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Results tagged with lo.logic
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user 28128
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.
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 …
- 16.6k
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 …
- 16.6k
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? …
- 16.6k
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 …
- 16.6k
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 …
- 16.6k
1
vote
2
answers
290
views
Compactness for countable models?
How and where is it proved that WKL$_0$ proves the compactness theorem for countable models? (This is a follow-up to a comment of F. Dorais.)
- 16.6k
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 $d:M^2→\mathbb …
- 16.6k
21
votes
9
answers
5k
views
Was the early calculus inconsistent?
This question does NOT concern the RIGOR, or lack thereof, of the early calculus. Rather the question is of its CONSISTENCY.
George Berkeley wrote in 1734 with reference to the early calculus that s …
- 16.6k
7
votes
4
answers
541
views
A conservative extension of Peano Arithmetic
Ulrich Kohlenbach makes the following intriguing comment here:
"In the 70s S. Feferman introduced a mathematically strong system S=restricted(PA^omega)+QF-AC+mu for classical mathematics (and in part …
- 16.6k
4
votes
0
answers
142
views
Can one formalize the prevalence of the Big Five systems of reverse math?
Simpson's systems of second order arithmetic turn out to be five in
number; to simplify notation let's denote them A, B, C, D, E. What
seems to be an empirical observation is that most theorems in
cl …
- 16.6k
15
votes
1
answer
807
views
Are key theorems finitistically reducible?
Simpson writes on page 378 of his Subsystems of Second Order
Arithmetic:
"For example, all of the following key theorems of infinitistic
mathematics are provable in WKL$_0$ and therefore, by theorem I …
- 16.6k
3
votes
0
answers
473
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 "standard" na …
- 16.6k
18
votes
2
answers
1k
views
New articles by Errett Bishop on constructive type theory?
Recently two formerly unknown articles by Errett Bishop (1928-1983) were posted online by Martín Escardó. One is entitled "A general language", deals with constructive type theory, and is 28 pages lon …
- 16.6k
5
votes
2
answers
486
views
How is compactness related to countable saturation?
By Cantor's intersection theorem every decreasing nested sequence of nonempty compact sets has a common point.
A superficially similar result holds that every decreasing nested sequence of nonempty …
- 16.6k
17
votes
1
answer
2k
views
What is the precise relationship between o-minimal theory and Grothendieck's "Esquisse d'un ...
I have seen various references in the literature to such a connection but they tend to assume that the reader is familiar with the connection, and limit themselves to providing additional detail. So …
- 16.6k