1. Is there a connection between Wittgenstein's argument against the "Theory of Types" and the proof of Godel's Incompleteness Theorem? Being only semi-knowledgeable, I will draw the connection of which I am thinking: it seems that Godel's proof relies on referring to symbols as numbers whereas Wittgenstein's argument is that you can do no such thing in Russel's Theory of types. Perhaps the connection is deeper than I imagine or perhaps I am off base by trying to make such a connection.

  2. Was it really Wittgenstein's argument against type theory that changed Logical Positivism so drastically? If not, then what was it that happened before Godel's proof which stopped Analytic Philosophy from thinking that it could axiomitize language? I ask this because, to me, it seems clear that Godel's incompleteness theorem would have stopped this project in its tracks.

  3. Should Wittgenstein be given some credit for Godel's incompleteness theorem?

  1. Yes, there is a connection, as you point out. In the Tractatus, Wittgenstein writes:

    3.332 No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the whole "theory of types").

    Gödel, as you know, proceeded to do precisely that.

  2. Wittgenstein's argument against type theory is one of many factors that changed Logical Positivism. Russell's "barber paradox" was another. If the history of Logical Positivism interests you, I'd recommend a delightful graphic novel called Logicomix which covers the territory nicely.

  3. Not really.

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    I thought Russell's type theory was intended to avoid paradoxes in set creation. Was W's argument against type theory inTLP or was it later? (the time line of the 'changes' is confusing to me? – Mitch Aug 14 '11 at 14:00
  • The short answer is that the history of Logical Positivism is also the history of its unravelling. Russell's "barber paradox" pointed to problems with set theory, so he invented the theory of types to try to address that. Wittgenstein found problems in the theory of types (in the Tractatus, and afterwards); Gödel later developed his own attack along these lines, etc. – Michael Dorfman Aug 17 '11 at 6:52
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    I think I should point out a technicality that many might not get by reading a lot of popular references on the incompleteness theorems: Gödel sentences don't explicitly refer to themselves. In arithmetic or number theory, for example, there is no symbol or symbolic way of saying "this formula." However, the proofs of the incompleteness theorems explicitly construct a Gödel sentence, and show that it will always be logically equivalent (in effect) to our informal interpretation of "this statement is not provable within the theory." – anon Aug 22 '11 at 0:10

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