FOM: Reply to Friedman on Arithmetic and Geometry

[Joe Shipman]

Harvey -- I'll let others answer your question for their own postings;
the main point of disagreement between Neil Tennant and myself is that
he sees a large epistemological asymmetry between arithmetric and
geometry and I see a smaller one.  Now that he has explained that he is
primarily concerned with the successor arithmetic rather than the theory
of + and * I don't have much to argue with him.

I look forward to seeing more from you on the foundational relevance of
a finite universe (either physical or set-theoretical).  There are nice
axiomatizations of the hereditarily finite sets equivalent to PA, as you
have pointed out.  If V_0={}, V_1={{}}, V_2={{},{{}}},
V_3={{},{{}},{{{}}},{{},{{}}}}, then V_4 has 16 elements, V_5 has 65536
elements, V_6 has 2^65536 elements, and V_7 has 2^2^65536 elements.  V_7
is the first "Universe" in which the elements are of infeasible size.
Someone who had an oracle for the first-order theory of the finite
structure (V_7, /in) could rule the world.

-- JS