*Notation*
Every
proof uses its own forms of notation, which has its own conveniences and hidden
assumptions. I set forth this paper’s notation here in order to simplify
critique. To quote the famous logician Humpty Dumpty, “when I use a word, it
means just what I choose it to mean – neither more nor less.”

This
paper discusses mathematical propositions, called ‘sentences’. They are defined
in terms of each other, using logical connectives such as ‘and’, ‘or’ and
‘no’. Call logically equivalent
sentences ‘equal’; denote equality by ‘=’. Let T denote anything obviously
true, and F denote anything obviously false. Therefore T = not F; the former
can mean ‘x=x’ or ‘1+1=2’ or ‘0 does not equal 1’; and F denotes the denial of
those statements.

Some
sentences are provable. The assertion that the sentence ‘S’ is provable is
another sentence; call it “prv S”. To assert prv S is to assert that S is necessarily
true, therefore true in all mathematical models.

Some
sentences are true on occasion. Such sentences are ‘possible’. The assertion
that the sentence ‘S’ is possible is another sentence; call it “poss S”. To
assert poss S is to assert that S is true in some mathematical model; and
therefore it cannot be proven false. Therefore poss and prv are conjugate to
each other:

poss
S = not prv not S

prv
S = not poss not S

The
possible is what you can’t prove false; and the provable is what you can’t
possibly deny. We can form combinations of these:

Poss
T = ‘truth is possible’

Prv
F = ‘falsehood is provable’

Poss
F = ‘falsehood is possible’

Prv
T = ‘truth is provable’

The
last two are easy to simplify. T is provable in all mathematical models, in
fact it is the standard of proof; therefore prv T = T. Similarly poss F is
false in all mathematical models, therefore poss F = F.

The
statements poss T and prv F involve a deep conundrum; namely, is our proof
system itself valid? Perhaps our reasoning methods contain fatal flaws. Is our
logic in fact consistent? Does it have a mathematical model? If so, then truth
is possible; if not, then falsehood is provable.

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