FOM: Nonstandard models of arithmetic
Joe Shipman
shipman at savera.com
Wed Mar 14 10:48:36 EST 2001
Todd Wilson writes:
>>As we know, every countable non-standard model of arithmetic has order
type
NN = omega + (Z * Q) ,
where Z and Q are the order types of the integers and rationals, and
where the successor function s : NN -> NN is obvious. Are there any
explicit definitions of + and * on NN that make (NN, 0, s, +, *) into
a model of arithmetic? If not, is there a proof that such definitions
are impossible (for example, a model of set theory in which all models
of arithmetic are standard)? In either case, where was this first
established? <<
Of course one can make a nonstandard model of Th(N) where + and * are
definable in set theory. The proof of the completeness theorem is
constructive for countable structures. What one cannot have is a
nonstandard model where + and * are computable. In fact, you can't even
have a computable nonstandard model of Peano Arithmetic, let alone one
for true arithmetic. I don't know who first published this, but Godel
must have known it. (One can get either + or * computable but not
both.)
-- Joe Shipman
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