**41**

votes

**16**answers

6k views

### How helpful is non-standard analysis?

So, I can understand how non-standard analysis is better than standard analysis in that some proofs become simplified, and infinitesimals are somehow more intuitive to grasp than epsilon-delta ...

**35**

votes

**1**answer

2k views

### Various flavours of infinitesimals

I'm not sure if this is a soft question, and whether it may be too broad or, on the contrary, too localized. Well, in Mathematics the concept of "infinitesimal" has been of extreme importance for ...

**34**

votes

**4**answers

3k 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 ...

**32**

votes

**2**answers

2k views

### Ergodic Theorem and Nonstandard Analysis

Here is a quote from Lectures on Ergodic Theory by Halmos:
I cannot resist the temptation of
concluding these comments with an
alternative "proof" of the ergodic
theorem. If $f$ is a complex ...

**29**

votes

**6**answers

5k views

### A remark of Connes

In an interview (at http://www.alainconnes.org/docs/Inteng.pdf) Connes remarks that
I had been working on non-standard analysis, but after a while I had found a catch in the theory.... The point ...

**24**

votes

**5**answers

3k views

### Nonstandard analysis in probability theory

I am quite new at nonstandard analysis, and recently I became aware of its use in probability theory mainly through the following two books:
Nelson (1987). Radically Elementary Probability Theory
...

**23**

votes

**4**answers

2k views

### In what ways did Leibniz's philosophy foresee modern mathematics?

Leibniz was a noted polymath who was deeply interested in philosophy as well as mathematics, among other things. From my mathematical readings I have the impression that Leibniz's stature as a ...

**23**

votes

**5**answers

1k views

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

This question does not concern the comparative merits of standard (SA) and nonstandard (NSA) analysis but rather a comparison of different approaches to NSA. What are the concrete advantages of the ...

**22**

votes

**9**answers

3k views

### What is… A Grossone?

Y. Sergeyev developed a positional system for representing infinite numbers using a basic unit called a "grossone", as well as what he calls an "infinity computer". The mathematical value of this ...

**20**

votes

**9**answers

3k views

### Was the early calculus inconsistent?

This question does NOT concern the RIGOR, or lack thereof, of the early calculus. Rather the question is of its CONSISTENCY.
George Berkeley wrote in 1734 with reference to the early calculus that ...

**20**

votes

**2**answers

2k views

### Euler's mathematics in terms of modern theories?

Some aspects of Euler's work were formalized in terms of modern infinitesimal theories by Laugwitz, McKinzie, Tuckey, and others. Referring to the latter, G. Ferraro claims that "one can see in ...

**18**

votes

**5**answers

1k 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 ...

**17**

votes

**5**answers

878 views

### What is the spectrum of possible cofinality types for cuts in an ordered field? Or in a model of the hyperreals? Or in a nonstandard model of arithmetic?

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. ...

**17**

votes

**2**answers

1k views

### Is non-existence of the hyperreals consistent with ZF?

I know that it is possible to construct the hyperreal number system in ZFC by using the axiom of choice to obtain a non-principal ultrafilter. Would the non-existence of a set of hyperreals be ...

**17**

votes

**2**answers

2k views

### Surreal numbers vs. non-standard analysis

What is the relationship between the surreal numbers and non-standard analysis?
In particular, is there a transfer principle for surreal numbers they way there is for NSA?
A specific situation in ...

**16**

votes

**2**answers

1k views

### Non standard Algebraic Topology

Let *$\mathbb R$ a field of non-standard real numbers (or any real closed field) equipped with its natural generalized metric $d(x,y)=|x-y|$. Equip *$\mathbb R^2$ and *$\mathbb R^3$ with the ...

**15**

votes

**10**answers

3k views

### nonstandard analysis book recommendation

I wish to learn nonstandard analysis. Are there any good book recommendations? I'm familiar with the ZFC system, and learnt analysis the classical way. I've found some undergraduate texts, but they ...

**14**

votes

**4**answers

1k views

### Non-ZFC set theory and nonuniqueness of the hyperreals: problem solved?

The reals are the unique complete ordered field. The hyperreals $\mathbb{R}^\*$ are not unique in ZFC, and many people seemed to think this was a serious objection to them. Abraham Robinson responded ...

**13**

votes

**2**answers

517 views

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

I've read these words: "How many ultra products $∏_Uℕ$ exist up to isomorphism, where $U$ is a non-principal ultrafilter over $ℕ$? If continuum hypothesis(CH) holds, then obviously just one ..."
i ...

