forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence ...
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 ...
13
votes
2answers
463 views
Woodin's unpublished proof of the global failure of GCH
An unpublished result of Woodin says the following:
Theorem. Assuming the existence of large cardinals, it is consistent that $\forall \lambda, 2^{\lambda}=\lambda^{++}.$
In the paper "The ...
7
votes
1answer
136 views
Determinacy from $\omega_1\rightarrow(\omega_1)^{\omega_1}$
Assuming the Axiom of Determinacy (abbreviated AD), Martin showed how to derive a rather strong partition on $\omega_1$, namely that $\omega_1\rightarrow(\omega_1)^{\omega_1}$. In "Infinitary ...
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 ...
4
votes
1answer
73 views
An ordinal not $\Sigma_1$ stable in alpha must be in the hull of smaller parameters in $L_\alpha$
This concerns an assertion of Sy Friedman in [1], Lemma 2, which claims that, under certain conditions, if $\beta$ is not 0 and not $\Sigma_1$-stable in $\alpha$, i.e. $L_\beta\prec_1 L_\alpha$, then ...
-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 ...
6
votes
0answers
176 views
Fragments of Morse—Kelley set theory
Morse—Kelley set theory (hereafter MK) is the impredicative counterpart of von Neumann—Bernays—Gödel set theory (NBG), where formulas containing class quantifiers are permitted in the comprehension ...
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 ...
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 $H$ ...
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 ...
7
votes
1answer
151 views
On free limits of iterations
Shelah isolates the notion of "$\aleph_1$-free iteration" in the first two sections of Chapter IX from Proper and Improper Forcing, and he proves there that properness is preserved by this sort of ...
1
vote
2answers
137 views
Are there better arithmetic on ordinals?
First fix the following notations:
$\mathcal{L}_{AR}:=$The first order language $\lbrace \overline{0},\overline{S},\overline{+},\overline{\times}, \sqsubset \rbrace$ which ...
3
votes
1answer
149 views
Large cardinals and mild extensions
It is known that for many large cardinals $\kappa$ (like weakly compact, measurable,...) , $\kappa$ remains large of the same type after forcings of size $<\kappa$. Now the questions are:
Question ...
