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袁萌 陈启清 1月25日
附件：此文发表于2016年8 月2日
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2016 08 02
Applications of Non-Standard Analysis to Number Theory and Topology?
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S.E friends,
I am wondering what might be some useful applications of the non-standard analysis to the number theory and topology? I am very interested in the set-theoretic topology and prime distributions, and I recently came across the field of non-standard analysis...The use of infinitesimal is interesting but I am not sure how can it be applied to my fields of interest.
Do infinitesimals offer
different perspective to the analytic number theory and topology?
general-topology number-theory soft-question analytic-number-theory nonstandard-analysis
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edited Aug 2 '16 at 15:19
Aweygan
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asked Aug 2 '16 at 14:54
MathWanderer
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The answer is "topology can be interpreted through the lens of non-standard analysis", but I don't know very much about how to do it, I'm afraid. – Patrick Stevens Aug 2 '16 at 15:10
1
Euclid's proof of infinitely many primes is easily rewritten to have a NSA flavor: Notice that, for
N
N
infinitely large,
N!+1
N!+1
has no standard factors. Thus, the smallest prime factor of this is infinitely large, meaning there is an infinitely large prime. And we are done, for this means every standard number has a prime above it, and, by transfer, every number has a prime above it, so there are infinitely many primes. – Akiva Weinberger Aug 4 '16 at 21:12
@Akiva, I don't follow this argument. The statement that every standard prime has a prime above it is not an elementary formula. Therefore one apparently cannot apply transfer to it. – Mikhail Katz Aug 5 '16 at 7:50
@MikhailKatz Perhaps… It's been a while since I've thought hard about this stuff. – Akiva Weinberger Aug 5 '16 at 19:27
1
@MikhailKatz I'm not sure how to phrase the proof in the Robinson sense, but there should be an equivalent: it's definitely true that a necessary and sufficient condition for the existence of infinitely many primes is the existence of a nonstandard prime. The argument above is fine for transfer in the IST sense. – GPhys Aug 5 '16 at 19:49
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For an application in topology, consider Tychonoff's theorem. Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. Once the basic machinery of the hyperreal framework is in place, the proof of Tychonoff's theorem is essentially a one-liner. The point is that there is an elegant an convenient characterisation of compactness as follows. A space
A
A
is compact if and only if every point of its natural extension
A
∗
A∗
is nearstandard. Here being nearstandard for
x
x
means that that there exists an infinitely close point to
x
x
which belongs to
A
A
.
Thus,
R
R
is not compact because its natural extension contains infinite numbers which are therefore not infinitely close to any real number.
The interval
(0,1)
(0,1)
is not compact because
(0,1
)
∗
(0,1)∗
contains infinitesimals which are not infinitely close to any real strictly between
0
0
and
1
1
.
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answered Aug 3 '16 at 8:06
Mikhail Katz
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In number theory, I've seen occasional good use made of the infinite numbers. If
P
P
is an infinite prime number, then the internal finite field of
P
P
elements is, externally, a field of characteristic zero.
I've used facts about finite fields to construct fields of characteristic zero with nice properties, and used facts about characteristic zero fields to prove things about sufficiently large finite fields.
The first time I've seen this method is in regards to the Ax-Kochen theorem: if Wikipedia's account is accurate, the main idea of the proof can be rephrased as saying that the field
Q
P
QP
of
P
P
-adic numbers and the field
F
P
((t))
FP((t))
of formal Laurent series over the finite field of
P
P
elements happen to be, when viewed externally, elementary equivalent as valued fields.
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