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Results tagged with lo.logic
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user 160347
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.
9
votes
Accepted
Weakly compact cardinals in $L$: how long do branches take to appear?
Here is a characterization of $\lambda$, but of course your question is somewhat ambiguous as to what counts as an answer.
For $n<\omega$, let $\alpha_n$ be the least ordinal $\alpha$ such that $\kapp …
- 10k
13
votes
Accepted
Is "every infinite set of strictly subnumerous sets is supernumerous to its union" equivalen...
Yes, this is like Tarski's result that if $\mathfrak{c}^2=\mathfrak{c}$ for every infinite cardinal $\mathfrak{c}$, then AC holds.
In fact, it suffices to suppose that $\mathfrak{c}\mathfrak{d}=\mathf …
- 10k
5
votes
Accepted
Can this semi-constructible structure satisfy existence of a measurable cardinal?
No. Under the hypotheses, there are limit ordinals $\eta$ such that $\mathcal{P}(\eta)\cap L\subseteq\mathfrak{L}_{\eta+1}$, and therefore, for example, $L_{\eta+2}\cap\mathfrak{L}_{\eta+1}\neq L_{\et …
- 10k
9
votes
Accepted
What is the consistency strength of "Singular worldly that is inaccessible in an inner model"?
To summarize the comments, the hypothesis is equiconsistent with (ZFC +) $0^\sharp$ exists + "there is a worldly cardinal". For as mentioned in the original post, under the hypothesis, $0^\sharp$ exi …
- 10k
8
votes
Accepted
Image-catching families in $\omega$
Original partial answer:
Here is some information:
An easy diagonalization shows that every image-catching family is uncountable. And more generally, MA implies that every image-catching family has ca …
- 10k
8
votes
Which are the hereditarily computably enumerable sets?
Re question 3: Not every arithmetic real is h.c.e.. In fact, every h.c.e. real is $\Sigma_2^\mathbb{N}$. For fix a program $e$ such that $x_e\subseteq\omega$. Let $n\in\omega$. Then $n\in x_e$ iff the …
- 10k
5
votes
Accepted
Are there outer models $V \subset W$ of $L$ such that $V$ is "far" from $L$ but $W$ is "not ...
One example: do some class forcing over $L$ to produce a proper class $V\models$ ZFC + "there is no set-forcing which forces that $V[G]=L[x]$ for a set $x$". Now do Jensen coding forcing over $V$ to p …
- 10k
14
votes
Accepted
Undefinable inner model
Here is an example from a transitive set model of ZFC + "there is a proper class of measurable cardinals". If there is one, then there is a countable one $N$, so fix such an $N$. Fix a sequence $\left …
- 10k
6
votes
Question regarding $W$ as not hyperarithmetic
Here is a proof that $\omega_1^{\mathrm{ck}}$ is admissible;
if someone knows where this fact is proved in the literature, please comment.
First note that there is no $\alpha<\omega_1^{\mathrm{ck}}$
w …
- 10k
13
votes
Accepted
Projective well-ordered sets, higher up
The theory $T_0$ = ZFC + "there is an inaccessible $\kappa$ such that every wellorder of a subset of $V_{\kappa+1}$ which is definable over $V_{\kappa+1}$ from parameters has length $<\kappa^+$" is eq …
- 17.7k
9
votes
Accepted
Precipitous ideal and inner model
No. Suppose otherwise. Let $G$ be the Levy collapse generic, and $D$ be the generic for forcing with the nonstationary ideal after that. Since $\mathrm{Ult}(L[U,G],D)$ is wellfounded, letting $i_D:L[U …
- 10k
17
votes
A Löwenheim–Skolem–Tarski-like property
Here is an upper bound:
Suppose $\kappa$ is $2$-fold supercompact. Then the property holds at $\kappa$. (Recall that $2$-fold supcompactness means that for each ordinal $\lambda$, there is $j:V\to M$ …
- 10k
4
votes
Accepted
Hyperarithmetically least elements in $\Pi^1_1$ sets
I suppose by $a\leq_Hb$, you mean that $a$ is $\Delta^1_1(\{b\})$. And I suppose in the question, $A=X$. Under this interpretation, the answer is no; in fact, there is a $\Delta^1_1$ set $X$ for which …
- 10k
6
votes
Exponentiation of Dedekind cardinals
ZF doesn't prove that the first implication reverses; i.e. it doesn't prove that $2^{\mathfrak{n}}=2^{\mathfrak{n}+1}\Longrightarrow \mathfrak{n}\geq\aleph_0$. For suppose $X$ is a set which is the di …
- 10k
8
votes
Accepted
Inner model for KP and a Well-Ordering of the Reals
Re the first question with KP, yes, $L_{\omega_1^{\mathrm{ck}}}$.
(Use the order of constructibility and Ville's Lemma.)
The second question is a bit vague. But for each ordinal $\alpha<\omega_1^{\ma …
- 10k