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 ...

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325 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 ...
4
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1answer
125 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, ...
3
votes
2answers
263 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 ...
1
vote
1answer
172 views

A stronger version of supramenability?

A group $G$ is supramenable iff for all $\varnothing\ne A\subseteq G$ there is a finitely-additive left-$G$-invariant measure $\mu_A$ on $G$ with $\mu_A(A)=1$. I'm interested in a seemingly stronger ...
2
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0answers
342 views

Deducing Skolem's nonstandard integers from downward Lowenheim-Skolem?

If one has a nonstandard model $\mathcal{N}$ of PA and adjoins to the first-order theory the countable list of axioms $1<H,\, 2<H,\, 3<H, \ldots$ (satisfied in $\mathcal{N}$) for all the ...
13
votes
2answers
349 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
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1answer
516 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 ...
5
votes
1answer
641 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 ...
13
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1answer
1k 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 ...
9
votes
4answers
470 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 ...
2
votes
3answers
689 views

Lapses of “the early proponents of the doctrine of limits”

I have a question that I have been wondering about for a long time without finding any answer. Concerning the period around 1900, Robinson commented in his 1966 book that "there is in the writings of ...
2
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1answer
262 views

differential geometry using Robinson's infinitesimals?

Is there a detailed treatment of differential geometry using Robinson's infinitesimals?
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9answers
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 ...
1
vote
1answer
96 views

Nonstandard definition for the generator of a standard Ito diffusion

For a standard Brownian motion, the generator of the diffusion is $$ L = \frac12 \frac{d^2}{dx^2}. $$ Is there a nonstandard definition of this generator?
5
votes
0answers
134 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 ...
5
votes
3answers
243 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 ...
4
votes
1answer
205 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 ...
2
votes
1answer
106 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 ...
3
votes
2answers
164 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 ...
7
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2answers
318 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 ...
19
votes
4answers
2k 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 ...
5
votes
1answer
185 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 ...
9
votes
1answer
911 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 ...
13
votes
4answers
960 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 ...
16
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2answers
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 ...
4
votes
4answers
906 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 ...
9
votes
2answers
1k 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 ...
11
votes
1answer
655 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 $P({\mathbb F}^{n+1})$ ...
19
votes
4answers
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 ...
4
votes
2answers
762 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 ...
10
votes
4answers
654 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. ...
29
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1answer
1k 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 ...
31
votes
2answers
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 ...
10
votes
6answers
2k 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 ...
27
votes
6answers
4k 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 ...
3
votes
1answer
293 views

Non Separability of the the Loeb Space

Let $X_i$, $i=1,2\ldots$ be finite sets of integers such that $|X_i|\rightarrow \infty$. Let $X$ be an ultraproduct of $X_i$. Let $\mu$ the Loeb measure associated with the the normalized counting ...
10
votes
1answer
549 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 ...
5
votes
1answer
686 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 ...
1
vote
1answer
525 views

Metrization of hyperreals

Hello, i was reading your article about non metrizability of *R. i was able to prove that the interval open topology is not metrizable by proving that the intersection of decreasing hyper-intervals ...
6
votes
9answers
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' ( ...
9
votes
9answers
2k 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 ...
33
votes
16answers
4k 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 ...
32
votes
4answers
2k 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 ...
16
votes
5answers
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 ...