FOM: on false bloopers

[Neil Tennant]

In my original posting on bloopers (statements mistakenly taken as
theorems, on the basis of incorrect `proofs'), I wrote

"... it follows from the very definition of `F-transcendent
blooper' that we could never know its truth value, or that it is
transcendent! Suppose P is an F-transcendent blooper. This means that
(unless we spot the error in the `proof') we shall never deduce
anything untoward by assuming P. Now, how can an apple be bad without
*coming to be seen as* affecting, if not the whole barrel, then at
least some other parts of it? Any result depending crucially on P as a
premiss would, itself, have to be F-transcendent."

The last sentence should have read "Any FALSE result depending
crucially on P as a premiss would, itself, have to be F-transcendent."
That's what was intended, but the context didn't make that clear.

But that observation raises further interesting questions.  Let us say
that a blooper P *spawns* a blooper Q if Q is `proved' by means of
crucial reliance on P.  The proof of Q with P as an assumption could
be perfectly in order; it is just that, if we were to stick in the
actual mistaken proof that we had for P, the resulting overall proof
of Q (whose assumptions now do not include P) would be mistaken.

Note the familiar point that valid proofs can have a false premiss and
a true conclusion.

Questions: 

1. Could an F-transcendent blooper spawn a T-transcendent one?
2. Do false bloopers (whether transcendent, eventual or available)
ever spawn T-bloopers?
3. More particularly, has any F-available blooper spawned a
T-available one?

If the answer to (3) is "Yes", then it may be that deductive *errors*
can end up playing a positive role in our search for mathematical truth.

As a reminder to the reader, here are the definitions once more of the
six kinds of blooper that a classical realist can countenance:

T-TRANSCENDENT
P is true, but there is no proof of P; that is, there is not even
a *possible* proof of P, in any conceivable extension of our axioms
and methods of proof that we would accept if we got to consider them
and their consequences. (We somehow have to rule out the silly case of
adopting P as a new axiom.)

T-EVENTUAL
P is true, and there is a proof of P; but such a proof is not
constructible within the system of axioms and methods of proof that we
currently accept. It requires adoption of some as-yet-unconsidered
axioms and/or methods of proof that we have not yet considered. (Same
caveat as above.)

T-AVAILABLE
P is true, and there is a proof of P in the system of axioms and
methods of proof that we currently accept. (The only trouble is, the
'proof' of P that we have at present is not a legitimate proof.)

F-TRANSCENDENT
P is false, but there is no refutation of P; that is, there is not
even a *possible* refutation of P, in any conceivable extension of our
axioms and methods of proof that we would accept if we got to consider
them and their consequences. (We somehow have to rule out the silly
case of adopting not-P as a new axiom.)

F-EVENTUAL
P is false, and there is a refutation of P; but such a refutation
is not constructible within the system of axioms and methods of proof
that we currently accept. It requires adoption of some
as-yet-unconsidered axioms and/or methods of proof that we have not
yet considered. (Same caveat as in F1.)

F-AVAILABLE
P is false, and there is a refutation of P in the system of axioms
and methods of proof that we currently accept.

Neil Tennant