20 votes

Was Cauchy prescient?

I found this paper by John Cleave, Cauchy, Convergence, and Continuity (1971) quite illuminating. According to our present-day (Weierstrassian) conception of the continuum, Cauchy's 1821 theorem ...
20 votes

Was Cauchy prescient?

After having read Katz' article, I must say I am not convinced and find that the standard interpretation, namely that of Cauchy making a mistake in 1821 and failing to acknowledging it or correcting ...
18 votes
Accepted

SPOT as a conservative extension of Zermelo–Fraenkel

In plain terms, the conservativity of SPOT over ZF means that if a particular statement S in the language of ZF is provable in SPOT, then ZF can already prove S (with a possibly different proof). Note ...
Ali Enayat's user avatar
  • 17.1k
13 votes
Accepted

Can nonstandard fields contain $\mathbb R$ in different ways?

Yes, this is possible. Let $F$ be a nonarchimedean strongly $\omega$-homogeneous real-closed field such that $\mathbb R\subseteq F$ (which exists by model-theoretic general reasons). Fix an ...
Emil Jeřábek's user avatar
13 votes
Accepted

Decidability of a first-order theory of hyperreals

Yes, the theory is decidable. If $F$ is an ordered field and $R\subseteq F$ a non-cofinal subfield, then $$O=\{x\in F:\exists u\in R\:(-u\le x\le u)\}$$ is a convex valuation ring of $F$, with maximal ...
Emil Jeřábek's user avatar
13 votes
Accepted

Is "All hyperreal fields $C(\mathbb{R})/M$ are isomorphic" independent of ZFC+$\lnot$CH?

In On ultra powers of Boolean algebras (Topology Proceedings 9 (1984) 269-291) Alan Dow proved (Corollary 2.3) that $\neg\mathsf{CH}$ implies there are two fields of the form $C(\mathbb{N})/M$ that ...
KP Hart's user avatar
  • 9,900
12 votes

What is... a grossone?

I would like to summarize some findings concerning the mathematics of Sergeyev's grossone. (1) Sergeyev's writing seems to contain confusion between the notions of ordinal and a cardinal numbers. ...
Mikhail Katz's user avatar
  • 15.1k
12 votes
Accepted

Has anything (other than what is in the obituary written by M. Noether) survived of Paul Gordan's defense of infinitesimals?

Paul Gordan's theses were published in De linea geodetica and digitised by Google, from which I reproduce the relevant page: Translation: I. The method of functional division, proposed by the ...
Carlo Beenakker's user avatar
12 votes

SPOT as a conservative extension of Zermelo–Fraenkel

Ali's answer contains all the required technical explanation. My answer is more "sociological background " in nature: For various reasons, there have been rather strong negative reactions ...
Sam Sanders's user avatar
  • 3,969
11 votes

Automorphisms of the hyperreals over the rationals and nontrivial automorphism groups

This note complement Joel's, who pointed out that every first order theory with an infinite model has a model with many automorphisms (this was first proved by Ehrenfeucht and Mostowski in their ...
Ali Enayat's user avatar
  • 17.1k
11 votes

Did Lagrange change his mind about infinitesimals?

Gert Schubring's Conflicts Between Generalization, Rigor, and Intuition, page 397 and following, gives a critical assessment of this issue:
Carlo Beenakker's user avatar
11 votes
Accepted

What is the Turing degree associated with an ultrafilter $U$?

Great question! This is something that Uri Andrews, Mingzhong Cai, David Diamondstone, and I looked at in a recent (still unpublished) paper. First of all, let's note that there's an important ...
Noah Schweber's user avatar
11 votes

Interpreting Conway's remark about using the surreals for non-standard analysis

I will provide some arguments why analysis is best developed using Nonstandard Analysis as opposed to any approach based on the surreals. If you are interested in the constructive/computable content ...
Sam Sanders's user avatar
  • 3,969
11 votes

Standard models of N and R: An Alice/Bob approach

At the request of Mikhail, I am turning a comment of mine into a partial answer, even though what I am going to write down are well-known, presumably by Mikhail as well. While I am not going to ...
Burak's user avatar
  • 4,135
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 ...
Joel David Hamkins's user avatar
9 votes

What is... a grossone?

