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Results tagged with lo.logic
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user 497904
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.
11
votes
2
answers
810
views
Undefinable inner model
What are some examples of a pair $M\subseteq N$ of transitive set models of $\mathsf{ZFC}$ with the same ordinals, such that $M$ is not a definable class (with parameters) in $N$? Is it possible that …
- 877
7
votes
0
answers
374
views
Is Borel cardinality the same as cardinality under determinacy?
Suppose $E,F$ are Borel equivalence relations on Polish spaces $X,Y$, respectively. Under strong enough determinacy axioms, is it true that $E$ Borel reduces to $F$ iff there is an injective map from …
- 877
3
votes
A better way to explain forcing?
Here is currently my favorite way to motivate forcing, and I conjecture that it works for most "real" mathematicians (non-logicians). A proof/disproof is left to the reader.
The forcing relation is in …
- 877
3
votes
0
answers
214
views
Basic cardinal arithmetic without choice
Do we know everything about addition and multiplication of cardinalities in choiceless set theory?
For example, let $M$ be a model of $\textsf{ZF}+\textsf{AD}+V=L(\mathbb{R})$, consider the sets $\mat …
- 877
11
votes
3
answers
785
views
When are two forcing posets "the same"?
Let $B$ and $C$ be complete Boolean algebras. To avoid triviality I may also want them to be atomless. For $b\in B$ nonzero, denote $B\upharpoonright b=\{p\in B:p\leq b\}$, which can be viewed as a co …
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6
votes
Who needs Replacement anyway?
Isn't replacement needed or at least the most natural way to construct projective/injective resolutions? Say we want a free resolution of $M$. Consider the free module $F_1$ with $M$ as the set of gen …
- 877