3 ms·
So, does that mean that one can assume this to be true and build a perfectly consistent theory, or conversely assume it to be false (with - say - at least one c
by ur-whale 7y ago
So, does that mean that one can assume this to be true and build a perfectly consistent theory, or conversely assume it to be false (with - say - at least one counter-example) and build another perfectly consistent theory?
- deleted 7y ago[deleted]
- karatinversion 7y agoYes, this is the exact meaning of “independent from” here.
- deepsun 7y agoWell, it's not possible to prove the consistency, thanks to Godel. Maybe one of your new theories would contain a statement, inconsistent with the rest of ZFC.
- CrazyStat 7y agoThis is incorrect. It absolutely is possible to prove consistency, what Gödel tells us is that in any consistent logic system there are true but unprovable (in that system) statements. For this particular list, the statements have been proven to both be consistent with ZFC and for their negations to be consistent with ZFC.
- ithinkso 7y agoThis is first incompleteness theorem. What deepsun was referring to is second incompleteness theorem - in a consistent system F the statement 'F is consistent' is in fact unprovable (in F).
- ginnungagap 7y agoWhen we say that a statement P is independent of ZFC what we're really saying is "if ZFC is consistent, then P is independent of ZFC (hence both ZFC+P and ZFC+not P are consistent)". This is the only sensible way to interpret claims of independency, since if ZFC is inconsistent it just proves every statement so there's no independent ones