One may avoid self reference in liar paradoxes by allowing ourselves infinite propositions with unlimited self-free reference. Consider the following infinite set of propositions:

  • Pi is the proposition "all of Pi+1, Pi+2, ... are false".

Now suppose P0 were true. Then P1, P2, ... would all be false. But P1 asserts that P2, P3, ... are all false, hence we've just shown that P1 is true! That's impossible.

So we have disproved P0, i.e. it is false. Clearly, the same argument works for P1, so P1 too is false; continuing, each Pi is false. In particular, this shows that P0 is true (and so are all the other Pi's).

Contradiction either way.

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