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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
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 …
- 16.6k
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 …
- 16.6k
22
votes
2
answers
2k
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 operat …
- 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
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
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
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
13
votes
6
answers
2k
views
Intuitionistic logic as quantization of classical logic?
A classically trained mathematician is more likely to be familiar (at least anecdotally) with an area of mathematical physics such as deformation quantization than with intuitionistic logic. It is he …
- 16.6k
8
votes
0
answers
1k
views
What's Reeb's take on naive integers?
Georges Reeb's "claim Q" is the statement that "naive integers don't fill up $\mathbb{N}$". To anyone familiar with model theory this could easily be interpreted as the existence of nonstandard models …
- 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
7
votes
2
answers
1k
views
Salvaging Leibnizian formalism?
Can one justify Leibniz's formalism in a suitable algebraic or topological context?
We have published some papers recently where we argue that Leibniz's formalism for the calculus wasn't inconsisten …
- 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
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 …
- 16.6k
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 …
- 16.6k