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I have only a vague understanding of nonstandard analysis from reading Reuben Hersh & Philip Davis, The Mathematical Experience. As a physics major I do have some education in standard analysis, but wonder what the properties are that the nonstandardness (is that a word?) is composed of. Is it more than defining numbers smaller than any positive real as the tag suggests? Can you give examples? Do you know of a gentle introduction to the nonstandard properties?

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I thought it is a way to avoid limits and make stuff more algebraic. But I'm a physicist too, so I'll wait for proper answers. – Yrogirg May 23 '12 at 9:39
Jerry Keisler’s freshman-level calculus text uses infinitesimals rigorously and would give you an idea of what non-standard analysis looks like in practice at ground level. For the theoretical underpinnings it really, really helps to know a little model theory (not to be confused with mathematical modelling). Keith Stroyan has another book of this type that goes a bit beyond freshman calculus. – Brian M. Scott May 23 '12 at 9:50
"Is it more than defining numbers smaller than any positive real" even in ordinary analysis there are lots of numbers smaller than every positive real :) A theory of NSA usually comes equipped with a notion of "standard", and the "infinitesimals" in that theory are those non-zero numbers smaller in absolute value than every standard number. I like Robert's book "Nonstandard analysis" as an introduction to the IST approach. – m_t_ May 23 '12 at 12:23
There is actually a companion book to the book mentioned by @BrianM.Scott and that book contains some material on the logical underpinnings in the first and in the last chapter. – Michael Greinecker Sep 9 '12 at 13:14
up vote 6 down vote accepted

The term "nonstandard" refers to nonstandard analysis using a nonstandard model. In nonstandard analysis, one usually intends to study the real numbers (and their functions and relations). The real numbers under certain operations model a field. In nonstandard analysis, the intended model consists of the real numbers, while the model used for study of the real numbers usually consists of a hyperreal field. Hyperreal fields are not isomorphic to the real number field. We can see this just by looking at finite hyperreal structures, that is just finite numbers, infinitesimals, and numbers infinitely close to finite numbers.

Given a hyperreal field with only finite elements, we do have a homomorphism from those finite hyperreals to the reals, at least once we take care of division. So, in the finite case, we can regard the "standard part" or "shadow" of a hyperreal number as a function, but we cannot correctly obtain an inverse function from the reals to the finite hyperreals. We also have a homomorphism from basically any structure of finite hyperreals to a given real structure.

For example, if we consider $(\mathbf{H}, +', *', -', /')$, that is the finite hyperreals under hyperreal addition, multiplication, subtraction, and division (where the denominator is not an infinitesimal) we have a map $\DeclareMathOperator{\SH}{SH}\SH$ to $(\mathbf{R}, +, *, -, /)$ such that

$$\begin{array}{} \SH(h&+'&i)&=&(\SH(h)&+&\SH(i))\\ \SH(h&*'&i)&=&(\SH(h)&*&\SH(i))\\ \SH(h&-'&i)&=&(\SH(h)&-&\SH(i))\\ \SH(h&/'&i)&=&(\SH(h)&/&\SH(i))\\ \end{array}$$

and for all finite hyperreal numbers $h$, there exists a real number $r$ such that $\SH(h)=r$.

No inverse of the shadow function from the reals to the finite hyperreals exists, or in other words no function $I$ exists such that $I(\SH(h))=h$ holds true. This comes perhaps as easiest to see if you consider that the shadow of each infinitesimal number equals $0$. But, $0$ does not associate with a unique hyperreal infinitesimal, since we could associate $0$ with one of an infinity of positive infinitesimals, or one of an infinity of negative infinitesimals in rather the same way.

See Pete L. Clark's last two comments below also.

Also, the Dedekind completeness property fails for the hyperreals, which indicates the hyperreals as a non-isomorphic structure to the reals under various operations. Keisler's text referenced by Brian M. Scott makes for a good reference to read, and it has some good examples.

