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Results tagged with lo.logic
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user 6085
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.
17
votes
Accepted
Coding a model of $0^\sharp$ from a $\Pi^1_1$ Gale-Stewart game
This problem is very much open.
Cheng Yong calls Harrington's $\star$ the assumption that there is a real $x$ such that all $x$-admissible ordinals are $L$-cardinals. From the work of Yong we know tha …
- 2,031
34
votes
Accepted
Latest status of core model theory?
${}$Hi Ioanna,
I.
The answer probably depends on how we define core model. At the level of "there are no Woodin cardinals in any inner model", we can finally show that core models exist, provably in $ …
- 2,520
30
votes
Counterintuitive consequences of the Axiom of Determinacy?
Let's see: $\mathsf{AD}$ implies that all sets of reals are Lebesgue measurable, have the Baire property, and the perfect set property (so, a version of the continuum hypothesis holds). It is conjectu …
- 1,446
19
votes
Accepted
Perfect set property for projective hierarchy
Analytic sets have the perfect set property, provable in, say, ZF+DC. This goes back to Suslin, and is discussed in Kanamori's book "The higher infinite" (Around section 12).
Large cardinals imply tha …
- 32.5k
32
votes
Is all ordinary mathematics contained in high school mathematics?
I believe this example may qualify. It is open whether Thompson's group $F$ is amenable.
We may present $F$ as $\langle A,B\mid [AB^{-1},A^{-1}BA]=[AB^{-1},A^{-2}BA^2]=\operatorname{id}\rangle$.
Amena …
- 12.9k
17
votes
Where is the Erdős–Rado theorem stated in Erdős and Rado's Bull AMS paper?
This is Theorem 39 in the paper (see Theorem 4.(i) for a user-friendly preview). But the fact that $(2^\kappa)^+\to(\kappa^+)^2_\kappa$ is older (1946) and due to Erdős, see here:
Paul Erdős. Some se …
- 32.5k
18
votes
Proof theoretic ordinal
This is a very interesting question (and I really want to see what other answers you receive). I do not know of any general metatheorems ensuring that what you ask (in particular, about consistency st …
- 4,713
12
votes
Accepted
Does Turing determinacy imply full determinacy?
This is open. In $L(\mathbb R)$ the answer is yes. Hugh has several proofs of this, and it remains one of the few unpublished results in the area. The latest version of the statement (that I know of) …
- 32.5k
29
votes
Accepted
Continuum Hypothesis
A classical reference is Hypothèse du Continu by Waclaw Sierpiński (1934), available through the Virtual Library of Science as part of the series Mathematical Monographs of the Institute of Mathematic …
- 4,713
11
votes
Accepted
Does "$X \not\to (\omega)^\omega_2$ for every infinite $X$" imply ${\sf AC}$?
The answer is no, the statement that for every set $X$ we have $$X\not\to(\omega)^\omega_2$$ does not imply the axiom of choice.
This was shown by Kleinberg and Seiferas in 1973, see
MR0340025 (49 #4 …
- 32.5k
15
votes
Accepted
Axiom of choice: ultrafilter vs. Vitali set
Stefan, "low" cardinalities do not change by passing from $L({\mathbb R})$ to $L({\mathbb R})[{\mathcal U}]$, so the answer to the second question is that the existence of a nonprincipal ultrafilter d …
- 47.8k
13
votes
Accepted
Is it consistent to have a function that is sensitive to subset relation from the power set ...
No, this is not consistent. Todorčević has shown in ZF that, in fact, there is no function $F\!:\mathcal W(S)\to S$ with the property you require. Here, $\mathcal W(S)$ is the collection of subsets of …
- 32.5k
23
votes
Who introduced direct limits?
As suggested by Gerald, the notion was first introduced for groups. Given a directed system of groups, their direct limit was defined as a quotient of their direct product (which was referred to as th …
- 32.5k
17
votes
Accepted
Smallest $\beta$ such that it is provable that $2^{\aleph_\beta} > 2^{\aleph_0}$
Perhaps the following may clarify the comments: for any ordinal $\delta$, there is a Boolean-valued extension of the universe of sets where $2^{\aleph_0}>\aleph_\delta$ holds. If you rather talk of mo …
- 32.5k
54
votes
Arguments against large cardinals
I do not know of any active set theorists who think large cardinals are inconsistent. At least, within the realm of cardinals we have seriously studied.
[Reinhardt suggested an ultimate axiom of the …
- 4,713