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Nonstandard analysis is a way of doing calculus and analysis with infinitesimals. The historical approach of Leibniz, Euler, and others to infinitesimal calculus was gradually replaced by epsilon, delta techniques in the context of a real continuum, in the 19th century. It was not until the 1960s that Abraham Robinson developed a theory of a hyperreal continuum that allows for a development of analysis procedurally akin to that of its founders.

70 votes
Accepted

A remark of Connes on non-standard analysis

...as soon as you have a non-standard number, you get a non-measurable set. Every nonstandard natural number $N$ gives rise to a nonprincipal ultrafilter $U$ on $\mathbb{N}$, by saying that a set $X …
Joel David Hamkins's user avatar
56 votes

How helpful is non-standard analysis?

The other answers are excellent, but let me add a few points. First, with a historical perspective, all the early fundamental theorems of calculus were first proved via methods using infinitesimals, r …
Joel David Hamkins's user avatar
47 votes
4 answers
4k views

Which topological spaces admit a nonstandard metric?

My question is about the concept of nonstandard metric space that would arise from a use of the nonstandard reals R* in place of the usual R-valued metric. That is, let us define that a topological sp …
Joel David Hamkins's user avatar
45 votes
Accepted

Does every series of hyperreal numbers converge to some hyperreal number?

The answer is strongly negative. Arbitrary extensions. The first thing to say is that whenever one extends $\newcommand\R{\mathbb{R}}\R$ to a larger ordered field $F$, one has immediately destroyed (e …
Joel David Hamkins's user avatar
31 votes
Accepted

Is non-existence of the hyperreals consistent with ZF?

The answer is yes, provided ZF itself is consistent. The reason is that the existence of the hyperreals, in a context with the transfer principle, implies that there is a nonprincipal ultrafilter on $ …
Joel David Hamkins's user avatar
31 votes

Why does CH imply that there is a unique ultrapower of $\mathbb{N}$?

The point is that the ultrapower of any structure $\mathcal{M}$ by a nonprincipal ultrafilter $\mu$ on $\mathbb{N}$ is countably saturated, that is, it realizes any finitely satisfiable $n$-type with …
Joel David Hamkins's user avatar
29 votes
Accepted

What are the advantages of the more abstract approaches to nonstandard analysis?

To my way of thinking, there are at least three distinct perspectives one can naturally take on when undertaking work in nonstandard analysis. In addition, each of these perspectives can be varied on …
Joel David Hamkins's user avatar
22 votes
5 answers
1k views

What is the spectrum of possible cofinality types for cuts in an ordered field? Or in a mode...

I am interested to know the full range of possibilities for the cofinality type of cuts in an ordered field and in other structures, such as nonstandard models of arithmetic. Definitions. Specifical …
Joel David Hamkins's user avatar
20 votes
5 answers
2k views

Isomorphism types or structure theory for nonstandard analysis

My question is about nonstandard analysis, and the diverse possibilities for the choice of the nonstandard model R*. Although one hears talk of the nonstandard reals R*, there are of course many non-i …
Joel David Hamkins's user avatar
17 votes
2 answers
2k views

Do the surreal numbers enjoy the transfer principle in ZFC?

The surreal field $\newcommand\No{№}\No$ is definable in ZFC, and it is easy to see that the surreal order is $\kappa$-saturated for every cardinal $\kappa$, precisely because we fill any specified ga …
Joel David Hamkins's user avatar
12 votes

Are hyperreal numbers isomorphic to formal power series?

I would like to point out that it is not true that every every hyperreal can be represented by a Laurent series in the way you describe. (Let me assume that by the term "hyperreals", you mean a nonst …
Joel David Hamkins's user avatar
10 votes

Automorphisms of the hyperreals over the rationals and nontrivial automorphism groups

As a general principle, every first-order theory with infinite models, such as the theory of real-closed fields, will have models with rich automorphism groups. The general reason is that one can ta …
Joel David Hamkins's user avatar
10 votes
Accepted

Surreal numbers, ultrapowers of $\Bbb R$, ordinal-valued functions and the slow-growing hier...

This kind of analysis is very well understood in ultrapowers, and one often sees this kind of thinking with ultrapowers, where one performs calculations with the representing function for an object. W …
Joel David Hamkins's user avatar
9 votes
Accepted

Isomorphisms between non-standard reals.

If the continuum hypothesis holds, then both of these ultrapowers are saturated models of cardinality $\omega_1$, and one can see that they are isomorphic by a back-and-forth argument. When the CH f …
Joel David Hamkins's user avatar
9 votes
Accepted

Metrization of hyperreals

I am not sure whom you are addressing in your question, but some of your remarks relate to issues brought up at this MO question. If not, could you let us know to which post you were referring? It is …
Joel David Hamkins's user avatar

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