Re:In Mathematics, Mistakes Aren’t What They Used to Be (2015)
by yorwba on Saturday August 27, 2016 @04:16PM
(#12373988)
> Are all unprovable theorems true?
"True" in this context means "conforms to my intuitive understanding of what true means".
Gödel sentences are an elaborate construction that express the concept "This sentence is unprovable from $AXIOMS" without using direct self-reference, but this construction only means what we think it means if the natural numbers are the "real" natural numbers.
Because no finite set of axioms can rule out all nonstandard models, you can't be sure that there isn't some "proof" corresponding to a nonstandard number.
If such a thing exists, the Gödel sentence is still unprovable (because no proof exists) but false (because it asserts that no "proof" exists, although there is a nonstandard one).
"True" in this context means "conforms to my intuitive understanding of what true means".
Gödel sentences are an elaborate construction that express the concept "This sentence is unprovable from $AXIOMS" without using direct self-reference, but this construction only means what we think it means if the natural numbers are the "real" natural numbers.
Because no finite set of axioms can rule out all nonstandard models, you can't be sure that there isn't some "proof" corresponding to a nonstandard number.
If such a thing exists, the Gödel sentence is still unprovable (because no proof exists) but false (because it asserts that no "proof" exists, although there is a nonstandard one).