FOM: finite axiomatization and conservative extension

[Neil Tennant]

Could any fom-er supply a reference for the earliest proof of a
theorem along the following lines?:

Any r.e. set of sentences in a first-order language L is, for some L*
extending L, the restriction to L of some finitely axiomatizable
theory in L*.

Second question:

Are there any interesting/natural conditions on such r.e. sets that
would force L* to be a language of higher order than L, rather than
merely a first-order extension of L?

Neil Tennant

___________________________________________________________________
Neil W. Tennant
Professor of Philosophy and Adjunct Professor of Cognitive Science
230 North Oval
The Ohio State University