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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Reference request: choiceless cardinality quantifiers
There is a substantial literature on the logic of cardinality quantifiers. (E.g., the quantifier $Q_\alpha$ where $M \vDash Q_\alpha x \, \varphi (x)$ iff $\vert \{a \in M : M \vDash \varphi[a] \} \...
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Does completeness of the theory of a bijection without finite orbits depend on choice?
Consider the following sentences in a first-order language with one unary function symbol $f$:
$\forall x \exists y (fy=x)$
$\forall y\forall z(fy=fz\to y=z))$
$\forall x (\underbrace{f\dotsb f}_{n\...
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Can the following definition of choice principle salvage the prior attempts?
In prior postings 1 , 2, I've presented a definition of choice principle as what is equivalent to a selection principle. However, it was proved that it is inadequate in the sense that it admits ...
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Does weak countable choice imply that the Cauchy reals are Dedekind complete?
Assuming the axiom of weak countable choice, is the set of modulated Cauchy reals Dedekind complete?
The second theorem on this ncatlab page claims something equivalent, but it doesn't contain a proof ...
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Is there a strict limit on choice principles in $\sf ZFC$?
Is there a principle $\sf P$ that $\sf ZFC$ [or some suitable extension of it] proves to be a strict limit on choice principles?
By a choice principle I mean a sentence (or scheme) that is equivalent ...
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How much Dependent Choice is provable in $Z_2$? And what about Projective Determinacy?
So, second order arithmetic, $Z_2$, is capable of proving quite a few things. One thing which would be of use is dependent choice for $\mathbb{R}$.
Basically, dependent choice on $\mathbb{R}$ says ...
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Is the Ordering Principle equivalent to a selection principle?
Working in the context of set theory $\sf ZF$, selection may be defined as a function from nonempty sets to their elements. Formally:
$\operatorname {selective}(c) \iff \operatorname {function}(c) \...
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Logical strength of a statement about vector spaces
[Apologies if this is a really trivial question, I know virtually nothing about set theory, and the following came up while preparing undergraduate linear algebra lectures.]
I'm asking about the ...
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Is the Class Well Ordering principle "CWO" the maximal choice principle?
In a prior posting, the Class Well-Ordering principle "$\sf CWO$" was presented, which simply states that there is a well-ordering over all classes of $\sf MK$. On the other hand, it is ...
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Reverse-mathematical strength of Banach-Tarski
What is the reverse mathematical strength of the Banach-Tarski paradox?
The usual proof of Banach-Tarski should carry out in $\mathrm{ZF}+\mathrm{AC}_\kappa$, where $\kappa$ is the supremum of the ...
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Is there a class choice principle over MK that is equivalent to class well ordering over MK?
$\sf MKCWO$ is the theory obtained by adding a new primitive binary relation $\prec$ to the signature of $\sf MK$ and axiomatize that $\prec$ is a well order on classes, that is:
$\textbf{Transitive:}...
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Does $\mathsf{ZF}$ prove $\operatorname{Col}(\lambda,\kappa)$ preserves cardinals below $\lambda$?
Let $\lambda<\kappa$ be cardinals and consider the forcing $\operatorname{Col}(\lambda,\kappa)$ adding a generic surjection $\lambda\to\kappa$. More formally, $\operatorname{Col}(\lambda,\kappa)$ ...
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Locally presentable and accessible categories without the axiom of choice?
Is there a good reference for the study of locally presentable and accessible categories without the axiom of choice? For instance, it seems one will need to understand:
What is a good notion of $\...
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An equivalent of the axiom of choice? [closed]
There is such a thing as a math course for relatively non-mathematically inclined people that is intended to challenge students' intelligence more than to teach them some mathematics. (It is true that ...
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Is Morse-Kelley set theory with Class Choice bi-interpretable with itself after removing Extensionality for classes?
Let $\sf MKCC$ stand for Morse-Kelley set theory with Class Choice. And let this theory be precisely $\sf MK$ with a binary primitive symbol $\prec$ added to its language, and the following axioms ...
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