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, ...

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116 views

What are current trends/questions in algebraic logic? [on hold]

What are current trends/questions in algebraic logic?I mean the research developed by Paul Halmos. And anyone could give some reference for overview of it's history? Also any overview of it's ...
4
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1answer
113 views

A question on the minimal ordinal for ZFC

If ZF has a standard model there is a least ordinal $\sigma$ such that $L_{\sigma}$ is a model of ZFC. What is $\sigma$ called?
5
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2answers
123 views

Transfinite recursion, collection and replacement in KP and KF

Kripke Platek set theory has collection instead of replacement, and it is a weakening of KP if one has replacement instead of collection. Call KP minus collection plus replacement KF for Kripke ...
10
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1answer
137 views

Idempotent ultrafilters and the Rudin-Keisler ordering

Short version: what can we say about the place of idempotent ultrafilters in the Rudin-Keisler ordering? Longer version: If $U$, $V$ are (nonprincipal) ultrafilters on $\omega$, then we write ...
8
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1answer
166 views

A question on an ordinal for ZFC-

ZFC-, which is ZFC minus power set, is modelled by $ L_{\delta}$ where $\delta$ is an admissible ordinal larger than any least $\Sigma_{n}$-admissible ordinal for n a natural number. Can some provide ...
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1answer
161 views

Ordinal Exponentiation in Genzen's Sequent Calculus

For Genzen's sequent calculus with PA axioms, why is the proof-theoretic ordinal $\epsilon_0$? This seems to hinge on what exactly it means for the level of a cut or CJ inference figure to be higher ...
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1answer
220 views
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1answer
308 views

Kadison-Singer problem

The Kadison-Singer problem is the following statement: for any $\epsilon >0$, there exists $r\in \mathbb N$ such that for any bounded operator $A$ on $\ell^2(\mathbb Z)$, there exists a partition ...
5
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1answer
243 views

Axiomatization of first order logic (finitely many variables)

Standard textbooks in mathematical logic will assume an infinite supply of variables. Their axiomatization of first order logic will typically contain an axiom of the form $\forall ...
5
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1answer
197 views

Why stationary sets were named such?

My question is about terminology: Do you know why stationary sets were named such? Going over the following MO question about the intuition behind stationary sets, the only compelling argument I can ...
2
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1answer
129 views

Can a class of arithmetical statements containing its own soundness condition be closed under negation?

Given a class $C$ of arithmetical sentences, an arithmetical theory $T$ is said to be $C$-sound if all the theorems of $T$ which are in $C$ are true. For instance, $T$ is $\Sigma_1$-sound if all ...
10
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2answers
594 views

What's a noncommutative set?

This issue is for logicians and operator algebraists (but also for anyone who is interested). Let's start by short reminders on von Neumann algebra (for more details, see [J], [T], [W]): Let $H$ ...
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1answer
147 views

Embedding of consistent subset still consistent in FOL (finitely many variables)

I am looking at FOL with no equality, no constant, no function symbol and the unique binary predicate $\in$ with variables in arbitrary sets $V$ or $W$. I have an injective map $i:V\to W$ which ...
6
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1answer
196 views

Intuition behind Pincus' “injectively bounded statements”

In David Pincus, Zermelo-Fraenkel Consistency Results by Fraenkel-Mostowski Methods, The Journal of Symbolic Logic, Vol. 37, No. 4 (Dec., 1972), pp. 721-743 Pincus introduces the notion of ...
3
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1answer
120 views

Question on a limit of admissible ordinals

Let there be an omega sequence of ordinals such that the first is the least $\Sigma_1$-admissible ordinal and the $n+1$st is the least $\Sigma_{n+1}$-admissible ordinal. What is the name, if any, of ...

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