An amusing example of Godel's Theorem This statement is true, but there is no way to prove it in any consistent formal system. Proving it will render the system inconsistent, and thus useless.

Related to Godel's Theorem is Epimenides Paradox, which is illustrated by the statement: "This sentence is false". Or the man from Sparta (I think it was Sparta) who said that all Spartans are liars.

This statement is true, since no arbitrary phrase unrelated to anything in the real world or mathematics (mathematics is not of the real world!) can be proven. So there.

And the level of proof in the real world is much lower than that of mathematics. After all, it can be proved, using Newtonian Mechanics that you can go faster than the speed of light...

Gödel proved his theorem by constructing (in the language of arithmetic) a statement which states "this statement cannot be proven in arithmetic".

Obviously, if arithmetic is consistent then the statement is true: if it were false, then it would be provable, and then arithmetic could prove a falsehood -- inconsistency! Hence it is true and unprovable in arithmetic (we've just seen it's true, but we weren't doing that bit in arithmetic but in some larger world).

So what's all the noise about?

That something like this can be said in the language of arithmetic. The point of formalism is to create a formal language free of the ambiguities that plague natural language. "Cute" statements like this node's title clearly show that self reference is such a harmful ambiguity, but who says we can't create a formal language that will bar the way to such ambiguity?

Gödel does. Specifically, he showed that any language which is powerful enough to state even mildly interesting things about arithmetic (about at the level of defining the basic properties of multiplication!) can formulate this statement; in fact, he gives a precise construction of this statement.

Sufficiently simple systems of formal logic are still immune to this disease. Unfortunately, they're so simple as to be useless -- they cannot say anything interesting.

In fact there exist other types of unprovable statements, with no overt self reference, but it took more than 30 years after Gödel's work to find the first "natural" examples.

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