A definable nonstandard model of the reals

Abstract

We prove in ZFC the existence of a definable, countably saturated elementary extension of the reals. It seems that it has been taken for granted that there is no distinguished, definable nonstandard model of the reals. (This means a countably saturated elementary extension of the reals.) Of course if V=L then there is such an extension (just take the first one in the sense of the canonical well-ordering of L), but we mean the existence provably in ZFC. There were good reasons for this: without Choice we cannot prove the existence of any elementary extension of the reals containing an infinitely large integer. Still there is one. Theorem (ZFC). There exists a definable, countably saturated extension R* of the reals R, elementary in the sense of the language containing a symbol for every finitary relation on R.