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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
by drdeca 6y ago
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 ?
- jlokier 6y agoOk. 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_completeness 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".
- drdeca 6y agoOh cool, I was (at least partially) wrong. (Or, I was missing something important at least.) I appreciate the correction/elaboration.