Another one of my topics in the Logic & Philosophy of Mathematics section.

Wittgenstein as a logician stated that the sum total of logic consists of tautologies, contradictions included. He also said that quintessentially because logic is tautological, logic takes care of itself.

Now, both Wittgenstein and Gödel met at the Vienna Circle, a meeting-place for the then acclaimed logical positivists. After Wittgenstein published his Tractatus-Logico-Philosophicus, the Vienna Circle studied it not only once but twice. Gödel held a positive opinion of the work. The only point of contention between Gödel and Wittgenstein was one single instance of disagreement about the implication of Gödel's work on logic and mathematics. Namely, Gödel's Incompleteness Theorems is what Wittgenstein claimed to be logical "tricks". Some people say that Wittgenstein didn't understand Gödel, however, I think it is the other way around. Let me explain in brief.

What Gödel proved was a truth that can only be seen in a formal system such as Peano Arithmetic. The rules governing formal systems do not apply to informal systems, where pretty much any instance of applying a new rule to an informal system, can be integrated into the system itself. This feature of informal systems, such as language, is not possible in formal systems like Peano Arithmetic. Wittgenstein went to great lengths in the Philosophical Investigations to try and see what rules govern language, and was not able to provide a clear answer. Just to point out my own understanding, the problem of what governs rules in language is a subject best framed, not in terms of logic, but according to syntax and grammar. Grammar is not subject to the constrains of logic, it takes part in the very structure of how language is able to be transmitted and understood between people. One day we will be able to understand universal grammar, which would be able to answer the question that grappled or still grapples many analytic philosophers since Bertrand Russell and his protege, Wittgenstein.

To return to the topic of Wittgenstein and Gödel, if one looks even at nature or physics, which describes logic, which the system of nature and physics is in fact complete and whole. There is no incompleteness in nature (As a side note, Einstein had discussions with Gödel about how singularities and indeterminism could partake in physics and nature, which made him think nature was "incomplete," for lack of a better word). This ad hoc argument is formulated against the Incompleteness Theorems regarding how nature and the universe are unitary and whole. None of this would make any sense to readers who think that logic is independent of reality. I don't think this is possible. Logic manifests itself from human intuition and the laws of nature describing the relations logic can posit, through human understanding.

Not to sound grandiose; but until greater understanding is attained about the relation between logic and mathematics, which was an aspiration of many mathematicians and logicians during the war period, which Gödel had negated with his Incompleteness Theorems, then we will not have the scaffolding to venture past the conclusions of Gödel. Hilbert's program was intended for such greater understanding. Again, logic takes care of itself and its relation to mathematics is quite possibly something we would be able to better understand if we can get past the conclusions of Gödel's Incompleteness Theorem's.

I would like to end this post with a thought that generated, mostly, these corresponding thoughts about Gödel and Wittgenstein, and physics.

Gödel's incompleteness theorem applies to formal languages with countable alphabets. So it does not rule out the possibility that one might be able to prove 'everything' in a formal system with an uncountable alphabet OR expand the alphabet to account for new variables*. — Shawn

*-Variables could stand for new understanding itself, within a system of thought or a model of reality.

Edit: Added and fixed some things.