Gödel's Incompleteness Theorem

Imagine there is a library containing a record of every possible fact, under every conceivable situation. That is, what would be correct logically under one set of axioms, as opposed to a differing set, as well as every truth of history, the present, the future and well, everything!

A man walks into the library, talks to the librarian and asks for the fact of a matter regarding a scribble in his notepad. The scribble reads... "The library will never assert that this statement is true."

What a quandary!

If the library has the 'fact' as true or false, what then? If the library has a book stating the falsity of the statement, then the statement is true, and can not be false. If there is a record in the stacks somewhere that the statement is true, then it has contradicted itself.

This shows us that either;

  • 1) We know something the library doesn't. (That the statement is an 'exhibit A' for incompleteness)
  • 2) The library can not know either way. Therefore it's 'complete knowledge' is incomplete.

Of course, the corresponding statement, "The library will always assert this statement is false" has a similar outcome. If the library says "true", it has contradicted itself. Yet again, if it states it is false, the statement is true.

So do we know something that the library can never know? What if the library knows that it can never know it?

Of course this has been seized upon to claim that God is an inconsistency, because it is impossible for God to know everything. Therefore God is not perfect.

So of our statement: if the decidability of a statement equates to whether it is provable or not, then the lack of the ability to reason one way or another in a system is not to state the falsehood of the system: rather the system itself must be inclusive of some statements that the language may not be able to show correct or untrue: But in the example above, is the statement on the notepad a proper statement? Surely if it is not, we could state that it is not a case of true or false, but one of some form of nullity.

Nullity, as a conjunction can prove any disjunction we choose: that the language may not decide the statement or its opposite as true or false would indicate that somehow the complement of the empty set within the language itself is in itself a contradiction and therefore also empty. Then the language can not describe the statement: or rather the statement can not be formalised in the language of the system. (analogous to "not nothing" = nothing)

But if God is an inconsistent subject we would need to affirm that some truly positive statements were indescribable in any language. Positive statements result from the negation ("opposite of positive") and this somehow privates the perfection of God, rather than contradicting a language. Perfection appears in a non-principal ultrafilter in Anselm's argument. We need to show there is a consistent language with a correct model with which our language permits the existence of God in a coherent manner.

Nullity would have the result in the language that the perfection of God may not account for some statements - i.e. if and only if we find a disjunction where both sides are positve or both negative statements. The library example is just such a case, that either the library can not answer "The library will never assert that this statement is true." or "The library will always assert this statement is false". both are "negative" to the theist's hope for an answer.

However is there some form of statement where these form either side of a single disjunction? We would have to imply from the empty set to show such a disjunction "provable" or that it exists. The lack of a conjunction aside from nullity to imply this could only show of statement A v B the non decidability of the statement. ("not nothing" = nothing again.) If we may eliminate the requirement in the language for an empty set, and we have no need to imply from it to account for any decisively positive properties, we must admit there may be non-decidable ones also.

We should be prepared to assert that positive properties or perfections have their opposite or negation always a privation in some form, and so the library should assert "the statement is arrived at from nullity" rather than "true" or "false".

Then axioms may exist to assert truth or falsehood in their own setting: but even the axiomatic method fails in "completeness" when it is tested with itself: it is a closed system, but if there is a null axiom, then there is a null statement, and a null system in many forms.

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