differences between cardinal (1, 2, 3...) and ordinal (1st, 2nd, 3rd,...) numbers that don't arise with finite numbers. The smallest infinite ordinal number is ω and ω + 1 ≠ 1 + ω; addition is not a commutative operation.

In fact, there are many subjects in mathematics that are perfectly self-consistent, and free from contradiction, but counterintuitive. It's foolish to dismiss them as irrelevant to the real world because so much of what was thought of as 'pure mathematics' has found applications in the real world already. Things like, for example, complex numbers and group theory, are now firmly established as fundamental to physics. Other subjects like Lie algebras and quaternions (a hypercomplex number system) have also found real world applications.

Where is that demonstrated in nature. He seemed to be equating infinity with what is the largest amount of numbers with zero. That intuitively seems a bad comparison but the chief problem here is evidencing either.

The comparison was simply that some arithmetic operations are not applicable to them. There are plenty of other systems that some operations do not apply to. In fact, if you restrict yourself to natural number arithmetic, division (in the normal sense) doesn't apply to them, either, because rational numbers are outside the domain. Also, in the real number system, the square root operation cannot be applied to negative numbers.

Your recently acquired obsession with the need for direct evidence in nature, is in total contradiction with your theism, not to mention being logically silly.

[Walt Zingmatilder]

Missed this nugget of nonsense....

How many zebras are there in the room with you right now?

Only by using incorrect 'working out'. If you use incorrect maths you'll get incorrect answers. If you use zero incorrectly, then you can prove that any number is equal to any other number.

What's more, I also showed that you got the same two answers by using the correct set theory approach to the problems stated, without any hint of contraction because the two situations are different when stated in those terms. Infinities are not really numbers in the same way as finite numbers are. It was trying to use them as such that produced the different answers to the 'same' questions, because the questions only looked the same if you simplistically used the infinities in the same way as ordinary numbers.

In fact, strictly speaking, all numbers are derived from set theory in modern mathematics, but we are so used to normal arithmetic, which works well for finite numbers, that it doesn't matter for most problems. Similarly, the concept of cardinality is not used for normal numbers because 'size' or 'quantity' work just fine for finite numbers. Also, with infinities, there are very significant differences between cardinal (1, 2, 3...) and ordinal (1st, 2nd, 3rd,...) numbers that don't arise with finite numbers. The smallest infinite ordinal number is ω and ω + 1 ≠ 1 + ω; addition is not a commutative operation.

In fact, there are many subjects in mathematics that are perfectly self-consistent, and free from contradiction, but counterintuitive. It's foolish to dismiss them as irrelevant to the real world because so much of what was thought of as 'pure mathematics' has found applications in the real world already. Things like, for example, complex numbers and group theory, are now firmly established as fundamental to physics. Other subjects like Lie algebras and quaternions (a hypercomplex number system) have also found real world applications.

The comparison was simply that some arithmetic operations are not applicable to them. There are plenty of other systems that some operations do not apply to. In fact, if you restrict yourself to natural number arithmetic, division (in the normal sense) doesn't apply to them, either, because rational numbers are outside the domain. Also, in the real number system, the square root operation cannot be applied to negative numbers.

Your recently acquired obsession with the need for direct evidence in nature, is in total contradiction with your theism, not to mention being logically silly.

And this post establishes actual infinities how exactly? What is the indirect evidence for actual infinities given you’ve talked about maths which is not the focus.

[Walt Zingmatilder]

A box full of sand, and nothing else, contains exactly zero zebras.

zero zebras here gives rise to all kinds of absurdities here. There are no actual Zebras. What on earth is a zero zebra? They can’t be actual. But beyond all of this. How does it help you establish direct or indirect evidence for actual infinities.

[Walt Zingmatilder]

Both untrue

So we arrive at you being able to provide indirect evidence for actual infinities.

logically absurd. Oh, and by the same 'argument' we can deduce that no God exists.

Nobody is claiming God is physical whereas the claim is there Is evidence for actual physical infinities.

