Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with lo.logic
Search options answers only
not deleted
user 3106
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.
96
votes
What if Current Foundations of Mathematics are Inconsistent?
Contrary to popular opinion, there is no single foundation for mathematics. Probably you're referring to ZF or ZFC, but most mathematics can be developed on the basis of axioms that are logically muc …
91
votes
The unification of Mathematics via Topos Theory
Topos theory provides a dictionary between (certain areas of) logic and (certain areas of) geometry. As such, it provides all the benefits that mathematical dictionaries do: It lets you translate bet …
- 82.7k
69
votes
What does it mean to suspect that two conjectures are logically equivalent?
Noah Schweber's answer is in some sense the "right" answer, and I would have said something similar if he hadn't beaten me to it. However, I think that it's worth pointing out that reverse mathematic …
- 82.7k
68
votes
What are some reasonable-sounding statements that are independent of ZFC?
Harvey Friedman has devoted a large portion of his career to finding "natural" statements that are unprovable in ZFC. One example is given at the end of Martin Davis's article "The incompleteness the …
68
votes
Situation with Artemov's paper?
The essential issue is the same as one that has been discussed many times here on MO, for example here and here. Consider the following string $S$.
$$(\exists x \exists y \exists z : xxx + yyy - zzz = …
- 82.7k
57
votes
Can we prove set theory is consistent?
Your question certainly makes sense and it is a point that I feel is too often glossed over in textbooks.
Let me rephrase your question. Goedel's second theorem says that, assuming that a certain fo …
- 82.7k
55
votes
Category theory and set theory: just a different language, or different foundation of mathem...
I think that Penelope Maddy's article What Do We Want a Foundation to Do? is a good starting point if you want to read some literature. I don't agree with all of Maddy's conclusions but the terminolo …
- 82.7k
49
votes
Is PA consistent? do we know it?
EDIT: I have written a paper that greatly expands on my answer here, and that in particular contains sketches of Gentzen's proof and Friedman's proof, as well as a discussion of formalism.
I have alr …
47
votes
In what respect are univalent foundations "better" than set theory?
This is a question that has been discussed a lot on the Foundations of Mathematics mailing list (unfortunately with more polemics than necessary IMO—though I confess that I may have been guilty of sto …
46
votes
Bourbaki's definition of the number 1
These calculations have been carried out by José Grimm; see [1] as well as [2].
According to one version of the formalism in the original Bourbaki, Grimm gets
$$16420314314806459564661629306079999627 …
- 82.7k
46
votes
Logic in mathematics and philosophy
I agree with the commentators that the question is rather too broad, but here's an attempt to answer it anyway.
Readers of MO will likely have less familiarity with non-mathematical logic, so it might …
- 82.7k
45
votes
Accepted
Automatically solving olympiad geometry problems
Arguably, the so-called "area method" of Chou, Gao and Zhang represents the state of the art in the field of machine proofs of Olympiad-style geometry problems. Their book Machine Proofs in Geometry …
- 82.7k
42
votes
Accepted
Given a polynomial-time algorithm, can we compute an explicit polynomial time bound just fro...
[Edit: A bug in the original proof has been fixed, thanks to a comment by Francois Dorais.]
The answer is no. This kind of thing can be proved by what I call a "gas tank" argument. First enumerate …
- 82.7k
42
votes
Knuth's intuition that Goldbach might be unprovable
Others have mentioned several theorems (Goodstein, Paris-Harrington, Robertson-Seymour, Kruskal) that are unprovable from the axioms of first-order Peano arithmetic. However, these theorems are prova …
- 82.7k
40
votes
Top-down mathematics, or "Where it all begins"
One approach, mentioned by Pace Nielsen in the comments, is to start with what I call strict formalism. The only substantive assumption required for strict formalism is that you are capable of recogni …
- 82.7k