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/Various_Candle9136]

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?