[FOM] Ordinals as hereditarily transitive sets

[Parsons, Charles]

On May 7, 2015, at 12:27 PM, Aldo Antonelli <antonelli at ucdavis.edu> wrote:

> Sorry to be coming late to this thread.
> 
> Concerning the various definitions of the ordinals, I believe they are
> not all equivalent in the absence of the Axiom of Foundation
> (Regularity), and this is something people in AFA and related theories
> might care about (one might still want the ordinals around for
> transfinite recursion, for instance, even in the presence of the
> Anti-Foundation Axiom).
> 
> I seem to recall asking Larry Moss about this (longer ago than Larry
> or I probably care to remember), and he pointed out that Robinson's
> 1937 definition (an ordinal is a transitive set linearly ordered by
> epsilon) works in non-well-founded contexts as well.
> 
> Larry can correct me if this is not quite accurate.
> 
> -- Aldo
> 

It's even later for me to intervene in this discussion.

The definition of an ordinal number as a transitive set all of whose elements are transitive occurs in Paul Bernays' letter to Gödel of 5 May 1931. (See Gödel, Collected Works, vol. V, pp. 112-13.) He explicitly assumes the axiom of foundation. Probably Bernays should be credited with seeing that, given Foundation, the ordinals can be characterized in this way.

However, he doesn't continue to use this definition in his series of JSL papers, "A system of axiomatic set theory." There he follows Robinson. (See p. 19 of the reprint in G. H. Mueller, Sets and Classes.)

Charles Parsons