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I noticed that a big part of nonstandard analysis aimed to work with ordinary real valued functions. It is a bit strange for me because when we expand rational numbers to reals, we do not formulate theorems for functions $\mathbb Q \to \mathbb Q$ - we study $\mathbb R \to \mathbb R$ functions.

Why don't we use $^* \mathbb R$ as the main model of what we mean saying "numbers"? There is no people's congenital love to $\mathbb R$. We use $\mathbb R$ only because it is not the worst model for continuous processes. What the problem to replace $\mathbb R$ with $^* \mathbb R$? I mean to "forget" all $\mathbb R$-based mathematics and rewrite it in terms of $^* \mathbb R$. We will lose some theorems (like intermediate value theorem) but do we really so need them all?

Sorry for mistakes, english is not my native language.

TheWildPalms
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  • What is "$^*\mathbb R$" in this context ? The set of the rational numbers ? – Peter Dec 31 '23 at 10:08
  • I learnt the way to construct the real numbers via Cauchy sequences of rational numbers. The Dedekind cut is another way I have no experience with. – Peter Dec 31 '23 at 10:10
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    $^* \mathbb R$ is hyperreal numbers. Construction on free ultrafilter $U \subset 2^ \mathbb N$ – TheWildPalms Dec 31 '23 at 10:15

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Robinson introduced nonstandard analysis in a 1961 article and then a 1966 book. A decade later, the program you outlined was in fact carried out independently by Hrbacek and Nelson. In their approach called "axiomatic nonstandard analysis", mathematics takes place over $\mathbb R$ (rather than an extension thereof), and infinitesimals and unlimited (informally: "infinite") numbers are found within $\mathbb R$. Here we retain all classical analysis including the Intermediate Value Theorem. The theories developed by Hrbacek and Nelson are conservative over ZFC and therefore prove exactly the same results as ZFC. The difference is the availability of new tools that enable more transparent proofs of results, and also proofs whose non-infinitesimal equivalents may be too complex for a human reader to parse.

Mikhail Katz
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  • I know about Hrbacek and Nelson. The problem is - they construct new set theory which is very unpleasant to me. I like ZFC and don't want to go beyond it. I want to look at $^* \mathbb R$ simply as extension of $\mathbb R$. – TheWildPalms Dec 31 '23 at 11:03
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    @TheWildPalms, It is generally held that real analysis is about the real numbers rather than the hyperreal numbers. The latter is a convenient tool in studying real objects: real functions, real operators, etc. For example, infinitesimals are useful in proving basic results such as intermediate value theorem for real functions, by enabling more transparent proofs. You can't do any of that without the transfer principle. If you are willing to forego the transfer principle, there are much simpler systems with infinitesimals, such as the field of Laurent series. – Mikhail Katz Dec 31 '23 at 11:10
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    @TheWildPalms, some of the axioms of IST are hard to follow, but today we have a new theory SPOT whose axioms are completely transparent and moreover it is conservative over ZF (without the axiom of choice); see this introduction. – Mikhail Katz Dec 31 '23 at 11:14
  • Yes, I want to construct analysis "from zero", but not on the "framework" of real numbers (but in ZFC). I need a set of "numbers" for this. I even don't limit myself by fields. I'm considering all options: Laurent series, Hardy field, transseries. May be something else? So, the main question is: is it generally possible? – TheWildPalms Dec 31 '23 at 11:36
  • Paolo Giordano developed an approach in this direction. @TheWildPalms – Mikhail Katz Dec 31 '23 at 11:39