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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.
49
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1
answer
2k
views
Producing finite objects by forcing!
It is a trivial fact that forcing can not produce finite sets of ground model objects. However there are situations,
where we can use forcing to prove the existence of finite objects with some proper …
- 32.2k
34
votes
5
answers
2k
views
Forcing as a replacement of induction and diagonal arguments
Let me give some examples motivating the question.
The use of forcing instead of induction: For this consider Cantor's theorem:
Theorem 1. Any two countable dense linear orders $I, J$ without end po …
- 32.2k
34
votes
4
answers
3k
views
Is it possible to define higher cardinal arithmetics
In number theory there are several operators like addition, multiplication and exponentiation defined from $\omega\times\omega$ to $\omega$. Each of them is defined as an iterat …
- 32.2k
29
votes
2
answers
2k
views
On the probability of the truth of the continuum hypothesis
First note that there exists a natural measure $\mu$ on $P(\omega \times \omega)$, inherited from the Lebesgue measure on the reals (by identifying the reals with $P(\omega)$ and $\omega$ with $\omega …
- 32.2k
20
votes
5
answers
7k
views
Applications of set theory in physics
In the introduction of the paper "Links between physics and set theory", the following quote of Eris Chric is stated:
"Set theory perhaps is too important to be left just to mathemat …
18
votes
2
answers
628
views
Is the notion of fixed point property for topological spaces an absolute notion?
Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point.
Is the notion of FPP for topological spaces an absolute notion? More p …
- 32.2k
18
votes
3
answers
2k
views
Scott-Solovay unpublished paper on ``Boolean valued models of set theory''
I have read some papers from 1970$^{th}$, and in some of them, the paper of Scott and Solovay on ``Boolean valued models of set theory'' is given as a main reference, with many references to the resul …
- 32.2k
17
votes
0
answers
903
views
Souslin trees and weakly compact cardinals
In Souslin trees on the first inaccessible cardinal it is asked if it is consistent that there are no $\kappa-$Souslin trees at the least inaccessible cardinal $\kappa$.
In this question I would like …
- 32.2k
17
votes
0
answers
555
views
Gitik's work on Shelah's weak hypothesis
It seems that Moti Gitik has recently refuted some variants of Shelah's weak hypothesis. For this see the title and abstract of his talk at the Set Theory, Model Theory and Applications conference.
I …
- 32.2k
16
votes
0
answers
768
views
Ideas behind Gitik's solution of PCF conjecture
Recently Moti Gitik has refuted Shelah's PCF conjecture (see Short extenders forcings II ) by proving the following theorem:
Theorem. Assuming the consistency of infinitely many strong cardinals, one …
- 32.2k
16
votes
2
answers
1k
views
Von Neumann's consistency proof
In the paper Zur Hilbertschen Beweistheorie, John Von Neumann has proposed a consistency proof for
a fragment of first-order arithmetic (the fragment without induction and with
the successor axioms on …
- 32.2k
15
votes
3
answers
851
views
Undefinability of $\mathbb{Z}$ in the reals
It is a well-known fact that $\mathbb{Z}$ is not definable in the structure $\mathcal{R}=(\mathbb{R}, +, ., < , 0, 1)$. This follows from Tarski's quantifier elimination, and in fact, we can conclude …
- 32.2k
15
votes
1
answer
1k
views
Characterization of Cohen reals
The following is a well-know fact:
Theorem The real $r$ is Cohen over $V$ iff if it does not belong to any meager Borel
set coded in $V$.
Now suppose that $\kappa$ is an uncountable cardinal and l …
- 32.2k
15
votes
6
answers
2k
views
The origins of forcing in mathematical logic and other branches of mathematics
As everyone knows, forcing was created by Cohen to answer questions in set theory.
Question 1. What are the first applications of set theoretic forcing in other branches of mathematical logic, like n …
15
votes
0
answers
1k
views
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