4 ms·
It's "possible" presumably only in the sense in which one can assign a True/False value to any proposition. I guess one could see a logic as deciding if a set
by mjburgess 4y ago
It's "possible" presumably only in the sense in which one can assign a True/False value to any proposition.
I guess one could see a logic as deciding if a set of propositions is inconsistent. And one could imagine a "trivial logic" which says "Yes" for any set, consistent?(props | logic) = Yes.
Then as we add more structure to the logic, more sets become inconsistent: consistent?(CountableReals | LawExMiddle) = No.
What value is there to saying the reals are "possibly" countable when it seems we're basically enforcing no consistency in the set of propositions which would constitute a "proof" of such a thing?
- chriswarbo 4y agoEuclidean geometry is a common example of this: if we remove the parallel postulate, we get a weaker theory; unable to prove various "facts". However, this weaker theory is also more general, applying to curved spaces too; where those facts are not true. It turns out that the physical universe is curved, and hence those original "facts" don't hold in the 'real world'. It seems like a similar thing is happening here: we can prove that the set of reals is uncountable, but only if we use classical axioms which don't account for computability. Using a weaker (intuitionistic/constructive) theory prevents us proving that the reals are uncountable; that turns out to be necessary, because some of the alternatives cause the reals to be countable. Since the physical universe is computable (or at least, computably indistinguishable from being computable), these alternatives may be better descriptions of the 'real world' than classical logic.