FOM: Further comments and clarifications

[Joe Shipman]

>>It's too bad that you can't bootstrap these results.  You have several

>>results of the form
>>
>>"Con(ZFC+kappa_i) implies Thereexists a Q-system satisfying i implies
>>Con(ZFC+lambda_i )"
>>
>>where kappa_i<lambda_i, but the intervals (kappa_i,lambda_i) don't
>>overlap enough that you can combine these to get a result of the form
>>
>>"Con(ZFC + something small) implies Thereexist Q-systems satisfying
>>1,2,...,i implies Con(ZFC+something big)".
>Such "results" are always impossible. You can never go from Con(ZFC +
>something small) to Con(ZFC + something big). This is because of
Godel's
>2nd incompleteness theorem and the fact that Con(ZFC + something big)
>always implies Con(ZFC + Con(ZFC + something small)). This is in turn
>because "something big" always implies Con(ZFC + something small).

I think you didn't parse my question properly.  The added strength comes
from the conjunction of the Q-system statements.  You identified
statements Qi, of the form "There exists a q-system satisfying
[something involving definability]".

Then you stated a number of the results of the form "Qi iff Con(Xi)",
where Xi ranged from PA through ZFC through "ZC plus infinitely many
strong inaccessibles", proved over ACA (or WKL_0 for i=1).

Then you stated several more results of the form Con(Yi) implies [Qi
implies Con(Zi)], where Yi and Zi were large cardinal axioms with Yi<Zi,
proved over ACA.

My point is simply that if there were overlaps in the intervals (Yi,Zi)
or (0,Xi) you could link some of your results together and get
statements of the form Con(ZFC+something small) implies that *some
conjunction of Qi's* implies Con(ZFC+something big).  This is *the same
form* as some of the results you already stated (Yi="small" and
Zi="big"), so it's not cheating or violating the incompleteness
theorems.

Not that I'm suggesting you should find some further "in-between" Qi's
to accomplish this, what you've got is most impressive; I was just
curious about the appearance of large cardinals in the left-hand-side of
some but not all of your results.


    >I don't fully agree with this. They seem to have interest
independently of
    >each other; both are valuable. Walks in Z^k are fundamental enough
in their
    >own right. There is a huge mathematical tradition of automatically
doing
    >things in k dimensions.

    >Incidentally, you can get incomprehensible growth by looking at
even just
    >binary sequences. Look at the results about the functions m(k) and
B(k) at
    >the end of special posting 24.

I agree; but for "high school" it really helps to stick to dimension
<=3, while the difference between binary and ternary sequences isn't as
much of a stumbling block for "high school" as the difference in
complexity between the definition of n(k) and the definitions of m(k)
and B(k).

    >By the way, is "predicatively infeasible" better English than
    >"predicatively unfeasible"?

"Infeasible" is much more commonly used than "unfeasible".

-- Joe Shipman