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What do you mean when you say that a hyperreal field is not isomorphic to the real field by nature? – user23211 May 23 '12 at 14:32
Just to pre-empt any confusion, it's the external Dedekind completeness property that fails for the hyperreals. The hyperreals satisfy the internal version, as they must by the transfer principle. – Hurkyl May 24 '12 at 1:21
-1: There is no homomorphism from the hyperreals (a non-Archimedean ordered field) to the reals (an Archimedean ordered field). Hint: what would the shadow of an infinitely large hyperreal be? – Pete L. Clark Sep 8 '12 at 5:18
@Doug: Your fix is a good one. In any ordered field, the family of finitely large elements -- i.e., the ones which are in absolute value less than $n$ for some $n \in \mathbb{Z}$ -- form a subring of the field, and there is a well-defined ring homomorphism from the finitely large elements to a non-Archimedean field; we identify two elements iff their difference is infinitesimal. – Pete L. Clark Sep 8 '12 at 18:25
Here is a slightly better description of the above homomorphism: in an ordered field $F$, let $R_f$ be the subring of finitely large elements of $F$. Let $\mathfrak{m}_f$ be the subset of infinitesimal elements of $F$. Then $\mathfrak{m}_f$ is a maximal ideal of $R_f$ and the ordering on the ring $R_f$ induces an Archimedean ordering on the field $R_f/\mathfrak{m}_f$. – Pete L. Clark Sep 8 '12 at 18:28

It might help to understand a tiny little bit of the background leading Abraham Robinson to developing his model of nonstandard analysis. There have been several approaches to incorporating infinitesimals into analysis. One such approach was developed by Schmieden and Laugwitz. They considered sequences of real numbers under the equivalence relation where two sequences are equal if from some point onwards all their coordinates agree. This leads to an extension of the real numbers but does not yield a field (but rather a ring with zero divisors). Nonetheless, one can still do quite a lot of analysis in this larger ring which includes proper infinitesimals.

Roughly the same time Skolem had used fundamental results in logic (model theory) to construct models of the natural numbers that included infinitely large natural numbers. These models were not intended to have any applications but rather used for logic oriented investigations. Skolem called these models of the naturals nonstandard models of arithmetic.

Robinson was well aware of the work of Schmieden and Laugwitz and of Skolem's work. Robinson's great contribution was in the realization that the Schmieden-Laugwitz model can be greatly improved utilizing Skolem's techniques. It is thus almost certain that Robinson chose the term nonstandard analysis due to the existing nonstandard models of arithmetic.

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To complement the fine answers given earlier, I would like to address directly the question of the title: "What exactly is nonstandard about Nonstandard Analysis?"

The answer is: "Nothing" (the name "nonstandard analysis" is merely a descriptive title of a field of research, chosen by Robinson). This is why some scholars try to avoid using the term in their publications, preferring to speak of "infinitesimals" or "analysis over the hyperreals", as for example in the following popular books:

Goldblatt, Robert, Lectures on the hyperreals. Graduate Texts in Mathematics, 188. Springer-Verlag, New York, 1998

Vakil, Nader, Real analysis through modern infinitesimals. Encyclopedia of Mathematics and its Applications, 140. Cambridge University Press, Cambridge, 2011.

More specifically, there is nothing "nonstandard" about Robinson's theory in the sense that he is working in a classical framework that a majority of mathematicians work in today, namely the Zermelo-Fraenkel set theory, and relying on classical logic.

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"Nonstandard" is really referring to the nonstandard numbers, including the infinitesimal numbers and hyperreal numbers. Foundationally, it can be shown that if the structure we call "the real numbers" and denote $\mathbb{R}$, exists, then so does another structure, $\mathbb{R}^{*}$ that, in certain useful ways, behaves just like that structure, but has an element $\epsilon$ such that $0<\epsilon < 1/n$ for every positive integer $n$. This new element is called nonstandard, or more specifically infinitesimal, since it is nonzero yet smaller than every positive standard real number.

From the existence of just this one nonstandard element can be proven the existence infinitely many more, even hyperreals. For example, since the sentence $\forall x\in \mathbb{R}, \exists y \in \mathbb{R}[0<y<x \vee x<y<0]$ holds of $\mathbb{R}$, it also holds of $\mathbb{R}^{*}$, so we get infinitely many infinitesimals (after applying that sentence to $x=\epsilon$). Similarly, the sentence $\forall x \forall y [x<y \rightarrow 1/x > 1/y]$, with $x=\epsilon$ and $y=1/n$ shows that $1/\epsilon$ (or the reciprocal of any infinitesimal) is bigger than any standard real; i.e. it is hyperreal. These two sentences are examples of the "useful ways" that $\mathbb{R}^{*}$ behaves like $\mathbb{R}$. (This is really the transfer principle.)

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A gentle introduction would be Wikipedia I guess. They also describe the criticisms of nonstandard analysis there.

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