FOM: sharp boundaries/tameness

Kanovei kanovei at
Thu Feb 14 01:12:12 EST 2002

>Date: Wed, 13 Feb 2002 20:13:32 -0500
>From: Harvey Friedman <friedman at>

The argument does show that it is not provable in ZF that there is a proper
elementary extension of the field of reals with a predicate for being an

The problem (in ZFC version) was given to me long while ago 
by V.A.Uspensky, it appears as an open problem in my paper 
(with M.Reeken) in Math Japonica 1996. 

What can we say about the niceness of a proper elementary extension in the
case of "tame" expansions of the real field?

There is another problem which I learned from Peano people 
also long while ago, which has some connection to this 

Problem. Does there exist a Borel nontandard model of PA 
whose Scott set is complete, in the sense that for any 
standard binary sequence f\in 2^N there is a hyperfinite 
binary sequence s of infinite length such that f=s\restriction N. 

I guess this one is still open, and both ones are amazingly 


More information about the FOM mailing list