[FOM] Re: Progressions of theories (was: Consistency of formal systems)
torkel at sm.luth.se
Wed Jun 18 13:57:29 EDT 2003
Richard Zach says:
>> I would say that autonomous progressions are highly relevant to
>> mathematical epistemology, and they play a central role in my book.
>I looked at the introduction to your book on your website, but
>unfortunately there's not a lot there on the philosophical parts of the
>book. I guess I'll just have to wait--when will it be out?
I would guess this fall. There's been a delay in production because I
produced the book using software with a dubious equation editor, and
the ms is now being "translated" into LaTeX through the efforts of
an editor in the US.
About autonomous progressions, the autonomous notations do not
constitute any proof-theoretically reasonable system of notations for
ordinals. For example, if we consider the canonical sequence of length
epsilon_0 of iterated consistency extensions T_alpha of PA (obtained
by using a canonical system of notations for the ordinals smaller than
epsilon_0), Turing's construction can be modified to show that for any
alpha<epsilon_0, we can define an autonomous notation a (given that our
base theory extends PA) with |a|=omega+1 such that T_a proves the
consistency of T_alpha. But for the purposes of a discussion of
mathematical epistemology (involving qualitative rather than
quantitative arguments), what matters is only that we can argue that
for autonomous notations a, our (implicit or potential) mathematical
knowledge encompasses the theorems of T_a.
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