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Jan 16, 2023 at 20:16 comment added Chill2Macht @Neil I can't speak for the answerer, but I believe the following sentence ("truth in V cannot be defined in V due to a result of Tarski") in the answer is the explanation. I.e. the result of Tarski is what proves "the fact that V is a model of ZFC cannot be proven formally within ZFC". The result of Tarski being referred to is presumably Tarski's theorem on the nondefinability of truth, which is closely related to Godel's incompleteness theorem (if you're more familiar with the latter).
Aug 4, 2019 at 21:23 comment added Neil "However, the fact that V is a model of ZFC cannot be proven formally within ZFC" where can it be proven? (sincere question).
Feb 2, 2010 at 2:27 history answered Norman Lewis Perlmutter CC BY-SA 2.5