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 ...

Community wiki

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 ...

Community wiki

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 ...

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 ...

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 ...

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 ...

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. ...

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 ...

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 ...

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 ...

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:

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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-...

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)...

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^\...

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 ...

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 $...

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 ...

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 ...

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 ...

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 ...

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 $z$...

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 ...

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 ...

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 ...

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