G=This sentence is not provable in T — TheMadFool
Q is enumerable and so is the set of finite sequences of members of Q. The set of infinite sequences of members of Q and R are not enumerable. Okay, some things are enumerable and others aren't.
— TonesInDeepFreeze
I know all that, already said it, spent years writing proofs for professors. Not asking random internet guy about the basics of analysis. Tried to ask you about 'subjective' (maybe philosophical) responses to all the symbols that swim like fish in those textbooks you mentioned. You gave a disappointing response, like you are deaf and mute to anything that isn't mere chatbot correctness. I have loved math as a meaningful 'science' of form(s) with some intuitive validity. I care about various formalisms only because they strive to mean something, capture something beyond them. The continuum is a endlessly fascinating beast that great thinkers have wrestled with for centuries. I don't know if you know or care much about mathematical history, but I love the drama. But I'll save that for others who aren't satisfied with the relatively trivial (however difficult at times ) syntactical part. — plaque flag
Shouldn't G be in the form of arithmetic calculus propositions for the incomplete theorem to apply?1. G is provable. So G is unprovable
2. G is not provable
So, there is G in the theory T
Have I got it right? — TheMadFool
Could you demonstrate and prove the provability and unprovability of G in real arithmetic sentences in T?We adduce a sentence G that is is true (to be more precise, it is true in the standard model for the language of arithmetic) if and only if G is not provable in T.
Then we prove that G is not provable in T. So G is a true sentence that is not provable in T. Moreover we show also that ~G is not provable in T. So T is incomplete. — TonesInDeepFreeze
We adduce a sentence G that is is true (to be more precise, it is true in the standard model for the language of arithmetic) if and only if G is not provable in T.
Then we prove that G is not provable in T. — TonesInDeepFreeze
Could you demonstrate and prove the provability and unprovability of G in real arithmetic sentences in T? — Corvus
If all it proves is that every T has the true and unprovable sentence "this sentence is true and unprovable" then it seems vacuous. — Michael
Or does it prove that every T has a "natural" example of a true and unprovable sentence, like the strengthened finite Ramsey theorem in Peano arithmetic? — Michael
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.