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Results tagged with lo.logic
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user 33842
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.
6
votes
Computability complexity of the first-order theory of arithmetic?
The theory of arithmetic is $\Delta^1_1$, thus it cannot be $\Pi^1_1$-complete (since $\Delta^1_1$ is a strict subset of $\Pi^1_1$ and these classes are closed under recursive preimages).
- 2,442
5
votes
Basic results with three or more hypotheses
The HSP-theorem from universal algebra: If a class of algebraic structures (over a given signature) is closed under homomorphic images, substructures and products, then it is defined by a set of equat …
4
votes
3
answers
819
views
Impact of the axiom of replacement on finite sets
The axiom of replacement is usually used to prove the existence of large sets, to provide a reflection principle, for transfinite recursion… However, I am wondering how it affects finite sets. Let me …
- 2,442
2
votes
What would be some major consequences of the inconsistency of ZFC?
Notice that you can drop the axiom of replacement or replace it by a weaker reflection principle. Without this axiom you have less consistency strength—it might still be consistent even if ZFC is not …
- 2,442
2
votes
Existential quantification over regular predicates
That is a central point about automatic structures: By projection (“existential quantification”) you get another regular predicate, and regular predicates are also closed under intersection and comple …
- 2,442
2
votes
Accepted
How many subsets of $\mathbb{R}$ are order isomorphic to $\mathbb{Q}$?
There are continuum many countable subsets of the continuum (because $\mathfrak{c}^{\aleph_0}=2^{\aleph_0}$). Thus the answer is $\mathfrak{c}$. See this question.
- 2,442