Questions tagged [axiom-of-choice]
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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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 \...
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(AC) and existence of basis of $\{0,1\}^X$ for any set $X$
For any set $X$, let $\{0,1\}^X$ be the collection of all functions $f:X\to\{0,1\}$. We make it into a vector space over the field $\mathbb{F}_2$ by endowing it with pointwise addition modulo $2$ and ...
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How big can determined sets be?
I vividly remember seeing an affirmative answer to this question presented in seminar, but I can't track down a citation nor can I prove it myself, so now I'm doubting it's actually true:
Working in $...
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Are there higher amorphous sets in the determinacy world?
Assume $\mathsf{AD}^++V=L(\mathcal{P}(\mathbb{R}))$ as usual for this kind of problem. My question is motivated by the observation that there is no $\omega_1$-amorphous set, i.e., an uncountable set ...
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Cardinal arithmetic inequalities according to ZF
Suppose $\kappa$, $\lambda$, $\mu$, and $\nu$ are cardinals which may or may not be ordinals. Can we prove without resorting to the axiom of choice either of the following:
$\kappa + \lambda \...
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Is every set of cardinals bounded?
Let ZFU$_\text{R}$ be ZF (formulated with Replacement) modified to allow a proper class of urelements. A cardinal $\mathfrak{b}$ is an upper bound of a set $X$ of cardinals if $\mathfrak{a} \leq \...
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When is $\Theta$ a (weakly) Lowenheim-Skolem cardinal?
The base theory is $\textsf{ZF}$. The following definitions are due to T. Usuba.
Definition 1: An uncountable cardinal $\kappa$ is a weakly Lowenheim-Skolem cardinal if for every pair of ordinals $\...
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Does ZF alone prove that every complete, atomless Boolean algebra has an infinite antichain?
Here by "antichain" I mean a set of elements that have pairwise-trivial meets, not merely ones that are pairwise-incomparable. Clearly, every atomless Boolean algebra has antichains in this ...
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Theories yielding $\mathit{Con}(\mathsf{ZF+\neg AC})$ without forcing
My question is:
What are some examples of consistent (relative to large cardinals) extensions of $\mathsf{ZFC}$ within which there is a forcing-free proof of the consistency of $\mathsf{ZF+\neg AC}$?
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About the consistency strength of a ubiquitous Perfect Set Property with a singular $\omega_1$ [duplicate]
Kanamori writes in the Higher Infinite on page 135 that
"Specker had already made the conceptual move to inner models;
through a sequence of implications he had in effect established in ZF that ...
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Set size comparison via non-existence of surjections
If $X, Y$ are sets, let us say that $X$ is strictly smaller than $Y$, in symbols $X \prec Y$, if $Y$ is non-empty and for every map $f:X\to Y$ we have $Y\setminus\text{im}(f) \neq \varnothing$.
Our ...
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Is Zorn's Lemma equivalent to the Axiom of Choice for individual sets?
It is well-known that in $\mathsf{ZF}$, the Axiom of Choice and Well-ordering Theorem are equivalent. What is perhaps less well-known is that there is a "local" version of this equivalence.
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Comparability of power sets and (AC)
For sets $X, Y$ we write $X \leq Y$ if there is an injective map $f:X\to Y$.
Let (S) be the statement:
For any sets $X, Y$, either ${\cal P}(X) \leq {\cal P}(Y)$, or ${\cal P}(Y) \leq {\cal P}(X)$, ...
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Does $\mathsf{AD^+}$ prove "well-ordered union of well-orderable sets is well-orderable"?
Theorem 1.4 of [1] says if $\mathsf{AD^+}$ holds and either $V=L(T,\mathbb{R})$ for some set $T$ of ordinals or $V=L(\mathcal{P}(\mathbb{R}))$, then we have the following principle, which is in some ...
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When is "anomalously few substructures" possible?
Given a variety (in the sense of universal algebra) $\mathscr{V}$ axiomatized by a finite set of equations $E$, say that $\mathscr{V}$ is consistently gappy iff it is consistent with $\mathsf{ZF}$ ...
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