FOM: sharp boundaries/tameness

[Kanovei]

>Date: Wed, 13 Feb 2002 20:13:32 -0500
>From: Harvey Friedman <friedman at math.ohio-state.edu>

>
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
integer.
>

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 
question. 

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 
tough. 

V.Kanovei