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I know that nonstandard analysis is analysis plus the existence of infinitesimal numbers. Does it mean that nonstandard analysis is the same theory as $ZF+\exists$infinitesimal numbers?

From what I read about it on Wikipedia there seem to be a few different approaches to it. There is "Robinson's method" which is "based on studying models". There is "Nelson's method" which is "Internal set theory" which is an extension of ZF. Are they the same? Or are there theorems that are true in one but not in the other?

If IST is an extension of ZF how much stronger than ZF is the "theory of nonstandard analysis"?

Mikhail Katz
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Student
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3 Answers3

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To answer your first question, one can indeed think of nonstandard analysis, in a first approximation, as entailing the existence of infinitesimal numbers as you write, but to be more precise one would have to elaborate further. I like to think of it in terms of three approaches (other editors may disagree), as follows:

(1) the most straightforward approach (in my opinion) is through the construction of a proper field extension $\mathbb R^\star$ (notation varies, but let's stick with this one which is used in Keisler's book, see http://www.math.wisc.edu/~keisler/calc.html) of the real field $\mathbb R$. This field extension can be constructed in a way somewhat similar to constructing $\mathbb R$ out of the rationals $\mathbb Q$. Namely, one starts with sequences of real numbers, introduces a suitable equivalence, and obtains the hyperreal field $\mathbb R^\star$ as the quotient. This construction is called the ultrapower construction. A serious undergraduate algebra course provides enough background to understand this construction; namely what is required is the existence of a certain maximal ideal in a suitable ring.

(2) A more sophisticated route (and the one taken by Robinson in his 1966 book, see http://www.google.co.il/books?id=OkONWa4ToH4C&source=gbs_navlinks_s) is to invoke the compactness theorem from mathematical logic (more specifically, model theory) so as to prove the existence of such an $\mathbb R^\star$ (in Robinson's book this is denoted $^\star\mathbb R$).

(3) Edward Nelson's approach, called IST (internal set theory) is a reformulation of Robinson's approach where, instead of extending the field $\mathbb R$, Nelson takes the "syntactic" route. This means that the language of ordinary set theory is enriched by the addition of a unary predicate "Standard". Then infinitesimals are found within the "ordinary" $\mathbb R$ itself; so to speak they have been there all along, it's just that we haven't noticed (because the ordinary syntax of set theory is insufficiently rich). An infinitesimal number is NOT "Standard".

All three approaches are equivalent (at least at a basic level), so one proves the same theorems in all of them. No new axioms are needed beyond those of ZFC.

A more detailed summary by Joel David Hamkins appears here.

Since ZF was mentioned in your question, I add the following. Recently it turned out that one can do axiomatic nonstandard analysis conservatively over ZF (for a discussion of the meaning of conservativity see https://mathoverflow.net/a/401076/28128). The relevant publications can be consulted at https://u.math.biu.ac.il/~katzmik/spot.html .

Mikhail Katz
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    Your answer is great, thank you. Then ZF should be replaced with ZFC in my question? – Student May 29 '13 at 12:59
  • Certain versions of an infinitesimal-enriched number system can be constructed without any form of the axiom of choice at all (not even ultrafilter lemma). Thus, Skolem in 1933/34 developed such a system in a purely constructive way. But to get an extension of the reals satisfying the transfer principle, one does need some form of choice. – Mikhail Katz May 29 '13 at 13:06
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$\mathsf{IST}$ is a conservative extension of $\mathsf{ZFC}$: any internal formula that can be proved within internal set theory can be proved in $\mathsf{ZFC}$. This is proved in the Appendix to Edward Nelson’s original $1977$ paper in the Bulletin of the American Mathematical Society, ‘Internal Set Theory: A New Approach to Nonstandard Analysis’.

Student
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Brian M. Scott
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  • Is IST strictly stronger than ZFC? – Student May 29 '13 at 13:00
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    It depends what you mean by "stronger". IST provides a richer syntax so that certain proofs can be stated in a much more lucid way than the traditional ZFC framework. See, for example, the recent discussion here: http://math.stackexchange.com/questions/405497/i-would-like-to-know-an-intuitive-way-to-understand-a-cauchy-sequence-and-the-ca/405515#405515 On the other hand, as Brian mentioned, IST is a conservative extension of ZFC, so in a purely formal sense neither is stronger than the other. Just think of writing a computer program in binary. – Mikhail Katz May 29 '13 at 13:18
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To understand the theory nonconstrutivly of Robinson, it's sufficient to use the model theory, in this link: https://hal.inria.fr/hal-01248379/ , we can find a new approach to nonstandard analysis without using t he ultrafiltries.

saghe
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  • This work seems to be unpublished. Since you are the author, can you clarify whether the sine function extend to the omicron reals? – Mikhail Katz Jun 02 '16 at 06:54
  • the idea of the omicran-reals is to construct the hyppereals by using a subset of IR^IN, because we can find many problem in the stabilisation of the field if we use IR^IN, for exemple the hyperreal defined by the sequence((-1)^n) is negatif or positive, but we can determine his signe because the order relation is unknown from the Robinson's method, in the new method we can't find this problem – saghe Jun 05 '16 at 17:30
  • So what's the sign of $(-1)^n$? How do you prove that your system has an order on it? You realize folks here may have limited time to invest in a paper that has not been published. – Mikhail Katz Jun 05 '16 at 17:36
  • the sequence (-1)^n is not defined in the new method, we can't find a holomorphic function in neiberhood of zerooo such that f(1/n)=-1^n, in the field of omicran-reals every sequence (f(1/n)) which defined a (nonstandrd number!!!) is a constant sign from a certain rank. – saghe Jun 05 '16 at 17:44
  • If you only work with analytic functions then such a theory was developed by Levi-Civita a century ago. This was studied in particular in a book by Robinson and Lightstone in the '70s. Have you looked at the history of the question? – Mikhail Katz Jun 05 '16 at 17:46
  • my theory is based to find a new approach without using the model theory, and i find the applications of this new theory we can see the link https://hal.archives-ouvertes.fr/hal-01252675, and https://hal.archives-ouvertes.fr/hal-01265296 , when i'm restart to the work, i don't know that exist an mathematician called abraham robinson, I tell the truth. – saghe Jun 05 '16 at 17:58
  • saghe, there is a principle in western law that "ignorance of the law is no excuse". Similarly, ignorance of earlier work is no excuse. A famous case is discussed in detail here. I would suggest familiarizing yourself with earlier work before you make grandiose research announcements, so as to avoid Sergeyev's embarrassment. – Mikhail Katz Jun 05 '16 at 18:00
  • i' don't undertand your advise, i find a new method which different from Robinson and Nelson, we can't find another which is similar to my approach, if we find a mistake in the construction, I have to thank you because I'm not found someone to give the criticism of my approach. – saghe Jun 05 '16 at 18:28
  • i' don't undertand your advise, i find a new method which different from Robinson and Nelson, we can't find another which is similar to my approach, if you find a mistake in the construction, I have to thank you because I'm not found someone to give the criticism of my approach. – saghe Jun 05 '16 at 18:29
  • In mathematics as in any academic field a researcher needs to take into account prior work and place his own research in perspective with regard to such prior work. He needs to explain what he accomplishes that previous methods don't, and in what way his methods are different from his predecessors. – Mikhail Katz Jun 06 '16 at 07:02