**12**

votes

**1**answer

692 views

### Can nonstandard analysis be used to prove results in constructive or computable analysis?

Nonstandard analysis is a useful tool which can be used to prove a number of results in analysis.
Question
Can it also be used to prove results in computable or constructive analysis?
If ...

**11**

votes

**3**answers

683 views

### Bibliographic request concerning an article by Bernstein and Robinson

Concerning the article "Bernstein, Allen R.; Robinson, Abraham.
Solution of an invariant subspace problem of K. T. Smith and
P. R. Halmos. Pacific J. Math. 16 1966 421-431" I am interested in
finding ...

**11**

votes

**1**answer

716 views

### Non standard algebraic geometry: shadows of varieties

In this question $\mathbb F$ is a field and $P({\mathbb F}^{n+1})$ is the projective space of dimension $n$ over $\mathbb F$. The term algebraic variety means a subset of $P({\mathbb F}^{n+1})$ ...

**10**

votes

**8**answers

3k views

### Leibnizian calculus textbook

Where can I find a calculus textbook that emphasizes differentials?
Is there such a book that I could realistically require my calculus students to use?
I want a textbook that supports me when I tell ...

**10**

votes

**1**answer

631 views

### Non-standard enlargements, $\zeta(s)$ and analytic continuation

Consider an extension of the Riemann zeta function $\zeta(s)$ where $s$ now runs over a non-standard enlargement of the complex plane.
Observe that if $s=\sigma + it$ with $\sigma>1$ real and ...

**9**

votes

**4**answers

586 views

### dense orders are saturated

In a FOM msg of Fri May 21 19:59:44 EDT 1999,
http://www.cs.nyu.edu/pipermail/fom/1999-May/003149.html
Simpson gave a short proof of the following theorem:
Theorem. Let F be a real closed ordered ...

**9**

votes

**3**answers

412 views

### Is a model of arithmetic contained in a model of arithmetic an initial segment?

It's easy enough to show that if $\mathbb{N}_1$ is a non-standard model of the Peano axioms, then there is a canonical embedding $\mathbb{N} \to \mathbb{N}_1$, and we have a theorem that if $x \in ...

**9**

votes

**2**answers

2k views

### Defining the slowest divergent series

This question might seem too fuzzy, and if so, I will be happy to withdraw it. Until then, here it is:
I know that a method of slowing a divergent series of positive reals is to replace the $n$-th ...

**9**

votes

**1**answer

310 views

### Pontryagin dual of the surreal numbers?

Has any work been done on the Pontryagin dual of the surreal numbers (suitably topologized)? I have not been able to find anything and am not sure if this is still unknown.
Alternatively, has this ...

**9**

votes

**1**answer

395 views

### Are there any standard analysis facts that can be proven or arrived only by means of non-archimedean extensions of reals and non-standard analysis?

I wonder whether non-standard analysis, non-archimedean extensions of reals such as surreal or hyperreal numbers can help us in obtaining standard-analytic results which are not possible to obtain by ...

**9**

votes

**1**answer

239 views

### Literature that helps explain what the theory of numerosities contributes with

Since 2003 a group of Italian mathematicians (Benci, Di Nasso and Forti) has developed a new measure for infinite sets that satisfies the Euclidian principle: The whole is greater than the part. The ...

**8**

votes

**2**answers

532 views

### differential geometry using Robinson's infinitesimals?

Is there a detailed treatment of differential geometry using Robinson's infinitesimals?

**8**

votes

**2**answers

344 views

### Hyperreal finitely-additive measure on [0,1) assigning $b-a$ to $[a,b)$ or $(a,b]$ and infinitesimals to singletons

Is there a hyperreal-valued finitely additive measure on all the subsets of [0,1), or at least the Borel ones, that
assigns $b-a$ to $[a,b)$ and to $(a,b]$ for all $a\lt b,$ and
assigns an ...

**8**

votes

**1**answer

416 views

### Who said “the naive counting numbers don't exhaust $\Bbb N$”?

In the context of Robinson's framework, or more precisely its reformulation by Ed Nelson, one of the practitioners in the field expressed the sentiment something like "the naive counting numbers don't ...

**7**

votes

**9**answers

3k views

### Would Euler's proofs get published in a modern math Journal, especially considering his treatment of the Infinite?

I was wondering how mathematicians of today would treat, for example, Euler's proof of zeta(2).
In William Dunham's book 'Journey through Genius' ( ...

