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Does your uncountable set of propositions include some which cannot be written as a finite string of symbols? If "yes" - how might such as proposition be prove
by bell-cot 2mo ago
Does your uncountable set of propositions include some which cannot be written as a finite string of symbols?
If "yes" - how might such as proposition be proven true with a finite string of symbols?
(If your proof symbols are from an infinite character set, that has its own issues.)
- gradschool 2mo agoI'm assuming a finite alphabet and a finitely axiomatizable proof system per convention. I can't think of an uncountable set of propositions in which each can be written as a finite string of symbols, so that's what I was missing. Thank you for clearing this up for me.
- Xmd5a 2mo agoIf you substitute uncountability in your intuition with algorithmic incompressibility (too much information to be captured by shorter descriptions) you have Chaitin's incompleteness theorem. Kudos for the well-placed hunch!