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Your word salad is not even wrong. For example:

> Gödel number encodes a paradox

WTF do you mean by this?

A statement that can't be proved true or false and thus demonstrates incompleteness of logic.

How is that a paradox? And how does that imply the correct argument I gave doesn't work? (It doesn't imply that.)

Paradox is a statement that can't be proved true or false; "this statement is false" is an example of Gödel statement. The argument doesn't work, because execution of a Turing machine can't be expressed as a Gödel number.

> Paradox is a statement that can't be proved true or false

No it isn't.

> execution of a Turing machine can't be expressed as a Gödel number

Yes it can.

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