Math 89 Winter 2016

Set Theory

Set Theory

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The non-absoluteness of the aleph hierarchy. So we see that given a model, V, of ZFC set theory (equivalently that ZFC is consistent), then we must have a universe of discourse V* that satisfies the formal description of ZFC. Yet the universe V* is countable in the eyes of V. What does this really say about Cantor's diagonalization argument for the uncountability of the real numbers? Note that the argument holds in both V and V* since it is formally argued from the axioms ZFC. So who are the real numbers really, and what does it mean to say that they are uncountable in a given model? | Löwenheim-Skolem |