Re: question about completeness and incompleteness

On Sun, 05 Apr 2009 19:56:28 -0700, Ben Crowell
<crowell09@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> said:
Chris Menzel wrote:
On Sun, 05 Apr 2009 12:22:29 -0700, Ben Crowell
<crowell09@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> said:
I think another thing that was messing me up was that I was assuming
that nonstandard models of PA had to have a different cardinality than
the standard one, whereas actually PA isn't w-categorical.

Right, though, interestingly, all countable nonstandard models of PA
are isomorphic.

Really!? That's bizarre!

And, as Aatu points out, utterly false; it is a very simple consequence
of Gödel's first incompleteness theorem and the L-S theorem that there
are countable non-standard models of PA that are not isomorphic. (E.g.,
let G be the Gödel sentence for PA. Consider the theories PA+~G+H and
PA+~G+~H, where H is the Gödel sentence for PA+~G.) As Aatu correctly
surmised, the fact I had in mind when I blurted out the falsehood above
is that all countable nonstandard models of PA have isomorphic
*orderings*. Every nonstandard model of PA consists of a copy of the
standard natural numbers followed by a dense linear ordering (without
endpoints) of copies of the integers. The only countable DLO without
endpoints, up to isomorphism, is the rationals. Hence, every countable
nonstandard model of PA consists of a copy of the natural numbers
followed by a copy of the rationals, each point of which is a copy of
the integers.

Sorry for the confusion.

.

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