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Results tagged with mathematical-philosophy
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user 4600
Philosophical aspects of logic and set theory; truth status of mathematical axioms; Philosophy of Mathematics; philosophical aspects of mathematics in general; relation of mathematics to philosophy; etc. Consider also posting at http://philosophy.stackexchange.com/, where philosophy-of-mathematics is one of the most popular tags.
21
votes
Accepted
Question arising from Voevodsky's talk on inconsistency
Let $S$ be a first order definable Martin-Löf random set such as Chaitin's $\Omega$. If Peano Arithmetic, or ZFC, or any other theory with a computable set of axioms, proves infinitely many facts of t …
- 24.8k
4
votes
Gödel's ontological proof & Benzmüller's work
I know Benzmüller's work from a slightly different context, formalizing other interesting systems of modal logic. Hadn't heard of this specific project, but it seems very admirable.
Formalization of …
- 24.8k
13
votes
Interesting meta-meta-mathematical theorems?
You could let $\alpha_0$ be the statement Con(ZFC), and $\alpha_{n+1}$ be ZFC $\not\vdash\alpha_n$, and at limit ordinals $\alpha_\lambda$ is $(\forall \beta<\lambda)($ZFC $\not\vdash \alpha_\beta)$. …
- 24.8k
18
votes
Why is this new result such a big deal?
They show that $\DeclareMathOperator{\WKL}{WKL}\DeclareMathOperator{\RT}{RT}\DeclareMathOperator{\RCA}{RCA} \RT^2_2$ is $\Pi^0_3$-conservative over $\RCA_0$. Thus, there is no way that $\RT^2_2$ can b …
- 24.8k
4
votes
Can we define an "empirically generic" real number?
It sounds like you are talking about what in computability theory and set theory are known as Cohen generic reals (the lowest level of which in computability theory is 1-generic, then 2-generic and so …
- 24.8k
5
votes
What was Hilbert's view of Gödel's Incompleteness Theorems?
Here's the Logicomix account of Hilbert's reaction.
Perhaps a reference to Hilbert's Hotel?
- 24.8k