Questions tagged [set-theory]
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 relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
6 questions from the last 7 days
4
votes
1
answer
319
views
Higher analogues of Gandy basis theorem
For $n\in\omega$ and $x$ a real let $C_n^x$ be the canonical $\Pi^1_n(x)$-complete set. E.g. $C_1^x=\mathcal{O}^x$, etc. I recall seeing long ago the fact that, assuming large cardinals (precisely: ...
- 19.4k
1
vote
1
answer
193
views
Do we have ${\frak b} \leq {\frak s}$ in ZFC?
Let ${}^\omega\omega$ denote the set of functions $f:\omega\to \omega$. For $f, g \in {}^\omega\omega$ we define
$f\leq^* g$ if there is $N\in\omega$ such that $f(n)\leq g(n)$ for all $n\in \omega$ ...
4
votes
1
answer
255
views
Does anyone use measures that take values in real numbers and cardinal numbers?
The counting measure on $\mathbb{R}^n$ is a map that takes a subset $A$ of $\mathbb{R}^n$ and returns its cardinality if it is finite or the symbol $\infty$ if it is infinite.
So, if $A\subseteq\...
- 1,038
0
votes
0
answers
165
views
Formalizing the Completeness Theorem given languages of infinite cardinality
I am reading Kunen's books on set theory and logic. In his approach, the metatheory is finitistic (which can be approximated in PRA).
This implies that in the finitistic metatheory, one can do formal ...
- 225
-11
votes
0
answers
147
views
Decoding this recursive structure notation [closed]
I encountered this mathematical structure but need help
decoding the notation:
[
T = \prod_{i=1}^{\infty} \mathbb{N}
]
with hierarchy levels:
$!1, !2, !3\ldots$
$T1G, T1H, T1B\ldots$
$A1, A2, A3\...
- 1
4
votes
0
answers
115
views
Algebraic behaviour of "bounded functions" to topological abelian groups
It's well known (and not so hard to prove directly) that for any topological space $X$ the group $C^b(X, \mathbb Z)$ — locally constant functions taking only finitely many values — is a free abelian ...
- 5,926