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␈↓ ↓H␈↓αNote on comprehension

␈↓ ↓H␈↓␈↓ α_The␈αideal␈αset␈αtheory␈αwould␈αbe␈αCantor's␈αwith␈αan␈αunrestricted␈αcomprehension␈αaxiom␈αschema.
␈↓ ↓H␈↓With␈α∂such␈α∂a␈α∂schema,␈α∂the␈α∂axioms␈α∂of␈α∂␈↓↓pairing,␈↓␈α∂␈↓↓unions,␈↓␈α∂and␈α∂␈↓↓power␈α∂set␈↓␈α∂could␈α∂be␈α⊂dispensed␈α∂with,
␈↓ ↓H␈↓although␈α⊂the␈α⊂axioms␈α⊂of␈α⊂␈↓↓extensionality,␈↓␈α⊂␈↓↓infinity,␈↓␈α⊂␈↓↓regularity␈↓␈α⊂and␈α⊂␈↓↓choice␈↓␈α⊂would␈α⊂still␈α⊂be␈α⊂required.
␈↓ ↓H␈↓Unfortunately,␈αCantor␈αset␈αtheory␈α
is␈αinconsistent,␈αbut␈αwe␈α
would␈αlike␈αto␈αre-establish␈α
as␈αmuch␈αof␈αit␈α
as
␈↓ ↓H␈↓possible.  Here are some ideas:

␈↓ ↓H␈↓␈↓ α_We begin with questions about unrestricted comprehension terms {x|P(x)}.

␈↓ ↓H␈↓␈↓ α_1.␈αWhich␈αterms␈αcan␈αbe␈αshown␈αto␈αexist␈αin␈αZF,␈αi.e.␈αfor␈αwhich␈αformulas␈α␈↓↓P(x)␈↓␈αcan␈αwe␈αprove␈αthe
␈↓ ↓H␈↓existence of a set ␈↓↓a␈↓ satisfying

␈↓ ