Axiom of choice

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See wikipedia:Axiom of choice

The axiom of choice (AC) is an axiom, usually adjoined to the Zermelo-Frenkael (ZF) axioms of set theory. With this axiom the theory is known as ZFC. ZF and ZFC are equiconsistent.

Informally the axiom of choice says that given a family of non-empty sets we have a function selecting exactly one element from each set.

Equivalent formulations

The axiom has many equivalents, including but not limited to:

  • The assertion that all the infinite cardinals are $\aleph$-numbers.
  • The well-ordering principle, which states that every set can be well-ordered.
  • Zorn's lemma, stating that if the union of any chain in $x$ is itself a member of $x$, then $x$ must have some maximal element, with respect to inclusion.
  • Cantor's Law of Trichotomy, the statement that any two sets are comparable with respect to cardinality, that is, $|A|\lt|B| \lor |A|=|B| \lor |B|\lt|A|$

Weaker forms

Among weaker form are the axiom of dependent choice (AD) and the axiom of countable choice ($\text{AC}_\omega$).

Statements inconsistent with AC

Cardinals without the axiom of choice cannot all be equated with initial ordinals. Some sets cannot be well-ordered, and the Hartog number measures how well-ordered they can be. There can be e.g. infinite Dedekind finite cardinals and amorphous sets.

Reinhardt cardinals and Berkeley cardinals are known to be inconsistent with ZFC, but not known to be inconsistent with ZF.

Axiom of determinacy is inconsistent with AC too.

Without the axiom of choice $\omega_1$ can be e.g. measurable (this is even consistent with dependent choice), strongly compact, supercompact(Jech, 1968, Takeuti, 1970, after [1]) or huge [2]. More generally, e.g., if $κ$ is a weakly compact cardinal and $η < κ$ is a regular cardinal then there is a symmetric model in which $η^+$ is weakly compact (as Jech, 1968 showed). This applies to large cardinal notions preserved under small symmetric forcing, like measurability, weak compactness or Ramseyness, but not to Erdős cardinals (the partition property $κ → (α)^{<ω}_2$ is preserved, but the requirement that they are minimal such that a partition property holds is not necessarily preserved). ([3], chapter 3.3)

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