9. the Axiom of Choice and Zorn’s Lemma
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چکیده
§9.1 The Axiom of Choice We come now to the most important part of set theory for other branches of mathematics. Although infinite set theory is technically the foundation for all mathematics, in practice it is perfectly valid for a mathematician to ignore it – with two exceptions. Of course most of the basic set constructions as outlined in chapter 2 (unions, intersections, cartesian products, functions etc.) are part of a mathematician’s standard language. The second exception is the Axiom of Choice and Zorn’s Lemma. Zorn’s Lemma, sounds like a small theorem that is preparatory to a bigger theorem. That’s the usual meaning of the word “lemma”. In fact it’s not a theorem at all – it is an axiom. And the Axiom of Choice is also an axiom. These axioms are equivalent. That is, you can prove Zorn’s Lemma if you assume the Axiom of Choice and you can prove the Axiom of Choice if you assume Zorn’s Lemma. But this axiom, for indeed they are essentially just a single axiom, is consistent with and independent from the ZF axioms. It has the same status as the Continuum Hypothesis, which is also an optional axioms. In fact the Axiom-of-Choice-Zorn’s Lemma combination is consistent with, and independent from, the ZF axioms supplemented by the Continuum Hypothesis. This means that you logically can choose on of the following four set theories.
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