[Walt Zingmatilder]

Just to point out that if you take that position, it means all the times you've asked why is there something rather than nothing, you've just said are incoherent questions

“There may be no such thing as nothing” as far as aI can recall is your line. Mine was there is a reason why there is something rather than nothing. Nothing being the absence of anything physical.

Again, we seem to have strayed from establishing if there are infinities.

If it ends up that actual infinities can only be believed in, then where does that put the superiority of non belief in things like. God for instance?

In my opening post I merely allude to new thinking on infinities vis a sceptical approach based on a demand for evidence. If Stranger can make good his counter claim to” there is no indirect evidence tfor actual infinities” then he’s home and dry.

[Stranger]

I thought after I'd written post #40, and before I hit 'post', that it was going to be a classic case of 'casting pearls before swine', but I'd written it by then. Ho-hum.

And this post establishes actual infinities how exactly?

I'm not trying to establish actual infinities, I'm arguing against the silliness in the OP that they are contradictory and cannot exist.

What is the indirect evidence for actual infinities given you’ve talked about maths which is not the focus.

The fact that space-time behaves, to the limit of our ability to measure it, exactly like a continuum (infinite points between any two separate points).

The fact that the observed 'curvature' of space, on large scales, is 'flat' (as far as we are able to measure it). The simplest topology associated with 'flat' space is infinite.

zero zebras here gives rise to all kinds of absurdities here. There are no actual Zebras. What on earth is a zero zebra? They can’t be actual. But beyond all of this

You don't understand how numbers work? No wonder you're struggling so much.

Nobody is claiming God is physical whereas the claim is there Is evidence for actual physical infinities.

Going for the special pleading fallacy, okay. If infinities are inherently contradictory, why would it matter whether an actual, real infinity was physical or not? And you're misrepresenting the claim. The claim is that infinities are not contradictory and therefore cannot be ruled out in reality. It is a bit of a side issue that there is some weak but indicative evidence that there might be real infinities even before we get into speculative conjectures or hypotheses.

[Nearly Sane]

“There may be no such thing as nothing” as far as aI can recall is your line. Mine was there is a reason why there is something rather than nothing. Nothing being the absence of anything physical.

Again, we seem to have strayed from establishing if there are infinities.

If it ends up that actual infinities can only be believed in, then where does that put the superiority of non belief in things like. God for instance?

In my opening post I merely allude to new thinking on infinities vis a sceptical approach based on a demand for evidence. If Stranger can make good his counter claim to” there is no indirect evidence tfor actual infinities” then he’s home and dry.

Are we trying to establish that there are no such thing as real infinities? And even if we were why does that stop it being an issue that if your 'logic' is followed then you end up with taking contradictory positions being an issue?

[jeremyp]

Courtiers reply?

No. The analogy of this argument is the equivalent of you telling us that wool is a form of cotton. Regardless of whether the emperor is wearing wool cotton or nothing at all, that would be a false statement.

But the problem isn’t with it being mathematically coherent or logically coherent but you demonstrating it physically, continual preening over solving the maths does not equate with showing these things in nature.

Why would I need to demonstrate something that is logically invalid in the physical world? What would it tell us about the maths (spoiler: nothing)?

Where does something having something taken away but leavingq the same amount being left occur in nature?The author calls them contradictions exclamation mark

No he doesn't. He says they are apparent contradictions, but actually not in reality. Read the whole article.

[jeremyp]

Show me a zero zebra or infinite zebras for that matter.

Photo taken from my desk just now.

Zero zebras

[jeremyp]

you demonstrating it physically

Well, it looks like the Universe is infinite in extent according to measurements of its expansion. So, assuming that it is and that the number of stars in it is also infinite (it would have to be else, space would not be infinite in extent), it is a real life example of Hilbert's hotel. Let's take a couple of examples.

• I take all of the stars (there are infinitely many) and remove all of the red giants (there are infinitely many). There are still infinitely many stars left. ∞ - ∞ = ∞
• I take all of the stars and remove all of the stars more than 10 light years from Earth (there are infinitely many). There is a finite number of stars left. ∞ - ∞ = 12

Wow. Look, I seem to have the same contradiction as the article had. Except there is absolutely nothing wrong with or contradictory about what I did.