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Results tagged with lo.logic
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user 48041
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.
7
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0
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110
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On the optimal strength of Goodstein's theorem
Goodstein's theorem is a famous example of an arithmetical statement that is unprovable in $\mathsf{PA}$ but provable in a stronger theory. It is well-known that Goodstein's theorem implies the consis …
- 3,042
3
votes
1
answer
161
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Reverse mathematics on lightface $\Pi^1_1$-uniformization for unary relation
It is known that the following form of $\Pi^1_1$-uniformization is equivalent to $\Pi^1_1$-Comprehension over $\mathsf{ATR}_0$ (cf. VI.2.6 of Simpson's book) :
(Kondo's uniformization theorem) For ea …
- 3,042
4
votes
3
answers
394
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Hyperarithmetically least elements in $\Pi^1_1$ sets
My question is: Do we have a hyperarithmetically $\le_H$-least real in any $\Pi^1_1$ set? That is
Question. Suppose that $A$ is a non-empty $\Pi^1_1$ set. Then can we find a real $a\in A$ such that …
- 3,042
7
votes
1
answer
338
views
Does Mostowski's collapsing lemma prove $\Delta_0$-transfinite recursion?
Let $\mathsf{T}$ be the theory comprising Extensionality, Foundation (stating every set has an $\in$-minimal element), Pairing, Infinity, Union, $\Delta_0$-Separation, and the closure under rudimenta …
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9
votes
1
answer
473
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The relation between $\Pi_1$-Foundation and $\Sigma_1$-Foundation over Kripke-Platek set theory
Let $\mathsf{KP_0\omega}$ be Kripke-Platek set theory with Infinity but Foundation (or $\in$-Induction) restricted to $\Delta_0$-formulas. $\mathsf{ZF}$ proves $\in$-Induction holds for arbitrary form …
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9
votes
0
answers
271
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Can we have a 'universal class' for elementary embeddings $j\colon V\to V$
Work over $\mathsf{GB}$, Gödel-Bernays set theory (without choice). My question is the following:
Question. Is the following statement consistent with $\mathsf{GB}$? There is a universal class for el …
- 3,042
10
votes
3
answers
946
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Set theory determined by $V_\alpha$ for limit ordinals $\alpha>\omega$
Von Neumann hierarchy has a critical role in set theory. It is well-known that $V_\alpha$ is a model of $\mathsf{ZC}$ if $\alpha$ is a limit ordinal. Furthermore, $V_\alpha$ satisfies the cumulative h …
- 3,042
5
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0
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228
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Which very large cardinals are preserved under Woodin's forcing for $\mathsf{AC}$?
Woodin showed that we can force $\mathsf{AC}$ if there is a proper class of supercompact cardinals while preserving supercompacts, by forcing Easton-support iteration of $\operatorname{Col}(\kappa,<V_ …
- 3,042
8
votes
1
answer
455
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Reference request: proof-theoretic strength of $\mathsf{KP}$ with recursively large ordinals...
Large set axioms are notions corresponding to large cardinals on constructive set theories like $\mathsf{IZF}$ or $\mathsf{CZF}$. The notion of inaccessible sets, Mahlo sets, and 2-strong sets corresp …
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5
votes
1
answer
174
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Finding many subsets of $V_{\lambda+2}$ stable under $j:V\prec V$
Working over $\mathsf{ZF}$ with an embedding $j:V\prec V$ with a critical point $\kappa$. Take $\lambda=\sup_{n<\omega}j^n(\kappa)$. (You may assume $\mathsf{DC}_\lambda$ if you need, but I am not sur …
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6
votes
0
answers
243
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Models of $\mathsf{ZF^-_2}$ over $\mathsf{ZF}$
Let $\mathsf{ZF^-_2}$ be a second-order ZF without Powerset, and the second-order Collection in place of Replacement. It is easy to see that every transitive model of $\mathsf{ZF^-_2}$ is closed under …
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8
votes
1
answer
587
views
Is there an abstract logic that defines the mantle?
It is a known result by Scott and Myhill that the second-order version of $L$ yields $\mathrm{HOD}$.
Recently, Kennedy, Magidor, and Väänänen (Inner models from extended logics: Part I and II) investi …
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14
votes
2
answers
1k
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Is the existence of double complement of a set provable in Intuitionistic ZF?
In Powell's article [1] he introduces the axiom of double complement, which says a double complement $\{x : \lnot\lnot(x\in A)\}$ is a set for any set $A$.
I can't find similar axiom from other refe …
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7
votes
1
answer
315
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Consistency strength of the existence of a transitive model of $\mathsf{ZFC}^-$ with a $\kap...
Let $\mathsf{ZFC}^-$ be the Zermelo-Fraenkel set theory without power set axiom.
For a transitive model $M$ of $\mathsf{ZFC}^-$ and an cardinal $\kappa\in M$ in the sense of $M$, an unary predicate $U …
- 3,042
6
votes
1
answer
574
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Axiom of choice and the equality between second-order constructible universe and HOD
I try to prove $L_{SO}=\mathrm{HOD}$, where $L_{SO}$ is second-order constructible universe which has similar definition with $L$ but it uses second-order definability rather than the first-order defi …
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