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Results tagged with set-theory
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user 57888
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.
4
votes
Accepted
Can one satisfaction class code another?
In case of Peano Arithmetic the answer is yes (emphatically yes), if I understand your question correctly. This follows from Theorem 3.3 in Smith's "Nonstandard Definability" APAL 42 (1989) pp. 21-43 …
- 1,229
14
votes
Why is an internal proof of consistency satisfactory for some systems?
The original context of Gödel's Theorem was slightly different. Famously, Hilbert had asked for a purely combinatorial proof of consistency for strong set theories. Notice that this seems like a very …
- 1,229
7
votes
When is it okay to intersect infinite families of proper classes?
Here is an example of a family of classes of rings which you couldn't intersect: The class $A_n$ consists of all rings of power greater than n and the rings of cardinality $k≤n$ such that $ϕ_k$, where …
- 1,229
14
votes
Accepted
How can any theory prove well-foundedness of ordinals above $\omega_1^{\text{CK}}$?
Just slightly expanding the comment of Emil Jeřábek: on one hand, in ZFC we can define some objects we call ordinals and prove transfinite induction of each of them. This is not what we mean by the or …
- 1,229
3
votes
Why can we assume a ctm of ZFC exists in forcing
This is to slightly elaborate on point 2 of Noah Schweiber's answer, since in my opinion this approach is often presented in a somewhat confusing manner which ommits some key subtelties.
Forcing const …
- 1,229