Re: Question about universal set

Dave Seaman writes:
In place of that flawed concept, sometimes called "unbounded comprehension",
we have instead "bounded comprehension", which is another name for the axiom
of separation.

Maybe you found the term bounded comprehension in a book, but your
definition is not the one that is used in the contemporary literature.
I have seen the term ``restricted separation''.

In my mind, bounded comprehension is the kind found in Kripke-Platek
set theory (KP), a subtheory of ZFC that is important for studying
definability. Essentially, bounded comprehension is the comprehension
scheme is limited to fomulas with only bounded quantifiers, i.e.
quantifiers of the form (exists x in z) and (forall x in z).

If you google for "bounded separation" in quotes or 'kripke platek
"bounded separation"' with double quotes you will see what I mean.

You can look this up in any book on admissible sets. There is a list
of axioms for KP at
http://en.wikipedia.org/wiki/Kripke-Platek_set_theory . That page
accurately labels bounded comprehension Sigma_0 comprehension.

.

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