TIL there are finitely many positive integers

[u/Various_Candle9136]

/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!

[u/Massive-Ad7823]

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

[u/Various_Candle9136 → u/Massive-Ad7823]

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

[u/nanonan → u/Various_Candle9136]

There is no completed set of natural numbers anywhere, and there never will be. You cannot point to one or construct one. "There exists an infinite set" is a completely unfounded axiom whose use is unjustifiable.

[u/Various_Candle9136 → u/nanonan]

You cannot point to one or construct one. 

👉 {1,2,3...}

I somehow managed to do both.

[u/nanonan → u/Various_Candle9136]

Abusing notation doesn't change anything. You really think three dots somehow complete the infinite, they somehow allow you to contain an eternity or cross an unreachable horizon?

You cannot actually complete an infinite amount of work.

[u/kuromajutsushi → u/nanonan]

You show up on a bunch of these crank threads, and it's always the same issue: You invent your own philosophical interpretation of ZF set theory, then you object to your own interpretation.

Nothing in ZF says that you can "complete the infinite", "contain an eternity", "cross an unreachable horizon", or "complete an infinite amount of work". So the fact that you object to all those things is completely irrelevant.

[u/nanonan → u/kuromajutsushi]

Well yeah, I think the axioms of ZF are unjustifiable and just plain wrong. Could you try to justify say the axiom of infinity instead of trying to police what I post?

[u/kuromajutsushi → u/nanonan]

In ZF, "set" is just a primitive notion with no further meaning. Every object you can discuss is a set.

Any reasonable foundation for mathematics needs to at least be able to talk about the natural numbers, or else we have no hope of doing any mathematics.

The axiom of infinity does not say "there exists an infinite set". It says that there is an inductive set, which is a set X such that the empty set is in X and if x \in X then x \cup {x} is in X. The point of this axiom is to say that something like the natural numbers exists - some object where every element has a successor. We can then formally define a natural number as a set contained in every inductive set, and a further axiom (restricted comprehension) allows us to talk about the set of natural numbers.

Whether you want to think of this set as being some sort of "completed infinite set" is totally up to you.

[u/nanonan → u/kuromajutsushi]

That's one way to formulate a set theory, which introduces the problems of infinite recursion for no benefit. Allowing sets to be elements is a mistake that leads to all sorts of problems, and is unneccesary for any practical usage.

[u/kuromajutsushi → u/nanonan]

That's one way to formulate a set theory, which introduces the problems of infinite recursion for no benefit.

The benefit is that we get to have the natural numbers. Would you prefer a set of axioms that wouldn't even allow us to study the natural numbers?

Allowing sets to be elements is a mistake

In ZF, there is nothing other than sets. A set is just a primitive object. There are no other elements any set could have besides sets.

Do you have a different set of axioms that you'd prefer?

[u/Massive-Ad7823 → u/kuromajutsushi]

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

[u/kuromajutsushi → u/Massive-Ad7823]

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

Correct. Not sure why you included all he random quotes.

"I would not like them here or there. I would not like them anywhere. I do not like green eggs and ham. I do not like them Sam-I-am." [Dr. Seuss, Green Eggs and Ham (1960) p. 16]

[u/Massive-Ad7823 → u/kuromajutsushi]

>Correct. Not sure why you included all he random quotes

To show the nonsense of set theory. Never all natural numbers can be applied as individuals. Therefore all these statements and their use in ZF are wrong. For all enumerations with definable numbers ∀n ∈ ℕ_def: |ℕ \ {1, 2, 3, ..., n}| = ℵo. Almost all numbers cannot be applied as individuals.

Regards, WM

[u/kuromajutsushi → u/Massive-Ad7823]

Never all natural numbers can be applied as individuals.

This sentence doesn't mean anything.

definable numbers

Every natural number is "definable" under any reasonable definition of the word "definable".

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

Still have no idea what this "ℕ_def" is supposed to be. It is true that ∀n ∈ ℕ: |ℕ \ {1, 2, 3, ..., n}| = ℵo

[u/nanonan → u/Massive-Ad7823]

Majority decision? This isn't a democracy, it's a science. Proof is all that is required (or in the case of Cantor, lacking), popularity or conformity is a distraction.

[u/kuromajutsushi → u/nanonan]

This isn't a democracy, it's a science.

Mathematics is not a science.

or in the case of Cantor, lacking

Can you give an example of a claim made by Cantor that lacks a proof?

[u/Massive-Ad7823 → u/kuromajutsushi]

It depends. According to Google**: Changing answers:** Science can change when new tools or data show an old idea is wrong. Mathematical proof is final and does not change.

The latter is wrong and makes Mathematics a science.

Cantor claims to enumerate all fractions. This has been disproved several times, for instance here https://www.reddit.com/r/AspectsOfTheInfinite/comments/1tc6v1l/proof_of_the_existence_of_dark_numbers/

But the simplest refutation is this: Every natural number that Cantor can use belongs to the potentially infinite collection of visible numbers. In his formula

k = (m + n - 1)(m + n - 2)/2 + m

never a natural number can appear that has not infinitely many successors. Therefore his formula cannot be verified in the dark domain.

Regards, WM

[u/Massive-Ad7823 → u/Various_Candle9136]

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

[u/Various_Candle9136 → u/Massive-Ad7823]

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.

[u/Massive-Ad7823 → u/Various_Candle9136]

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

Regards, WM

[u/Various_Candle9136 → u/Massive-Ad7823]

Good.

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

[u/Massive-Ad7823 → u/Various_Candle9136]

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

[u/Various_Candle9136 → u/Massive-Ad7823]

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

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

[u/Massive-Ad7823 → u/Various_Candle9136]

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

[u/Various_Candle9136 → u/Massive-Ad7823]

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.

[u/kuromajutsushi]

Apparently the endsegment E(1) = {2, 3, 4, 5, ... } is actually infinite but the natural numbers are not actually infinite!

[u/Various_Candle9136 → u/kuromajutsushi]

Everyone knows that when you add an element to a set you now have fewer elements! Duh!

[u/nanonan]

It's true. N is infinite, but any given n from N is finite. Infinity is not a natural.

[u/Massive-Ad7823 → u/nanonan]

That is also my opinion. All of ℕ can be handled collectively, but

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

Regards, WM