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first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.

15 votes
1 answer
807 views

Are key theorems finitistically reducible?

Simpson writes on page 378 of his Subsystems of Second Order Arithmetic: "For example, all of the following key theorems of infinitistic mathematics are provable in WKL$_0$ and therefore, by theorem I …
45 votes
6 answers
8k views

Situation with Artemov's paper?

Artemov's paper on Goedel's theorem has been on the arxiv since 2019. There was a (less than fully friendly) discussion of this on FoM. At stackexchange, I found only a brief mention at this MSE pos …
4 votes
0 answers
95 views

Explicit superexponential growth for Presburger Arithmetic

Fischer and Rabin proved a superexponential bound $2^{2^{cn}}$ for the worst-case length of a proof of a proposition of length $n$ in Presburger arithmetic. The result is in Michael J. Fischer and M …
5 votes
1 answer
411 views

Constructive compactness for countable models?

The compactness theorem for countable (Tarski?) models is equivalent to the weak König's lemma by a result of H. Friedman and others as noted here, in the context of classical logic. The weak König's …
4 votes
0 answers
142 views

Can one formalize the prevalence of the Big Five systems of reverse math?

Simpson's systems of second order arithmetic turn out to be five in number; to simplify notation let's denote them A, B, C, D, E. What seems to be an empirical observation is that most theorems in cl …
2 votes
0 answers
113 views

Robinson's views on Heyting's work?

Abraham Robinson and Arend Heyting had mutual respect (though holding differing philosophical views on the nature of mathematics). Heyting repeatedly expressed admiration for Robinson's work; see for …
4 votes
1 answer
437 views

Paris-Harrington via overspill?

I saw in an old logic paper that the Paris-Harrington theorem can be proved via Overspill. The presentation is unfortunately too technical for me to follow. Does somebody have any insight into this? …
7 votes
4 answers
541 views

A conservative extension of Peano Arithmetic

Ulrich Kohlenbach makes the following intriguing comment here: "In the 70s S. Feferman introduced a mathematically strong system S=restricted(PA^omega)+QF-AC+mu for classical mathematics (and in part …
8 votes
1 answer
586 views

Con(PA) via non-well-foundedness?

Lumsdaine made the following interesting comment: if Con(PA) fails in a non-standard model, it means it contains a “proof of non-standard length” of a contradiction from PA. With a little work, one …
6 votes
5 answers
2k views

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

This is a question about a comment in a recent publication by Roman Kossak. Kossak wrote: "Nonstandardness in set theory has a different nature. In arithmetic, there is one intended object of study …
6 votes
1 answer
350 views

Quantifier complexity of definition of compactness

This question is inspired by the post on quantifier complexity of continuity. We work with metric spaces M considered as two-sorted first-order structures (M,$\mathbb R$,d,+,⋅,<) where $d:M^2→\mathbb …
38 votes
6 answers
3k 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 ab …
35 votes
9 answers
14k 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 see …
7 votes
6 answers
3k views

Looking for a source for Intended Interpretation

Hao Wang writes: "The originally intended, or standard, interpretation takes the ordinary nonnegative integers $\{0, 1, 2, \ldots \}$ as the domain, the symbols $0$ and $1$ as denoting zero and one, a …
2 votes
1 answer
256 views

Transfer with minimal choice

Let FUF postulate the existence of a Free UltraFilter on $\mathbb{N}$ and ACC the axiom of countable choice. Consider the superstructure on $\mathbb{R}$ and its inclusion in the bounded ultrapower. I …

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