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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.
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
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
7
votes
Accepted
Strong partition property + DC + existence of non-principal ultrafilter on $\omega$
The key reference for this is
MR0799042 (87d:03141). Henle, J. M.; Mathias, A. R. D.; Woodin, W. Hugh. A barren extension. In Methods in mathematical logic (Caracas, 1983), C. A. Di Prisco, editor …
- 32.5k
7
votes
Accepted
Is $Z_3+{\rm AD}$ equiconsistent with $\text{ZFC+AD}^{L(\mathbb{R})}$
The only reference I know for precisely these matters is the handbook chapter
MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, …
- 32.5k
13
votes
Is ZFC+(negation of a large cardinal axiom) arithmetically sound?
Yes. This is obvious if there are no such cardinals. (I assume that the natural numbers of the universe of sets are the true natural numbers. Otherwise, the answer is no, and there is not much else to …
- 32.5k
13
votes
Accepted
Conflating reals and sets of countable ordinals "nicely"
The technique of almost disjoint forcing was introduced in
MR0289291 (44 #6482). Jensen, R. B.; Solovay, R. M. Some
applications of almost disjoint sets. In Mathematical Logic and
Foundations …
- 32.5k
7
votes
Accepted
"Weakly" Woodin cardinals
(As I pointed out in a comment) yes, partial Woodinness is common in arguments in inner model theory. Accordingly, you obtain determinacy results addressing specific pointclasses (typically, well beyo …
- 32.5k
7
votes
Accepted
Is there a natural inner model of AD$_\mathbb{R}$?
A Wadge initial segment (of $\mathcal P(\mathbb R)$) is a subset $\Gamma$ of $\mathcal P(\mathbb R)$ such that whenever $A\in\Gamma$ and $B\le_W A$, where $\le_W$ denotes Wadge reducibility, then $B\i …
- 32.5k
14
votes
Accepted
Is the Martin's axiom number $\mathfrak m$ regular
Not necessarily. That $\mathfrak m$ is consistently singular is proved in
MR0947850 (89m:03045) Kunen, Kenneth. Where $\mathsf{MA}$ first
fails. J. Symbolic Logic 53(2), (1988), 429–433.
Ther …
- 32.5k
13
votes
Accepted
Woodin on Posner-Robinson for the hyperjump and sharp
MR2449474 (2009j:03067) Woodin, W. Hugh. A tt version of the
Posner-Robinson theorem. Computational prospects of infinity. Part II.
Presented talks, 355–392, Lect. Notes Ser. Inst. Math. Sci. N …
- 32.5k
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
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
8
votes
Accepted
Is a model of arithmetic contained in a model of arithmetic an initial segment?
Let me address the first question.
We can have models of $\mathsf{PA}$, $M\subsetneq N$ with $M$ cofinal in $N$. In fact, $M$ and $N$ do not even need to have the same cardinality. However, one can p …
- 32.5k