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.
5,866 questions
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Why Zermelo postulated the existence of a set with no finite limit to the ranks of its elements?
The original Zermelo set theory explicitly allowed for urelements.
What was the reason that led Zermelo to formulate the Axiom of Infinity in terms of the existence of a set of the kind that has an ...
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What is the strength of projective determinacy?
Consider the following theories:
$T_1$: $\mathsf{ZFC+PD}$ where $\mathsf{PD}$ is stated as a schema.
$T_2$: $\mathsf{ZFC+PD}$ where $\mathsf{PD}$ is a single sentence in the language of set theory.
$...
- 1,519
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Is the product of "non-coding" forcings also "non-coding"?
Say that a forcing notion $\mathbb{P}$ is slow iff there is some $f:\mathbb{R}\rightarrow\mathbb{R}$ (in $V$) such that for every $\mathbb{P}$-name for a real, $\nu$, we have $\Vdash_\mathbb{P}\exists ...
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Without AC can there be a function with linearly ordered fibers from the powerset of an infinite set to the set?
Given AC, it is easy to see there is no function $f$ from ${\cal P}X$ to $X$ for an infinite set $X$ such that $f(A)\ne f(B)$ for incomparable $A$ and $B$ (i.e., $A\not\subseteq B$ and $B\not\subseteq ...
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What can be computed without collapsing $\omega_1$
Below work in $\mathsf{ZFC+CH}$ for simplicity.
Say that a (set) forcing notion $\mathbb{P}$ captures a map $f:\mathbb{R}\rightarrow\mathbb{R}$ iff there is some $\mathbb{P}$-name for a real $\nu$ ...
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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\...
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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 $n\in \omega$ ...
- 54.1k
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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: ...
- 19.5k
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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 ...
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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 ...
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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 $...
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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 ...
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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-...
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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 ...
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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 ...
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