[FOM] countable saturation

John Goodrick goodrick at Math.Berkeley.EDU
Sat Jan 14 17:19:55 EST 2006

On Sat, 14 Jan 2006, Ben Crowell wrote:

> What's throwing me for a loop is that K&S say in the main theorem on the
> first page of the paper, "There exists a definable, countably saturated
> extension *R of the reals R..." But C&K's definition of countably
> saturated says the model has to be countable, whereas the hyperreals
> K&S are constructing would be an uncountable set. Am I completely on
> the wrong track here? Does "countable model" not mean what I think it
> means?

I think there's a very simple answer for this: by "countably saturated" 
I'm pretty certain that Shelah means what Chang and Keisler would call 
"omega-saturated."  I think that C&K's usage of "countably satured" here 
is either idiosyncratic or out of date; in the couple dozen or so recent 
papers in model theory I've looked at I can't remember anyone ever using 
that phrase to mean a model that must be countable (model theorists 
tend to say "a countable omega-saturated model" these days if that's what 
they mean.)


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