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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.
6
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1
answer
215
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On the number of complete Boolean algebras
In their 1972 paper On the number of complete Boolean algebras Monk and Solovay showed that if $\lambda$ is an infinite cardinal, then there are $2^{2^\lambda}$ many isomorphism types of
complete Boo …
- 32.2k
5
votes
0
answers
191
views
Product of nice proper forcing notions
Question Are there forcing notions $P$ and $Q$ such that $P$ is proper and $\aleph_2$-cc, $Q$ is proper and satisfies the $\aleph_2$-pic (pic=properness isomorphism condition) such that $P \times Q$ i …
- 32.2k
12
votes
2
answers
906
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Bernstein's proof of the continuum hypothesis
In the paper The Continuumproblem, Felix Bernstein introduces a new axiom and uses it to conclude the continuum hypothesis.
(1) As the paper is relatively old and the writing style is somehow informal …
- 32.2k
15
votes
0
answers
1k
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Condensed mathematics and independence results
I recently saw a paper on ``condensed mathematics'', in which I found the following quote interesting (see Condensed Mathematics: The internal Hom of condensed sets and condensed abelian groups and a …
- 32.2k
4
votes
0
answers
202
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PFA for cardinal preserving forcing notions and the CH
Let $FA_{\aleph_1}$(cardinal preserving proper forcings) be the forcing axiom: if $\mathbb{P}$ is a cardinal preserving proper forcing notion and $(D_\xi)_{\xi<\omega_1}$ are dense subsets of $\mathb …
- 32.2k
12
votes
2
answers
580
views
Forcing notions adding minimal reals
I am looking for a comprehensive list of known forcing notions which add a minimal real into the ground model. I know some of them like the Sacks forcing, or the Judah-Shelah's example of a c.c.c. for …
- 32.2k
13
votes
0
answers
696
views
Applications of Set theory vs. model theory in mathematics
I have a question that has occupied my mind for some time.
Let's first consider applications of set theory and model theory in mathematics.
Major applications of set theory are in topology, Banach spa …
- 32.2k
8
votes
0
answers
182
views
Topological Vaught's conjecture for special theories
As is know, Vaught's conjecture is a special case of topological Vaught's conjecture.
On the other hand, the Vaught's conjecture is true for the following theories:
1- $\omega$-stable theories (Shel …
- 32.2k
11
votes
2
answers
705
views
ZFC applications of Shelah's creature forcing
Shelah's creature forcing is a very powerful method, with wide range of applications. The method also has some applications in ZFC, let's quote a few of them that I am aware of:
(1) In A partition the …
- 32.2k
14
votes
0
answers
498
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The Ax-Kochen isomorphism theorem and the continuum hypothesis
Let's recall that:
(1): The Ax-Kochen principle says that if $\mathcal{U}$ is a non-principal ultrafilter over prime numbers, then $\prod_{\mathbb{U}} \mathbb{F}_p((t)) \equiv \prod_{\mathbb{U}} \mat …
- 32.2k
7
votes
1
answer
326
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Effective set= ordinal definable set
I just today realized that the concept of ordinal definability is defined in a different way by vopenka-Balcar-Hajek ``The notion of effective sets and a new proof of the consistency
of the axiom of …
- 32.2k
6
votes
0
answers
315
views
measure of generic reals in forcing extensions
It is well-known that if $V[G]$ is a generic extension by adding a Cohen real, then
the set $\{r \in V[G]: r$ is Cohen generic over $V\}$ has measure zero.
On the other hand, if $V[G]$ is a generic …
- 32.2k
14
votes
0
answers
402
views
O-minimality and forcing
It is well-known that the structure $(\mathbb{R}, +, \cdot, <, 0, 1)$ is an o-minimal structure and hence the set of integers $\mathbb{Z}$ is not definable in it.
In an ongoing project with Will Bria …
- 32.2k
2
votes
1
answer
274
views
A variant of Radin forcing
Suppose $\kappa$ is a large cardinal (strong cardinal seems to be enough). Is there a forcing notion $\mathbb{R}$ with the following properties:
$(1)$ Forcing with $\mathbb{R}$ adds a club $C$ into $ …
- 32.2k
10
votes
2
answers
1k
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Examples of set theory problems which are solved using methods outside of logic
The question is essentially the one in the title.
Question. What are some examples of (major) problems in set theory which are solved using techniques outside of mathematical logic?
- 32.2k