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C00002 00002	maximal set theory
C00004 00003	.require "memo.pub[let,jmc]" source
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maximal set theory

Consider a class of set theories resembling that of Zermelo except
that we may vary the comprehension axiom.

Naive set theory forms terms {x|wff} for any wff and is inconsistent.
Zermelo set theory allows only {x|xεA ∧ wff1}, i.e. wff is restricted
to have the form  xεA∧wff1.

We consider arbitrary effective classes of allowed wff1, and consider
which of them lead to consistent set theories or at least theories that
can be shown equiconsistent with Zermelo set theory.

Conjecture 1 - There is am effective procedure for getting stronger set theories,
such that if T is a set theory T' = f(T) is stronger but equiconsistent.

Conjecture 2 - If two set theories are consistent, so is their union.  If
this isn't true, we may have to restrict the effective classes we admit
so as to make it true.

Conjecture 3 - The union of all consistent effectively given set theories
is a set theory but is no longer effective.  The idea is that the set of
all intuitively meaningful comprehension terms is not r.e.

Conjecture 4 - Although the above maximal set theory is not r.e., it will
have interesting mathematical properties.

These conjectures are perhaps overbold and should really be called
speculations.
.require "memo.pub[let,jmc]" source
.bb Note on comprehension

	The ideal set theory would be Cantor's with an unrestricted
comprehension axiom schema.  With such a schema, the axioms of
⊗pairing, ⊗unions, and %2power set%1 could be dispensed with, although the
axioms of ⊗extensionality, ⊗infinity, ⊗regularity and ⊗choice would still
be required.  Unfortunately, Cantor set theory is inconsistent, but
we would like to re-establish as much of it as possible.  Here are
some ideas:

	We begin with questions about unrestricted comprehension
terms α{x|P(x)α}.

.item←0
	#. Which terms can be shown to exist in ZF, i.e. for which
formulas ⊗P(x) can we prove the existence of a set ⊗a satisfying

!!a1:	%2∀x.(x_ε_a_≡_P(x))%1?

	#. For which ⊗P(x) is ({eq a1}) consistent with any consistent
extension of ZF?  I guess this must be the same as the answer to the
previous question.

	#. Which sets of ⊗P(x) are consistent with ZF?

	#. Which sets of ⊗P(x) are consistent with just extensionality?

	#. Is there a nice way of extending the comprehension axiom
that obviates the need for ⊗pairing, ⊗unions and %2power set%1?

	#. Is there any nice maximal set of comprehension terms, such
that adding more will produce inconsistency?  My guess is that there
are such maximal sets, and they can be profitably studied, but
such a maximal set is not recursively enumerable.

This is SET.NOT[W78,JMC] and was pubbed on {date} at {time}.