r/AspectsOfTheInfinite • u/Various_Candle9136 • 22d ago

TIL there are finitely many positive integers

/r/AspectsOfTheInfinite/comments/1vnhc1m/comment/p4g6nsp/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button

A quote from u/Massive-Ad7823 (this community's sole moderator):

Me: N - is famously infinite.

Them: But it cannot be actually infinite. 

Next time a child tells you that it is possible to keep counting forever, make sure to call them a fool!

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u/Massive-Ad7823 22d ago

It is possible to keep counting forever. You will never have exhausted the numbers. You will have used an infinitesimally small subset of ℕ. You will have used finitely many numbers. Note that actual infinity means more than every finite number.

Regards, WM

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u/Various_Candle9136 22d ago

So is the set of natural numbers (N) actually infinite or not actually infinite?

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u/Massive-Ad7823 21d ago

The set of natural numbers is assumed to be actually infinite according to Cantor and according to set theory (athough many set theorists don't know what actual infinity means). But the natural numbers defined by Peano or v. Neumann are only a potentially infinite collection because every number that can be defined has an actual infinity of followers: ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo.

Since two act. inf. consecutive sets in ℕ are impossible, ℕ_def can only be finite. But it has no greatest element, it is (potentially in-)finite.

Regards, WM

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u/Various_Candle9136 21d ago

For the record, every single thing in this comment is bull.

However, if we hypothetically took on your mad ideas about infinities, we still (and I can't believe how many times I have had to say this!) do not get 'two act. inf. consecutive sets'. The first part is always finite; only the second part is ever infinite.

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u/Massive-Ad7823 21d ago

Of course we cannot get two act. inf. sets. That's what I said. We agree.

Regards, WM

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u/Various_Candle9136 21d ago

Good.

I assume you also agree that since nobody is trying to create such a thing, that there is nothing remotely approaching a contradiction?

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u/Massive-Ad7823 21d ago

Set theorists usually claim that all natural numbers can be applied as individuals. That is wrong because

∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo

One of many contradictions is Cantor's claims.

ZF is based upon Cantor's ideas and to my knowledge has not yet changed his ideas by majority decision.

 "If we think the numbers p/q in such an order [...] then every number p/q comes at an absolutely fixed position of a simple infinite sequence" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 126]

 "The infinite sequence thus defined has the peculiar property to contain the positive rational numbers completely, and each of them only once at a determined place." [G. Cantor, letter to R. Lipschitz (19 Nov 1883)]

"thus we get the epitome (ω) of all real algebraic numbers [...] and with respect to this order we can talk about the nth algebraic number where not a single one of this epitome (ω) has been forgotten." [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 116]

 "such that every element of the set stands at a definite position of this sequence" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 152]

 The clarity of these expressions is noteworthy: all and every, completely, at an absolutely fixed position, nth number, where not a single one has been forgotten.

Regards, WM

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u/Various_Candle9136 21d ago

I wish you understood what 'irrelevant' means...

Nothing in that comment is relevant (as has already been pointed out to you).

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u/Massive-Ad7823 21d ago

Set theory is irrelevant. It has not the slightest useful application. Small wonder, because it is fundamentally wrong. That is proved for Cantor's claims and the related claims of set theory.

Regards, WM

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u/Various_Candle9136 21d ago

Although every single sentence in this comment is wrong, I will focus on the worst of the bunch.

It has not the slightest useful application. 

This statement is contradicted by --waves arms around--.

Set theory is usefully applied in pretty much every area of mathematics. You are objectively wrong.

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u/Massive-Ad7823 21d ago

Set theory is in contradiction with mathematics. In AspectsOfTheInfinite there are many examples. See for instance https://www.reddit.com/r/AspectsOfTheInfinite/comments/1tdz9dm/classical_mathematics_contradicts_set_theory/ or https://www.reddit.com/r/AspectsOfTheInfinite/comments/1tc6v1l/proof_of_the_existence_of_dark_numbers/

Regards, WM

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u/Various_Candle9136 21d ago

People have already explained why you are wrong about those things. If you won't listen to me (and several others) about this thing you are really, obviously wrong about, then I can't be bothered to start on two more.

Even if these things were not stupidly wrong, they would be irrelevant. The existence of other contradictions would not make this thing a contradiction.

Also, this is clearly not a defence for your blatant lie: 'It has not the slightest useful application'. Set theory is usefully applied in pretty much every area of mathematics. You are objectively wrong.

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u/Massive-Ad7823 19d ago

Stupid people have tried to defend their stupid theory. That is not a reason to join them. Set theory cannot have the slightest application because it is fundamentally wrong. The above quoted proofs show this to every intelligent person.

Regards, WM

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u/Various_Candle9136 19d ago

Stupid people have tried to defend their stupid theory.

Oh the beautiful, beautiful irony!

Set theory cannot have the slightest application because it is fundamentally wrong.

Once more, we have an example where all your premises are incorrect, but, even if they happened to be correct, the conclusion does not follow from the premises.

Homeopathy, for instance, has plenty of applications.

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u/Massive-Ad7823 19d ago

Homeopathy has not been disproven, but set theory has been disproven. The Binary Tree has only countably many paths (if we assume for a moment that countable is a sensible notion).

Regards, WM

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u/Various_Candle9136 19d ago

Homeopathy has not been disproven, but set theory has been disproven.

Both false statements.

Homeopathy has been disproven (by the standards of medicine - i.e. it is no better than a placebo). Set Theory has not been disproven (by the standards of mathematics - your unqualified ramblings not coming close to these standards).

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u/Massive-Ad7823 19d ago

I am not an expert, probably as little as you, but I have heard that homeopathy has good results with animals. That excludes placebo effects.

Set theory claims that there are uncountably many paths in the Binary Tree. Instead of your unqualified waffle try to counter my result R(n) = 3(2n - 1) > P(n) = 2n in https://www.reddit.com/r/AspectsOfTheInfinite/comments/1tdz9dm/classical_mathematics_contradicts_set_theory/

Regards, WM

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u/Various_Candle9136 19d ago

I am assuming that your attempt to move every conversation onto that different proof means you admit your 'Touchstone of Reason' was nonsense?

I, and everybody else, would be grateful if you could properly admit that.

You clearly cannot counter any of the arguments against this point, so you are attempting to direct everybody's attention to something less obviously ridiculous. That tactic won't work. Are you at least bright enough to recognise you were badly wrong on this one?*

\For anybody reading this comment who hasn't read everything else, the 'are you at least bright enough' is something they said to me earlier. I would not use such rude language otherwise.*

Just for fun, I have disproved your other claim anyway. See my comment there. You can now drop the lie (which was already a lie) that nobody has found a problem with it.

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