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Results tagged with lo.logic
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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.
15
votes
Examples of mathematical theories that are naturally written in exotic logics
It is up for debate how "natural" their approach is, but Smullyan and Fitting use modal logic to develop forcing, in their book, Set Theory and the Continuum Problem. See Michael Weiss's notes for a d …
- 82.7k
5
votes
Why should I believe Martin's Maximum++?
Although it's not really a scholarly publication, the Quanta Magazine article, To Settle Infinity Dispute, a New Law of Logic, gives a good introduction to the topic. That article mentions a conferenc …
- 82.7k
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 $S$.
$$(\exists x \exists y \exists z : xxx + yyy - zzz = …
- 82.7k
7
votes
Can the Riemann hypothesis be undecidable?
This is more of a comment than an answer, but I think it is important to point out that the book Equivalents of the Riemann Hypothesis: Volume 3 by Kevin Broughan (Cambridge University Press, 2023), a …
- 82.7k
7
votes
What's the earliest result (outside of logic) that cannot be proven constructively?
A somewhat different type of example, not as early as the ones in Andrej Bauer's answer, but perhaps a bit more resistant to "moving the goalposts," is an ineffective result in number theory.
For exam …
- 82.7k
10
votes
In what sense does the sentence $\operatorname{con}(\mathsf{PA})$ "say" that $\mathsf{PA}$ i...
There is really nothing peculiar about Con(PA) in this regard. Let's take a simpler statement, such as
$$(\exists x \exists y \exists z : xxx + yyy - zzz = 114) \vee (\exists x \exists y \exists z : …
- 82.7k
15
votes
Why is an internal proof of consistency satisfactory for some systems?
The answer by user57888 is correct, but let me emphasize two things. The first is that much of the interest in this type of question predates Gödel's theorems. So if you want to understand the origina …
- 82.7k
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 …
- 82.7k
7
votes
Accepted
MIP*=RE theorem and its impact on logic and proof theory
You wrote:
maybe there is some undecidable problem on which now we can shed some more light …
Depending on what you mean by "shed some more light," the answer is yes; the original paper already expl …
- 82.7k
6
votes
Accepted
What is lost in General Relativity without Hahn-Banach axiom in the ZF+HB set theory?
As Ryan Budney mentioned in a comment, there is some ambiguity about what exactly you mean by "general relativity." General relativity is primarily a physical theory rather than a mathematical theory. …
- 82.7k
13
votes
Are there any undecidability results that are not known to have a diagonal argument proof?
[Edited slightly for (hopefully!) greater clarity.]
This is more of a comment than an answer, but I think it is relevant. In the context of computational complexity theory (rather than computability …
- 82.7k
5
votes
An overview of mathematical-logical approaches in formalizing natural languages
This is a really a question about linguistics rather than mathematics, so you might be better off asking your question on Linguistics StackExchange rather than here. But anyway, I think what you're a …
- 82.7k
8
votes
Real reverse mathematics
Regarding your question of why we don't just take $\Phi_M$ to be the foundation, there's a well-known "inexhaustibility" difficulty coming from Gödel's theorem. I quote from the first page of Torkel …
- 82.7k
13
votes
Does the original 1931 proof of Gödel’s incompleteness rely on the completeness theorem, or ...
Certainly the incompleteness theorem can be proved purely syntactically and without reference to the completeness theorem. There is another MO question with more details.
I haven't worked through the …
- 82.7k
12
votes
Causality, if any, in mathematics itself
The short answer, as you surely suspected, is that there is no rigorous notion of the type you're asking for, that would allow us to say that (in a particular case) that $X$ definitely causes $Y$, or …
- 82.7k