Comments on: Doing Calculus on the Rationals (with the help of Nonstandard Analysis) http://xorshammer.com/2008/11/14/doing-calculus-on-the-rationals-with-the-help-of-nonstandard-analysis/ Some things in mathematical logic that I find interesting Wed, 18 Jan 2012 18:13:09 +0000 hourly 1 http://wordpress.com/ By: Xamuel http://xorshammer.com/2008/11/14/doing-calculus-on-the-rationals-with-the-help-of-nonstandard-analysis/#comment-308 Mon, 06 Sep 2010 21:42:48 +0000 http://xorshammer.wordpress.com/?p=218#comment-308 Re: “Are there really finitely many numbers less than ω?”

No, externally we know there are infinitely many numbers less than ω, but the model doesn’t know it. The model thinks there are finitely many– namely, ω many (which the model thinks is a finite amount).

]]> By: Xamuel http://xorshammer.com/2008/11/14/doing-calculus-on-the-rationals-with-the-help-of-nonstandard-analysis/#comment-307 Mon, 06 Sep 2010 21:41:15 +0000 http://xorshammer.wordpress.com/?p=218#comment-307 Confused on two things. First, shouldn’t st(0) be taken as an axiom? Otherwise the standard model, plus st interpreted as an empty predicate and w interpreted as 0, would satisfy axioms 1-4…

Second, you say: “construct Z and Q from N as usual”. Could you clarify? How do we extend an arbitrary primitive recursive function from N to Z to Q? Especially if it’s defined using recursion, which leads to an infinite loop if given anything besides a natural…

]]> By: msuica http://xorshammer.com/2008/11/14/doing-calculus-on-the-rationals-with-the-help-of-nonstandard-analysis/#comment-267 Thu, 11 Mar 2010 12:07:31 +0000 http://xorshammer.wordpress.com/?p=218#comment-267 Are there really finitely many numbers less than ω? Then we should be able to count the natural numbers smaller than ω and say that there are n such natural numbers, where n is a finite natural number. But this would imply that ω = n, which would seem to be a contradiction.

Is there a subtlety in the axioms that I’m misunderstanding?

- max

]]> By: mkoconnor http://xorshammer.com/2008/11/14/doing-calculus-on-the-rationals-with-the-help-of-nonstandard-analysis/#comment-145 Thu, 27 Nov 2008 02:50:00 +0000 http://xorshammer.wordpress.com/?p=218#comment-145 Yes, you are correct.

]]> By: Kenny Easwaran http://xorshammer.com/2008/11/14/doing-calculus-on-the-rationals-with-the-help-of-nonstandard-analysis/#comment-144 Mon, 17 Nov 2008 04:08:14 +0000 http://xorshammer.wordpress.com/?p=218#comment-144 In the non-standard reals, we have the nice property that every bounded real is infinitesimally close to a standard real – I imagine that this is a property that fails here, right?

I suppose we can probably prove that just by considering \sin 1 or something similar – for any standard rational x, I should either be able to find some rational number less than 1 whose \sin is greater than x by more than some standard rational, or some rational number greater than 1 whose \sin is less than x by more than some standard rational. Since \sin is strictly increasing in this range, this means that \sin 1 is not infinitesimally close to x. Thus, \sin 1 is not infinitesimally close to any standard rational.

So we’ve somehow built in approximations to all the standard reals. I suppose we can see this if we consider the rationals in some non-standard model of the reals. Write out the full decimal expansion for some real y. Now consider the rational number that we get by taking the first \omega digits after the decimal point in this expansion, and repeating these, instead of continuing with the rest of y. This number is infinitely close to y, but is rational. I imagine the model we get by constructing the rationals directly will include all of these?

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