### Definability of initial segments

by Shelah and Tsuboi. [ShTs:767]

Notre Dame J Formal Logic, 2003

We consider implicit definability of the standard part
{0,1,...} in nonstandard models of Peano arithmetic (PA), and
we ask whether there is a model of PA in which the standard part is
implicitly definable.
In section 1, we define a certain class of formulas, and show that in
any model of PA the standard part is not implicitly defined by
using such formulas.
In section 2 we construct a model of PA in which the standard part is
implicitly defined. To construct such a model, first we assume a set
theoretic hypothesis diamondsuit_{S_lambda^{lambda^+}}, which
is an assertion of the existence of a very general set. Then we
shall eliminate the hypothesis using absoluteness for the existence
of a model having a tree structure with a certain property.

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