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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Are Berkeley cardinals easier to refute in ZFC than Reinhardt cardinals?
Kunen showed that Reinhardt cardinals are inconsistent in ZFC. But his proof is a bit technical for a non-set-theorist to follow. Berkeley cardinals are stronger than Reinhardt cardinals. You can ...
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Can well-ordering of the universe due to global choice survive extensive failure of Extensionality?
That axiom of global choice leads to the well-ordering of the universe given the other axioms of Zermelo set theory is a famous result.
Now, if we weaken the power set axiom to the axiom stating that ...
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Trading Choice for Comprehension (or Replacement)
This question is basically a request for clarification about a remark made by Sam Sanders in a comment to another question: IIUC what he's saying, there are statements that can be proved either with a ...
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Exponentiation of Dedekind cardinals
Question: Let $\mathfrak n$ be an infinite cardinal. In ZF (set theory without the axiom of choice) can either of the implications
$$\mathfrak n=\mathfrak n+1\implies2^\mathfrak n=2^{\mathfrak n+1}\...
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The Parity Principle and $\mathbf{C}_2$ (choice for $2$-sets)
The Parity Principle states that
if $X\neq \emptyset$ is a set, then there is $\mathcal B\subseteq \mathcal P(X)$ such that whenever $a,b\in \mathcal P(X)$ with $a\mathbin\Delta b = \{x\}$ for some $...
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Parity and the Axiom of Choice
Motivation. The three-dimensional cube can be formalized by $\mathcal P(\{0,1,2\})$ where vertices $x,y\in\mathcal P(\{0,1,2\})$ are connected by an edge if and only if their symmetric difference $x\...
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Models of ZF (without Inaccessible cardinals) where only the full Axiom of Choice fails, but the Axiom of Countable Choice remains true?
Solovay's model (which assumes $I$ = "existence of inaccessible cardinal") will be a well-known construction to produce a model of ZF where only the full Axiom of Choice ($AC$) fails, but ...
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What are some "easy" violations of $\mathsf{SVC}$?
By $\mathsf{SVC}$, I mean "small violations of choice", which is the statement
$$(\exists S)(\forall X)(\exists f)``f\colon S\times\text{Ord}\to X\text{ is a surjection}".$$
Such an $S$ is ...
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How much of mathematical General Relativity depends on the Axiom of Choice?
One of the cornerstones of the mathematical formulation of General Relativity (GR) is the result (due to Choquet-Bruhat and others) that the initial value problem for the Einstein field equations is ...
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How much of the axiom of choice do you need in mathematics?
Say we have DC-λ where λ is some inaccessible cardinal. Is that enough to develop all of ordinary mathematics? If not, is there a strengthening that is but that nevertheless does not assume full ...
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The monochromatic principle and the axiom of choice
For any set $A\neq\emptyset$, denote by $[A]^A$ the collection of sets $B\subseteq A$ such that there is a bijection $\varphi:B\to A$. If ${\cal S}\subseteq [A]^A$, we say that $B\in[A]^A$ is ...
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"Minimal-ish" Dedekind-finite cardinalities of models
Throughout, we work in $\mathsf{ZF}+$ "There is an infinite Dedekind-finite set."
Say that a Dedekind-finite cardinality $\kappa$ is $\Sigma^1_1$-isolated iff there is some first-order ...
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Posets such that the collection of principal down-sets does not have property ${\bf B}$
We say that a hypergraph $H=(V,E)$ has property ${\bf B}$ if there is $S\subseteq V$ such that for all $e\in E$ with $|e|>1$ we have $S\cap e \neq \emptyset \neq e \setminus S$.
Let $(P,\leq)$ be a ...
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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 ...
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Do the difficulties in generalising Henstock-Kurzweil still exist if every subset of $\mathbb R^n$ is Lebesgue measurable?
There are apparently some difficulties generalising the Henstock-Kurzweil integral from functions of signature $\mathbb R\to\mathbb R$ to functions of signature $\mathbb R^n \to \mathbb R$. One ...
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