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Results tagged with axiom-of-choice
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An important and fundamental axiom in set theory sometimes called Zermelo's axiom of choice. It was formulated by Zermelo in 1904 and states that, given any set of mutually disjoint nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets. The axiom of choice is related to the first of Hilbert's problems.
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What choice principles does "every set is in bijection with a transitive set" imply?
A set $X$ is transitive if $x\in y\in X\implies x\in X$. We shall say $\mathsf{TC}$ to mean the axiom "For all $X$ there is transitive $Y$ such that $|X|=|Y|$", where $|X|=|Y|$ means that there is a b …
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Can there be a proper class of Dedekind-finite cardinals?
To formalise the question: Does $\mathsf{ZF}$ prove the existence of an ordinal $\alpha$ such that, whenever $X$ is Dedekind-finite, there is an injection $X\to V_\alpha$? (Equivalently the existence …
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Small Violations of Choice: Can we force AC without collapsing the cardinalities of ordinals?
We say that a model $M$ of $\mathsf{ZF}$ satisfies Small Violations of Choice ($\mathsf{SVC}$) if all (any) of the following apply:
There is a model $V\subseteq M$ such that $V\vDash\mathsf{ZFC}$, an …
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Reference request: The non-productivity of Lindenbaum numbers
For a set $X$, the Lindenbaum number of $X$, $\aleph^*(X)$, is the least non-zero ordinal $\alpha$ such that there is no surjection $X\to\alpha$. It seems to be well-known that for infinite sets $X$ a …
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Does $\mathsf{SVC}^\ast$ exist?
$\mathsf{SVC}(S)$ is the assertion that for all sets $X$ there is an ordinal $\eta$ and a surjection $f\colon\eta\times S\to X$. I would like to denote by $\mathsf{SVC}^\ast(S)$ the same assertion but …
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References for the axiom of surjective comparability
The axiom $W_\kappa$, for $\kappa$ a cardinal, is the statement that for all sets $X$, either $|X|\leq\kappa$ (that is, there is an injection $X\to\kappa$) or $\kappa\leq|X|$. Is there literature on t …
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What are some "easy" violations of $\mathsf{SVC}$?
By $\mathsf{SVC}$, I mean "small violations of choice", which is the statement
$$(\exists S)(\forall X)(\exists f)``f\colon S\times\text{Ord}\to X\text{ is a surjection}".$$
Such an $S$ is called the …
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