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A friend of mine advocated Wittgenstein's criticism of Godel's proof. They hold that there is something nongrammatical about Godel's proposition that makes for nonsense and therefore falsehood. And, a professor of physics allowed that mathematics was a failure. I was trying to restudy the matter anyway, so let me formulate my present understanding.

I need a statement of Godel's proposition. I think it can be put like this. A proof, when encoded in its Godel number, does not exist for this very statement, also as encoded in its Godel number.

If I want to make this statement false, the weakness would be in the encoding. If I want to make this statement true, then it is only necessary to uphold a standard understanding of integer arithmetic and computer science.

A proof for this proposition can not be encoded in a Godel number, apparently because the Godel number is a language not made to accommodate the needed metamathematical notation. So, if the Godel language were adjusted to be more of a Spinozist language (and nothing prevents this adjustment), then Godel's proposition would then be provable in the extended language, but would remain not provable in the original language as required by the proposition itself. If the proposition is adjusted to refer to the expanded language, then it would be false.

The nature of integer arithmetic can be adjusted to falsify Godel's statement. The adjustment requires that consistent symbolism by integer numbers be no longer possible, at least in this case. When I consider that recursive infinities interfere with the encoding of both the proposition and a would-be proof, then this adjustment does not seem so extreme.

Or, one could ask what this has to do with integer arithmetic, since it is only used as typesetting. I suppose that the answer to this is that the logic language that Godel encoded in numbers is itself capable of generating integer arithmetic.

But, nothing compels me to denounce the standard model of arithmetic or Godel's code, nor has the case been made that the Godel encoding is wrong or unreliable. So, rejecting Godel's proof seems to be simply a restrictive redefinition of things. There is definitely not a failure of mathematics here. Instead, there is a multiplicity of extensions of integer arithmetic to invoke when they are needed.

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Michael J. Burns