This sentence is not provable

Theorem:

"Truth" for English sentences is not definable in English.

Proof:

Suppose "Truth" is definable. Then so is its complement "False". A sentence must then be either True or False, and certainly not both.

Let S be the sentence "This sentence is False". Notice that the phrase "This sentence" refers to S. That is, the sentence refers to itself.

We have then:

S is True
if and only if
"This sentence is False" is True
if and only if
"S is False" is True
if and only if
S
is False

That is: S is True if and only if S is False. An obvious contradiction. Done!