[FOM] RE: FOM Platonic 3 vs Dedekind 3
Matt Insall
montez at fidnet.com
Wed May 14 20:48:50 EDT 2003
Hodges:
(And even if there is a Platonic real number systems, to which
the various defined real systems are merely isomorphic, then a Platonic
3 is a Platonic 3, and it is not a Dedekind 3.)
Insall:
You are right. Platonic 3 is an urelement, and Dedekind 3 is not.
At least, that is how I view it. Then there is Platonic pi, again,
not the same as Dedekind pi, but also different from Platonic 3,
and located a bit to the right of Platonic 3 on the Platonic real
number line. The Platonic real number line is a structure, all of whose
members are urelements. The Dedekind construction merely demonstrates
that I am justified in viewing the real line as a special type of
ordered field (namely a Dedekind-complete ordered field), and that we have
recognized structural features of the Platonic line correctly enough to
have not made the mistake of writing down an inconsistent set of beliefs
about it. Moreover, we can show that anyone who agrees with a reasonable
collection of set theoretic assumptions, and who has in hand a model of
these axioms, the Platonic real line is isomorphic to their model of those
axioms. (All of the above is actually relative to con(ZF), which, as we
know from Goedel's second incompleteness theorem, is undecidable, but
mathematicians frequently do not worry about undecidability, since they
can ``see'' the Platonic structures with which they work.)
Other Platonic entities include various nonstandard lines. These are
lines that include the Platonic real line, but include also some elements
that are nonzero infinitesimals and some that are translates of these
infinitesimals, and some that are remote (infinitely far away from the
origin). We are justified in accepting the existence of these nonstandard
lines on the basis of the ultrafilter theorem for boolean algebras, which
is a consequence of the axiom of choice, but is strictly weaker than the
full strength of AC. (One constructs over the Platonic real line a
superstructure, as follows. Let S_0 be the Platonic real line, and
for each natural number n, let S_{n+1} be the union of S_n with the set
of all subsets of S_n. Then union this tower of sets, and call the result
S. S, together with the membership relation restricted to S, forms a
structure for the language of set theory with urelements. We call such a
structure a superstructure. An ultrapower of (S,epsilon) is a rich universe
in which to do all the analysis one may wish to do, using the appropriate
members of the ultrapower as infinitesimals.)
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