Re: Skolem's Paradox and why is math the way it is?

From: Torkel Franzen (torkel_at_sm.luth.se)
Date: 09/18/04 

Date: 18 Sep 2004 07:59:06 +0200

troubled6man@yahoo.com (J.E.) writes:

> Why do mathematicians like a nondescriptively complete axiom system as
> a basis? Why don't we use second order logic, or if that's too big a
> jump how about IF-logic (which is bigger than 1st order, but not as
> big as second order)?

  What do you mean? What would it be to "use second order logic"? Use
how?

> And for the mathematicians that think this is unneccissary, then how
> do we know everyone is doing the same math if the axioms don't
> describe "real numbers" uniquely?

  What do you mean? What is it to "do the same math"?

> How do we know which model is the "intended" interpretation of set
> theory?

  What set theory are you talking about? How does the "intended
interpretation" enter into mathematics?

> Why do we care about "uncountably many" real numbers when in reality
> there are only a countable number that we can "prove theorems
> about"?

  What do you mean? What is it to "prove a theorem about" a number?
Is "for every real x, x=x" a theorem about numbers?

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