The origin is with the so-called Whiteley Sentence.
See C.Whiteley, “Minds, Machines and Gödel: A Reply to Mr. Lucas (1962)”, Philosophy 37:61-62 :
It is possible to devise a formula which will trap a human mind —say, Mr Lucas's— in the same way that his application of Gödel traps the machine. Take, for instance, the formula
'This formula ...
See e.g. Gödel’s Incompleteness Theorems : Diagonalization: it is a general result of FOL that :
Let A(x) be an arbitrary formula of the language of F with only one free variable. Then a sentence D can be mechanically constructed such that
F ⊢ D ≡ A(⌈D⌉).
In Heck's paper, page 2, the author applies this general result to formula (9) above (of system PA)...
I'm not an expert, but I'd like to venture a response.
In classical logic, "this statement is always false" is equivalent to "this statement is false" because there's no intermediate truth value -- "true" in classical logic means the same thing as "always true".
There's more nuance here under the model theoretic interpretation of such statements:
Could we therefore argue that what he calls his 'self' is nothing but these sets of experiences and their mental deductions that led him to form a boundary between him and the world?<
Is not this assertion objectionable on the same ground as the assertion that is attributed to science at the beginning of the question?
I mean, one could ask : " what ...
It seems to me this is a useful paradox in the sense that where it arises we know we must be thinking incorrectly. But as phrased it seems easy to overcome. I feel there is a more interesting and real paradox underlying this one.
It is not thoughts that think. If you drop this idea and rephrase the paradox then it might have more bite.