I have read recently a chapter in Hao Wang: A logical journey: From Godel to Philosophy , where Wang mentions his discussions with Godel on intensional paradoxes, but I have no clue what exactly they mean by intensional paradoxes and what is their importance and how do they relate to concept of concept and concept of absolute proof and why would one get a clear proof that mind is not a machine. -->
Gödel was keen to distinguish the intensional from the extensional. According to his account, the intensional is concerned with concepts, which are independent of their expression in any particular language. Concepts are distinct from classes, which are extensional. Concepts do not presuppose any particular domain of application. Gödel even suggests that this difference lies at the heart of distinguishing logic from mathematics:
The subject matter of logic is intensions (concepts); that of mathematics is extensions (sets). (Wang, p. 274)
For Gödel, paradoxes are a sign of having a defective or incomplete understanding of the notions of concept or class. While we can deal with extensional paradoxes, like Russell's paradox, by maintaining that a class may not belong to itself, this is not an option with concepts, since concepts can apply to themselves. For example, Wang (section 8.6.3) quotes Gödel as follows:
... while no set can belong to itself, some concepts can apply to themselves: the concept of concept, the concept of being applicable to only one thing (or one object), the concept of being distinct from the set of all finite mathematical sets, the concept of being a concept with an infinite range, and so on. It is erroneous to think that to each concept there corresponds a set.
Gödel also distinguishes intensional paradoxes from semantic paradoxes, which are those that arise from the use of a particular language. Gödel proposes to dispense with the liar paradox as a semantic paradox. We can ask the question whether the sentence “This sentence is not true” is true or not. However, the property ‘true in language L’ cannot itself be a predicate of L, so the paradox is resolved. By contrast, conceptual paradoxes can be formulated without reference to language at all. (Wang, section 8.5.10).
Gödel, as a platonist, thought that concepts are real things that exist independently of us, so the intensional paradoxes are a substantial problem standing in the way of our having a proper understanding of concepts. Gödel seems to suppose that if only we could clear up all of these paradoxes, we would be left with a perfectly clear understanding of how concepts work, including the logical concept of proof, and this would somehow show that the mind is not a machine. Exactly what he had in mind is not clear (to me, at least). Perhaps he considered that computation is purely extensional in nature, and therefore if we could acquire a perfect understanding of concepts, it would show that our intuitions transcend what is computable. Others, famously J. R. Lucas, have tried to defend the thesis that minds are not machines on this basis, but the claim is widely thought to be dubious.
As to the concept of absolute proof, it is distinguished from relative proof. Often, we cannot prove in absolute terms that a theory is consistent, only that it is consistent relative to some other given theory. I.e., we may be able to prove that if theory A is consistent then theory is B is consistent, but not absolutely that B is consistent. Gödel seems hopeful that a clarification of the concept of absolute proof would allow us to prove the consistency of our own minds, and hence show that our mathematical intuitions transcend what is computable by machines. Again, the claim seems dubious.