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Results tagged with lo.logic
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user 11115
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.
3
votes
Examples of concrete games to apply Borel determinacy to
May look at some of Andrew Marks papers, in particular ``A determinacy approach to Borel combinatorics''.
- 32.2k
5
votes
Accepted
On the number of complete Boolean algebras
The answer is that there are still $2^\kappa$ many isomorphism types of complete Boolean algebras of power $\kappa$.
This is proved by Shelah, see Building complicated index models and Boolean algebr …
- 32.2k
14
votes
Accepted
Is it consistent that $2^{(\cdot)}$ is "surjective" on the class of uncountable ordinals?
By Konig's lemma $cf(2^\kappa) > \kappa,$ so for example $\aleph_\omega$ can never be of the form $2^\kappa$ for any $\kappa.$
- 32.2k
9
votes
Accepted
On the existence of a real which is not set-generic over $L$
Yes, there is indeed such a paper, see Mack Stanley's paper "Coding a generic extension of L".
- 32.2k
7
votes
Could groups be used instead of sets as a foundation of mathematics?
Somehow related to the question is Shelah's paper Interpreting set theory in the endomorphism semi-group of afree algebra or in a category. There are further papers on this direction, see for example …
- 32.2k
22
votes
Accepted
What is known about the consistency of $2^{\aleph_\alpha} = \aleph_{\alpha+\gamma}$ for all ...
By a result of Patai, $\gamma$ should be finite (this is exercise 5.15 in Jech's book).
For any finite $n>0, H_n$ is consistent, see Merimovich's paper A power function with a fixed finite gap every …
- 32.2k
16
votes
Accepted
Can $\kappa^\lambda$ be large if $2^\lambda$ is small and $\lambda<\mathrm{cof}(\kappa)$?
It is consistent that such a pair exists, see my paper Singular cofinality conjecture and a question of Gorelic.
To show that some large cardinals are needed, suppose for example $\lambda=\aleph_0 < \ …
- 32.2k
12
votes
Can GCH fail everywhere every way?
When working in ZF, one can have more freedom. See
An Easton-like Theorem for Zermelo-Fraenkel Set Theory with the Axiom of Dependent Choice and An Easton-like theorem for Zermelo-Fraenkel Set Theory …
- 32.2k
4
votes
Properness for uncountable models
The concept first appeared in Shelah's paper Independence results
The theorem you have stated should be folklore, but you may see Tapani Hyttinen and Mika Rautila, The canary tree revisited
for a pro …
- 32.2k
15
votes
Accepted
A proper class of ordinals without an infinite constructible subset
Stanley, M. C., A cardinal preserving immune partition of the ordinals, Fundam. Math. 148, No. 3, 199-221 (1995). ZBL0843.03028.
An infinite set (or class) of ordinals is said to be immune if it nei …
- 32.2k
3
votes
Accepted
Adding a closed unbounded set containing of only limit ordinals with a special property
Maybe the following idea works: Given a condition as above, also require the following:
(1) for each $\alpha, f(\alpha)$ is indecomposable,
(2) suppose $dom(p)=\{\beta_0 < \beta_1 < \cdots < \beta_n\} …
- 32.2k
11
votes
Accepted
Coding the universe into a real over better core models
For measurable cardinals, the answer is yes and is due to Sy Friedman. See Coding Over a Measurable Cardinal.
There is some difficulty to extend the result to the context of Woodin cardinals, see Gen …
- 32.2k
3
votes
Accepted
Properness for small forcings
The answer is no, as the following upcoming work of Shelah and Usuba shows:
Theorem (Shelah-Usuba):
The following theories are equiconsistent with ZFC:
ZFC+CH+ “there is an $\omega _1$-stationary pres …
- 32.2k
6
votes
Accepted
Tree property at weak inaccessibles
In his paper Boolean extensions which efface the Mahlo property William Boos proves the following consistency result:
Theorem. Assume GCH holds and $\kappa$ is weakly compact. Then there exists a card …
- 32.2k
14
votes
Non-set-theoretic consequences of forcing axioms
Indeed there is a vast of applications, for example:
Using Martin's axiom, Shelah showed that there is a non-free Whitehead group. The book ``
Consequences of Martin's Axiom'' contains many other exam …
- 32.2k