Large resplendent models generated by indiscernibles

Abstract

The motivation for the results presented here comes from the following two known theorems which concern countable, recursively saturated models of Peano arithmetic.(1) if is a countable, recursively saturated model of PA, then for each infinite cardinal κ there is a resplendent which has cardinality κ. (See Theorem 10 of [1].)(2) if is a countable, recursively saturated model of PA, then is generated by a set of indiscernibles. (See [4].)It will be shown here that (1) and (2) can be amalgamated into a common generalization.(3) if is a countable, recursively saturated model of PA, then for each infinite cardinal κ there is a resplendent which has cardinality κ and which is generated by a set of indiscernibles.By way of contrast we will also get recursively saturated models of PA which fail to be resplendent and yet are generated by indiscernibles.(4) if is a countable, recursively saturated model of PA, then for each uncountable cardinal κ there is a κ-like recursively saturated generated by a set of indiscernibles.None of (1), (2) or (3) is stated in its most general form. We will make some comments concerning their generalizations. From now on let us fix a finite language L; all structures considered are infinite L-structures unless otherwise indicated.