Is there is no proof equivalent to impossible to prove p? — Agent Smith
How to prove that the statement "no true statement can ever be proved true" is true.
Analytic statements
Quine in
The Two Dogmas of Empiricism distinguished between two kinds of analytic claims - i) logical truths, true no matter how we interpret the non-logical parts in the statement, such as "no not-X is X" - and ii) synonymous truths, such as comparing bachelor with unmarried man.
As regards logical truths, logical truths are necessary truths, and necessary truths are beyond proof, in that no argument can be found to establish the truth of a necessary truth, as the nature of a necessary truth is to be true.
As regards synonymous truths, consider the statement "the sun rises in the east". By definition, the "sun" is something that rises in the east. So, the statement "the sun rises in the east" is analytically true. IE, if one morning something rose in the west, rather than the east, then that something wouldn't be the "sun", it would be something else.
IE, logical truths are beyond proof, and synonymous truths don't need to be proved as they are true by definition.
Synthetic statements
As regards whether the statement "the sun rises in the east" is synthetically true, one needs to prove that the sun rises in the east rather than the west.
If Idealism is true, and there is no external reality and the world only consists of ideas, then there is no sun, and the question as to where it rises becomes irrelevant.
If Indirect Realism is true, and our conscious experience is not of the real world itself but of an internal representation, then impossible to prove something outside that which we have direct knowledge of.
If Direct Realism is true, and the senses provide us with direct awareness of the external world, then it would be possible to prove that the sun rises in the east.
But how do we prove which of Idealism, Indirect Realism or Direct Realism is true, something philosophers have debated for thousands of years. As my perception of the sun would be identical, regardless of whether Idealism, Indirect Realism or Realism was true, it would therefore be impossible to prove which of these is in fact the case.
IE, synthetic statements cannot be proved.
Summary
Given the statement S "no true statement can ever be proved true", if S is true, proving it true would result in a contradiction.
IE, some true statements cannot be proved true. Though, in practice, proof is secondary to what pragmatically works.