[FOM] RE: FOM ``Near Superincompleteness''

[Matt Insall]

[Insall (previously)]
Now, let me suggest a more formal version of some of my above discussion
of Friedman's conjectures.  For this purpose, let a_0 = 2^2^2^2^2^2^2^2,
and for each n, let a_{n+1} = 2^a_n.  Let S_0 denote ZFC, and for each n,
let S_{n+1} be ZFC+``sin(a_n)>0''.  Let P_0 be the statement of Friedman's
first conjecture above:

``sin(a_0) > 0 cannot be proved or refuted in S_0 even with large cardinal
axioms, without using at least 2^2^100 symbols, even if abbreviations are
allowed''

and, more generally, for each n, let P_n denote the statement

``sin(a_n) > 0 cannot be proved or refuted in S_n even with large cardinal
axioms, without using at least 2^2^100 symbols, even if abbreviations are
allowed''.

Then I expect that something like the following can be shown:

CONJECTURE:  Suppose that P_0 holds.  Then for each n, there is
m > n such that P_m holds.

I think that in a practical sense, this is fairly similar to what Todd
Eisworth
is looking for as ``Superincompleteness''.  Well, almost.


[Insall (now)]
I should comment that the above conjecture can be improved upon.
Specifically,
one should be able to arrange that the theories S_n form a tower, but more
care
must be taken to avoid contradiction slipping in through the cracks.
Specifically,
let the a_n be as before, let the P_n be as before, and let S_0 be ZFC, as
before.
Now, construct the S_n as follows:  Given n, let Q_n be the statement
``sin(a_m) > 0'',
where m > n is least such that sin(a_m) > 0 cannot be proved or refuted in
S_n even with
large cardinal axioms, without using at least 2^2^100 symbols, even if
abbreviations
are allowed, and then let S_{n+1} be S_n + Q_n.  This recursive construction
of a tower
of theories even more strongly resembles a form of the type of
``superincompleteness''
I think Todd Eisworth described.  For at each stage we extend to a larger
theory by
adding a nonmeta-mathematical axiom about the behaviour of the sine
function, and there
is never a point at which we must stop adding such axioms, since no one will
ever prove
that we fell into a contradiction by adding an axiom.  (We may have fallen
into a
contradiction, but we cannot be convicted of having done so, assuming
Friedman's
conjecture holds and my modified versions of his conjecture hold.)