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Results tagged with set-theory
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user 8991
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
76
votes
Accepted
How should a "working mathematician" think about sets? (ZFC, category theory, urelements)
Set theory provides a foundation for mathematics in roughly the same way that Turing machines provide a foundation for computer science. A computer program written in Java or assembly language isn't …
- 7,145
33
votes
Accepted
Is there a computable ordinal encoding the proof strength of ZF? Is it knowable?
Rom Maimon is describing the program of proof-theoretic ordinal analysis.
First, as you observed in your addendum, it isn't interesting to find some encoding of an ordinal whose well-foundedness im …
- 7,145
9
votes
Accepted
Does second-order arithmetic prove every expressible instance of Dependent Choice?
Carl has pointed out that my previous answer missed a clause in the theorem I cited.
Simpson's book, Subsystems of Second Order Arithmetic, does address this in section VII.6. He shows that dependen …
- 7,145
7
votes
AC and Euclidean Geometry
Tarski gave a first-order formulation of Euclidean geometry and proved it complete (without using the axiom of choice). In particular, that means that Euclidean geometry is the same in every model of …
- 7,145
7
votes
Interpretation of the Second Incompleteness Theorem
While it's not directly a philosophical benefit, the Second Incompleteness Theorem is quite useful for giving concrete unprovability results: if we want to prove that theory T does not prove theorem X …
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7
votes
What ordinals are definable relations in Peano Arithmetic?
The computable ordinals---that is, the ordinals below $\omega_1^{CK}$---are, by definition, represented by computable relations, all of which can be represented by formulas in PA, and indeed, even by …
- 7,145
6
votes
Accepted
Is $PRA$ + $TI({\epsilon_0})$ mutually interpretable with some theory in the language of set...
Yes, the consistency of "ZFC with the axiom of infinity replaced by its negation" is provable in "PRA + TI($\epsilon_0$)". Technically one has to also show that "PRA + TI($\epsilon_0$)" can prove the …
- 7,145
5
votes
Accepted
Absoluteness of completeness
The deductive closure of $\phi$ could be complete in one model and incomplete in the other.
Here's one way to see that (perhaps not the most elegant). Take any reasonable finitely axiomatized theory …
- 7,145
4
votes
What is the depth of the "provability hierarchy"?
I think you'll have to at least tweak the definition of your hierarchy, since a (strong enough) consistent theory can't prove that it doesn't prove something: $T$ can prove "$\neg Con(T)$ implies that …
- 7,145
4
votes
Applications of idempotent ultrafilters
There are applications of idempotent ultrafilters (often under the name "idempotent member of the enveloping semigroup") to finding and classifying the structure of topological dynamical systems. Aus …