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. -->

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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.

  • It's worth noting that only theories extending Robinson Q and is primitive recursively axiomatized cannot prove its own consistency absolutely per Godel. Many philosophy theories expressed in natural languages may only need Presburger arithmetic at most (if need some simple utilities), so most of them may have chance to be claimed to be absolutely consistent and complete, such as reflected in many ancient sutra titles end in Pāramitā (meaning complete perfection)... Nov 27 '21 at 23:20
  • True, though presumably Gödel would maintain that our minds are powerful enough to understand and use arithmetics at least as strong as Q. So unless we are prepared to accept some pretty severe limitations on what mathematicians are capable of doing, proving the absolute consistency of the mind lies beyond the capabilities of recursively axiomatised systems, and hence of computers.
    – Bumble
    Nov 28 '21 at 0:32
  • As for Godelian implication to mind, we'd better differentiate 2 kinds of mind first. One is the usual flawed human mind, which on the contrary may very well be inferior to machines, thus all the sages in those sutras urged us to strive to think, speak and act like a recursively axiomatized consistent system. Another is the speculated idealized angelic mind whose existence depends transcending usual computability. I remember Godel cautiously hold just a disjunctive proposition though he may be inclined to be for its existence since though he didn't go to church but read bible every Sunday... Nov 28 '21 at 0:55
  • "While we can deal with extensional paradoxes, like Russell's paradox, by maintaining that a class may not belong to itself" Isn't the resolution to ban unrestricted set comprehension in favor of the axiom of specification? Non-well-founded set theory is consistent. en.wikipedia.org/wiki/Non-well-founded_set_theory
    – user4894
    Nov 28 '21 at 1:40
  • @user4894 the axiom schema of specification resolves Russell's paradox by rejecting unrestricted (naive) comprehension, but it cannot ensure "a class may not belong to itself", axiom of foundation is needed to further reject a set containing itself such as {{{...}}}. There're some paraconsistent set theories accepting naive comprehension or a class/set may belong to itself, or via higher order logic one can have some intensional set/type theory. So for me it's not that clear-cut math only deals with extensions. Nov 28 '21 at 3:23

Intensional, as used here, is synonymous with self-referential, nothing more. For instance, Russell's paradox. Or, for that matter, any other paradox, or contradiction arising due to self-reference.

  • Russell’s paradox is not an intensional paradox but extensional paradox because it deals with set theory. Dec 1 '21 at 9:46
  • @bodhihammer That's not what extensional means, that's a very naive understanding of ex-in statements, a statement could be set-theoretic but still intensional. Dec 1 '21 at 21:53
  • It’s the distinction that, for instance, Ramsey made, and also Godel. They considered Russell’s paradox an extensional one. Set theoretic operators are extensional operators. Dec 2 '21 at 8:50
  • @bodhihammer look, IN CONTEXT set theory is extensional because it's an extention of logic - axioms of ZFC are all in the the second order predicate logic. IN CONTEXT, then Logic becomes intensional because there is nothing more fundamental than it except for Language, structure, and etc but that is in totality logic. Set theory can, however, be broken down further making it an extension of logic (in other words, logic is defined recursively, in terms of itself, qua intensional), set theory is not. Dec 2 '21 at 9:20
  • @bodhihammer In the context of OP's question, intensional paradoxes are those paradoxes that arise due to self reference (qua intensional), in one for or another. So when you look at it, in context, every paradox, due to self reference, is intensional. I hope this clarifies it for you. (Very, very roughly) Think of it like this intensional: in terms of itself; Extensional: in terms of other than itself. Dec 2 '21 at 9:25

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