Tarski's Indefinablity Theorem:

Kurt Goedel discovered his Incompleteness Theorems (1931) partly by showing how to represent syntax within arithmetic. Each expression of the language of arithmetic is assigned a number. So, set of expressions are assigned sets of numbers. It turns out that for various syntactic properties (being a formula, being a sentence, etc.) these sets of numbers are recursive, and this can be defined by arithmetic formulas. For example, there is a formula Sent(x) in the language of arithmetic which defines the set of sentences of the language of arithmetic.

Can the same be done for semantical concepts, such as truth? Tarski's discovered around 1933 that in the most interesting cases, the answer is no. Roughly, a sufficiently rich interpreted language cannot represent its own semantics. The meta-language must be richer than the object language.

For arithmetic, this is how it works (the proof can be generalized to any language which is at least as rich as the interpreted language of arithmetic). Let L be the first-order language of arithmetic. Let N be the standard structure for L. So, (L, N) is the interpreted first-order language of arithmetic. Let T be the set of truths in N. Let T* be the set of code numbers of sentences in T.

TARSKI'S THEOREM: There is no L-formula True(x) which defines T*.

Proof: Suppose that such a predicate True(x) existed. In particular, if A is a sentence of arithmetic, then True("A") is true in N if and only if A is true in N. Hence, for all A, the Tarski T-sentence True("A") <-> A is true in N. But, using Goedel's Diagonal Lemma, we could construct a Liar sentence S such that S <-> ~True("S") would be true in N. But this contradicts True("S") <-> S. Hence, no such L-formula True(x) exists in the language of arithmetic.

So, informally, arithmetical truth is not arithmetically definable.

Note: it is possible to define a formula True(x) whose extension is the set T of truths in the language of first-order arithmetic. However, this must be a richer language, such as the language of second order arithmetic.

However, Tarski's point is quite general. No sufficiently powerful language can represent its own semantics.