**7**

votes

**2**answers

629 views

### Survey of the history of calculus?

Boyer 1939 is a nice readable survey of the history of the calculus, but it's showing its age. Discussing the notion of instantaneous velocity, he has:
Mathematics knows no minimum interval of ...

**7**

votes

**1**answer

388 views

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

Philip Ehrlich's paper “The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small”, The Bulletin of Symbolic Logic 18 (1) 2012, pp. 1-45. claims as a theorem that, in NBG, ...

**7**

votes

**2**answers

2k views

### Which universities teach true infinitesimal calculus? [closed]

My colleague and I are currently teaching "true infinitesimal calculus" (TIC), in the sense of calculus with infinitesimals, to a class of about 120 freshmen at our university, based on the book by ...

**6**

votes

**2**answers

971 views

### Are all countable, nonstandard models of arithmetic given by ultrapowers?

Countable models of PA fall into two categories: the standard one $(\omega, S)$ and the nonstandard ones (all the rest). The only way I've seen to construct a nonstandard model is through taking an ...

**6**

votes

**2**answers

405 views

### “Lebesgue-measurable” cardinals and real-closed fields

I understand the motivation behind measurable cardinals is to ask the question: "is there any set large enough to admit a non-trivial measure on all of its subsets?"
Hence, it's also worthwhile to ...

**6**

votes

**1**answer

900 views

### Uncountable nonstandard models of PA

Standard techniques (no pun intended) can be used to show that countable nonstandard models of Peano Arithmetic are order isomorphic to $\mathbb{N} + \mathbb{Z} \cdot \mathbb{Q}$. Once we have used ...

**5**

votes

**4**answers

974 views

### Nonstandard Methods ( or Model Theory ) and Arithmetic Geometry

I hear that the nonstandard methods are applied to many problems in various fields of mathematics such as functional analysis, topology, probability theory and so on.
So, I have become interested in ...

**5**

votes

**1**answer

212 views

### Isomorphisms between non-standard reals.

Let $U$ and $V$ be two non-principal ultrafilters over N, and $R_1$ and $R_2$ the non-standard extensions of R given by $R_1=R^N/U$ and $R_2=R^N/V$. Are they always isomorphic (I think not, but could ...

**5**

votes

**2**answers

861 views

### Salvaging Leibnizian formalism?

Can one justify Leibniz's formalism in a suitable algebraic or topological context?
We have published some papers recently where we argue that Leibniz's formalism for the calculus wasn't ...

**5**

votes

**3**answers

285 views

### Translation of a non-standard analysis formulation

Usually, it is quite easy (if cumbersome) to translate a formula like "if $\varepsilon$ is infinitesimal, and if $f$ is differentiable at $a$, $f'(a)$ is the shadow of ...

**5**

votes

**1**answer

825 views

### Surreals and NSA: some foundational issues

Surreals and NSA: some foundational issues.
A.
Leaving aside the whole internal machinery of surreals (with funny questions like is $\omega$ an entire number and if yes is it odd or even, simple, a ...

**5**

votes

**0**answers

175 views

### Constructing black noise with non-standard analysis

With noise in the sense of i.i.d. random sequence,
a noise is black if it is not isomorphic to standard Gaussian white noise.
Tsirelson showed the existence of black noise through the scaling limit ...

**4**

votes

**1**answer

235 views

### tennenbaum phenomena for the reals?

Let $\mathfrak{M} = \langle R, +,\times,> \rangle$ be such that $R$ is the set of real numbers and $\mathfrak{M} \models RA^1$ (the first-order axioms for the reals). Do we have characterisations ...

**3**

votes

**2**answers

315 views

### Is there a source linking Robinson's work in wing theory with his theory of infinitesimals?

Abraham Robinson worked in applied mathematics for several decades. MathSciNet lists 12 articles by Robinson in wing theory. His production included the book
Robinson, A.; Laurmann, J. A. Wing ...

**3**

votes

**2**answers

201 views

### Hyperfinite set containing the reals, with specified upper bound on internal cardinality?

Is this true? For any hyperfinite $n$ that isn't finite, there is a hyperfinite set $A$ such that $\mathbb R \subset A$ and $|A|\le n$ (that's the crucial part, of course)? Intuitively it seems ...

**3**

votes

**1**answer

120 views

### Star-transfer of powerset

What is the difference between ${^\sigma}\mathcal{P}(\mathbb{R})$, ${^\ast}\mathcal{P}(\mathbb{R})$, and $\mathcal{P}({^\ast}\mathbb{R})$? I know that $\mathcal{P}({^\ast}\mathbb{R})$ is the powerset ...