Abstract
The ZF axioms allow us to assert the existence of any set whose members are selected according to some definable “rule”—this is essentially what the replacement schema says. However, we often want to assert the existence of a set without knowing a rule for selecting its members. Typically, the members are simply “chosen” from other sets, but not according to any “rule.” When infinitely many choices are required, we may not be able to guarantee the existence of the set without some axiom of choice.
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Notes
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A reason for writing σ on different sides in the two notations is that in σ ∗ b we use σ before seeing b, namely, on the empty sequence; and in a ∗σ we use σ after seeing (the first member of) a.
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The usual statement of Zorn’s lemma does not restrict the ordering to be set inclusion. However, this is the only case we need, and there is really no loss of generality.
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Stillwell, J. (2013). The Axiom of Choice. In: The Real Numbers. Undergraduate Texts in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-01577-4_7
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