1: Axioms that were strong enough to prove all true statements in arithmetic. (completeness)

2: The language was defined so that no contradictory statement could be proven within the language. (consistency)

In addition they wanted to eventually prove that their language satisfied these criteria. If this was proven then they would know that any statement in the language essentially could be sifted into two piles: provable (true) or unprovable (false). Because self reference was eliminated, things like "this statement is false" could not be written in the language and hence were not statements in the language.

However, Kurt Gödel discovered that because their language referred to numbers, numbers could then be used to refer to the language through a system called Gödel numbering, allowing statements such as: "this statement is not provable in this language" which in a complete system is essentially the same as "this statement is false".

If Gödel's statement were true it would be false and if it were false it would be true, but even further the statement merely refers to provability and hence we must essentially prove it before we can prove it. What is interesting about this statement is that while it is obviously not demonstrable true or false we can easily see that it is true.

This method of "seeing" I attribute to a conceptual ladder that Wittgenstein referred to in proposition 6.54 in his "Tractatus Logico-Philosophicus":

"6.54 My propositions are elucidatory in this way: he who understands

me finally recognizes them as senseless, when he has climbed out

through them, on them, over them. (He must so to speak throw

away the ladder, after he has climbed up on it.)

He must surmount these propositions; then he sees the world

rightly."

This is useful to a certain extent. It is true that even if we add another axiom to the formal language where Gödel's statement is defined as true, we must change the language in order to prevent it from being contradictory, hence "...not provable in this language" can no longer refer to "this" language which is where we get into further manifestations of self reference due to the increased complexity of the new language, but we can still see that the original statement in the original language was true. A sigh of relief follows.

But with my own nastier example, take the statement: "this statement cannot be proven as true through the methods of philosophy". It is obviously a philosophical statement but if it can be proven in philosophy it will be false and if it cannot be proven it is true. You may think that it is similar to the Gödel statement in that although we cannot prove it we can easily see that it is true because we can say philosophically that there is nothing about the statement that would determine its truth value without us first attempting to prove it since it merely refers to provability.

However this in turn causes it to be false because philosophy is any logical system we can think of and hence the same original statement will remain with us regardless of whether we add a further layer of abstraction to the methods of philosophy.

Is the statement false or is it true? Or is it undecidable which really does make it true? You decide... or decide not to decide... which isn't not deciding. Here Wittgenstein's ladder is infinite and we are forever climbing it by virtue of the statement referring to all possible logical systems of thought. Is this a higher form of paradox?

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