Mathematics is the search of truth by way of proof, as defined by Mac Lane. Mathematics is not a collection of opinions or hyphotheses about what someone has meant. Sergeyev defines his grossone ...
Semen Kutateladze's user avatar
9 votes

Standard models of N and R: An Alice/Bob approach

Like Burak, I am responding to the OP's request to promote my comments to an answer, with the caveat that I want to avoid wading too deeply into philosophical debates that I think are beyond the scope ...
Timothy Chow's user avatar
  • 78.3k
8 votes

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

You can quote me on this if you like. This is such an old issue that I am surprised it is still up for discussion. I know you like ultra-powers, etc, but I have always thought they were wrong-...
Larry Manevitz's user avatar
8 votes
Accepted

Are the definable hyper-reals, using quantifiers only over the standard reals and natural numbers, the same as the algebraic numbers?

$\newcommand{\st}{\textrm{st}}\newcommand{\bR}{{\bf R}}\newcommand{\bN}{{\bf N}}$ I think the limit of any definable convergent sequence $(a_n)$ (including $e$) is definable by the formula $$\varphi(x)...
tomasz's user avatar
  • 1,216
8 votes

Cofinality of infinitesimals

As pointed out in a comment by James Hanson, the cofinality of the infinitesimals is the same as the coinitiality (i.e., cofinality or the reverse order) $\mu$ of the nonstandard part of $\omega^\...
Andreas Blass's user avatar
8 votes
Accepted

Interpreting Conway's remark about using the surreals for non-standard analysis

Conway was of course correct in saying that NSA is irrelevant to the surreals, but like Mike I found Conway’s further remarks about NSA puzzling and I am not sure what he had in mind. What I think ...
Philip Ehrlich's user avatar
8 votes
Accepted

Transfinitely iterating the Levi-Civita, Hahn or Puiseux constructions

Let us work in NBG set theory with global choice. There is, up to non unique isomorphism, a unique real-closed field that is $\kappa$-saturated for all infinite cardinals $\kappa$. Let's denote it by $...
nombre's user avatar
  • 2,367
8 votes
Accepted

Is there a constructive version of internal set theory?

As you have proven (by a well-known construction), one cannot expect to have full Transfer in constructive NSA. For different but related reasons, full Standardisation is off the table, though its ...
Sam Sanders's user avatar
  • 3,969
8 votes

Quantifier complexity of definition of compactness

Often the way you prove that something isn't formalizable in first-order logic is (ironically enough) with a compactness proof. This is how you show, for instance, that there isn't a first-order ...
James Hanson's user avatar
  • 10.3k
7 votes

Did Bishop make those comments in his oral presentation?

We were able to obtain an audio file of Bishop's talk from the American Academy of Arts and Sciences. Our analysis will be published in Historia Mathematica and is available on the arxiv. We ...
Mikhail Katz's user avatar
  • 15.1k
7 votes
Accepted

On a completeness property of hyperreals

This is also called Cauchy-completeness, and it coincides for non-Archimedean ordered fields with the natural valuation to the valuation-theoretic notion of completeness. Also, this is the same as ...
nombre's user avatar
  • 2,367
7 votes

Realization of $\mathbb{R}((X))$ as a subquotient of a hyperreal field ${}^{*}\mathbb{R}$

Let $A$ denote the convex subring of hyperreal numbers $y$ for which there exists an $n \in \mathbb{N}$ with $-\varepsilon^{-n}<y<\varepsilon^{-n}$. This has the set $\mathfrak{m}$ of numbers $z$...
nombre's user avatar
  • 2,367
6 votes

differential geometry using Robinson's infinitesimals?

Somewhat belatedly we developed foundations for differential geometry using infinitesimal displacements here: Nowik, T.; Katz, M. "Differential geometry via infinitesimal displacements." Journal of ...
Mikhail Katz's user avatar
  • 15.1k
6 votes

How is compactness related to countable saturation?

As Todd Trimble pointed out in the comments, the use of the term "compactness" in the Compactness Theorem does refer to the topological notion: the Stone space of the Lindenbaum algebra of a theory is ...
François G. Dorais's user avatar
6 votes

Can $\mathsf{RCA}_0$ prove that every nonempty c.e. set $A \subseteq \mathbb{N}$ has a least element?

As Emil Jeřábek and James Hanson mentioned in comments, this is well-known in the literature of first-order arithmetic as the $\Sigma_1$ least number principle, $\mathsf{L}\Sigma_1$. Simpson doesn't ...
Jordan Mitchell Barrett's user avatar

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