Suppose for the sake of argument that we look at ZFC with the axiom of infinity removed.(adsbygoogle = window.adsbygoogle || []).push({});

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems#First_incompleteness_theorem

http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems#Second_incompleteness_theorem

We would then be in a position where the hypotheses of Gödel's theorems are not satisfied, correct? Basically, I want to remove, for the sake of argument, a minimal amount of axioms of ZFC so that ZFC minus some axiom(s) leads to a theory that does not include arithmetical truths and is not capable of expressing elementary arithmetic.

Is it possible that ZFC- (my shorthand for ZFC with some axiom(s) removed) is consistent and complete?

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# Gödel's incompleteness wrt weakend versions of ZFC

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