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Math Help - Must all models of ZFC (in a standard formulation) be at least countable?

  1. #1
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    Must all models of ZFC (in a standard formulation) be at least countable?

    Must all models of ZFC (in a standard formulation) be at least countable?


    Why I think this: there are countably many instances of Replacement, and so, if a model is to satisfy Replacement, it must have at least countably many satisfactions of it.


    Does my question only apply to first-order formulations of ZFC, or are there second-order formulations of ZFC that can be finite?


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  2. #2
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    Re: Must all models of ZFC (in a standard formulation) be at least countable?

    yes.

    i don't think you even need replacement, the axiom of infinity will suffice.
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