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Hm? I thought (in)completeness was just about whether or not , for each well-formed-formula, either there is a proof of it, or a proof of its negation.

The CH is a syntactically valid statement in ZFC.

So, shouldn't the fact that ZFC cannot prove or disprove CH, be an example of ZFC being incomplete, regardless of whether CH is in fact true, false, or not-a-proposition-that-has-a-truth-value ?



Ok. If we are being careful, there are different kinds of completeness and we should specify which one we're talking about.

Here's a good list: https://en.wikipedia.org/wiki/Completeness_(logic)#Forms_of_...

The existence of a proof within the logic system for every well-formed formula or its negation is "syntactical completeness".


Oh cool, I was (at least partially) wrong. (Or, I was missing something important at least.) I appreciate the correction/elaboration.




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