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, ...
1
vote
0answers
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
votes
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
votes
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
votes
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
votes
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 ...
1
vote
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 ...
3
votes
1answer
220 views
Suggestions on the best introductory Model Theory texts
Any recommendations on the best texts for introducing Model Theory?
-4
votes
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
votes
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
votes
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
votes
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
votes
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 $H$ ...
1
vote
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
votes
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
votes
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 ...