FOM: miniaturization
Jan Mycielski
jmyciel at euclid.Colorado.EDU
Tue Sep 14 19:47:04 EDT 1999
Dear fom Readers,
Could somebody explain to me why H. Friedman is building various
statements of finite (or infinite) combinatorics equiconsistent with
various large cardinal axioms? (In other words, why he is doing
miniaturizations of large cardinal axioms?)
I understand that it is interesting to know surprisingly simple
statements of finite combinatorics whose proofs require strong axioms.
But, for reasons which follow, I do not see why it is interesting to
miniaturize.
In JSL 51 (1986), 59 - 62, I gave a general method for turning the
statement of the consistency of any theory T into a statement S(T) of
finite combinatorics. Moreover S(T) reflects in a transparent way the
original (model theoretic) intuition supporting T (as opposed to the
purely syntactic meaning of consistency).
Thus it seems to me that the problem of miniaturization is solved
once and for all. The only aditional work which could be done is to give
some more appealing combinatorial statements equivalent to my S(T). While
the statements of Paris and Harrington (for PA), and those of Friedman
which I saw, do not appear to me more appealing than S(T) (since they hide
T using complicated combinatorial (technical) equivalemt forms of S(T).)
(My JSL paper was improved in an important way by J. Pawlikowski
in AMS Abstracts 10 (1989), p. 172. Upon request I could send you reprints
of both.)
[I am not advocating here the idea (of S. Lavine) that looking at
the theory FIN(T) (of my JSL paper) gives us a better understanding of T,
or that it could give us any independent evidence for Con(T). I believe
that T formalised in Hilbert's epsilon extension of first-order logic is a
more direct approach to Con(T). Namely, it defines mathematically the real
process of construction (and naming) of mathematical objects whose initial
steps are performed in our human imaginations when we read the axioms of T
and when we prove theorems of T. And, if this process appears to us
sufficiently regular or simple (so that we become convinced that it cannot
collapse), we are convinced that T is consistent. And I do not believe
that FIN(T) can add anything to that mental experiment with T.]
Regards to all,
Jan Mycielski
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