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.
5,862 questions
4
votes
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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
-11
votes
0
answers
147
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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
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
318
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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
4
votes
0
answers
115
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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
0
votes
0
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164
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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
15
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1
answer
679
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An infinite game played on $\{0,1\}^\mathbb{N}$ in which the players must avoid creating an algebraic dependency
This mathematical game occurred to me. It may be quite basic for experts, but it seems to lead to some interesting questions.
Turns are indexed by elements of $\mathbb{N} = \{1,2,\ldots, \}$. In turn $...
- 11.9k
2
votes
0
answers
139
views
Understanding tree normalization and inflations
I am trying to get myself familiar with normalization of iteration trees, and one technical concept which I find especially hard to parse is inflation. I am following the notation of Farmer ...
- 143
15
votes
1
answer
727
views
Is 0# still unique in ZFC without powerset?
Working in $ZFC$, the statement "$0^\sharp$ exists" is often liberally taken to be one of many known equivalent statements.
However, working in $Z_2$ or $ZFC^-$ (with collection, well-...
- 397
8
votes
1
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347
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On the class forcing used in Jensen's proof of Con(CH+SH)
It seems that Jensen's proof of the consistency of CH + SH used class forcing, but the revelant properties are not clearly verified. I haven't learnt about class forcing, so I wonder whether it is ...
- 83
7
votes
1
answer
290
views
Adding special trees
For a regular cardinal $\kappa$, a $\kappa$ tree $T$ is called special when there is a regressive function $f : T \to T$ (regressive in the tree order) so that the inverse image of every point is the ...
- 21k
1
vote
0
answers
168
views
Partition Principle with a left inverse
This is a follow-up to this recent question. We were reminded
that the partition principle is the statement that for every surjection $f : A \longrightarrow B$ there is an injection $g : B \...
- 1,172
6
votes
1
answer
182
views
Nonstationary support iterations
In Friedman and Magidor - The Number of Normal Measures, the authors use a nonstationary support iteration of posets, rather than the more customary countable or Easton support iterations: conditions $...
- 2,215
9
votes
0
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122
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Relation of two notions of n-ineffable cardinal
There are two notions of n-ineffable. One is standard, defined in Baumgartner's paper, used by Harvey Friedman: κ is n-ineffable iff for every 2-coloring of $[\kappa]^{n+1}$, there is a stationary ...
- 1,752
9
votes
1
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231
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Determinacy transfer $\mathbf\Delta^1_{2n}\text{-}\mathsf{Det}\to\mathbf\Pi^1_{2n}\text{-}\mathsf{Det}$ over a subsystem of second-order arithmetic
It is known that for $n\ge 1$, $\mathbf{\Delta}^1_{2n}$-Determinacy implies $\mathbf{\Pi}^1_{2n}$-determinacy, but as far as I know, this is a theorem in $\mathsf{ZFC}$. It brings the following first ...
- 3,426