That's not true. Set theory is based on a lot of assumptions and it only consistent if you assume all of them are true.
I could argue if you start at infinity and count down you never reach zero as there are uncountable many infinite numbers. Worse there are uncountable many digits in an infinite number so any operation based on an infinite sequence never finishes. Making that argument and much of set theory meaningless.
PS: There are many Axioms which are chosen simply because it makes math more interesting. One of the more fun ones is to do set theory but allow for some undefined relationships. Aka x is in set A but it's undefined if it's in set B. Now quick draw a venn diagram.
Since you've edited your comment, I've deleted my reply, and here's a replacement ...
> That's not true. Set theory is based on a lot of
> assumptions and it only consistent if you assume
> all of them are true.
All of mathematics is based on the idea of taking some axioms, and then seeing what follows. We usually use axioms that somehow align with our intuition or experience, and then see if things develop as we expect. Consistency is nothing to do with "Truth".
> I could argue if you start at infinity and count
> down you never reach zero as there are uncountable
> many infinite numbers.
This doesn't make sense. To "start at infinity," what does it mean to count down at all? Certainly I can't interpret what you are saying in any meaningful way. Perhaps you could restate it in a more precise way.
> Worse there are uncountable many digits in an
> infinite number ...
There is no such thing as "an infinite number."
> ... so any operation based on an infinite sequence
> never finishes.
With care, operations on infinite sequences can be defined precisely and shown to be well-defined, so it's not clear what you're trying to say here.
> Making that argument and much of set theory
> meaningless.
??
> PS: There are many Axioms which are chosen simply
> because it makes math more interesting. One of the
> more fun ones is to do set theory but allow for
> some undefined relationships.
So you are creating a completely different math? Good, but not relevant here.
> Aka ...
I don't think you mean "aka" - that means "Also known as ...". I suspect you mean something more like "For example" which is "E.g."
> ... x is in set A but it's undefined if it's in
> set B. Now quick draw a Venn diagram.
You haven't defined any of your terms, and you're doing something completely different from standard set theory. Sure, it might be fun, but it's more likely a toy problem in a toy theory. Can you give more precise definitions for what you're doing? Perhaps you should write it up and submit a link.
It's not that I am creating a different Math. Just point out that branches of math are not just exploring different topics, they often use different Axioms.
With care, operations on infinite sequences can be defined precisely
Sure, but let's suppose your describing a finite state machines. Now, they can't generally deal with infinity.
The Venn diagram bit is actually part of an old CS joke. "Everything is working perfectly, and the database returns NULL."
Edit: Bringing back to my first post, the "Axiom of Choice" is not the only place you can disagree.
I could argue if you start at infinity and count down you never reach zero as there are uncountable many infinite numbers. Worse there are uncountable many digits in an infinite number so any operation based on an infinite sequence never finishes. Making that argument and much of set theory meaningless.
PS: There are many Axioms which are chosen simply because it makes math more interesting. One of the more fun ones is to do set theory but allow for some undefined relationships. Aka x is in set A but it's undefined if it's in set B. Now quick draw a venn diagram.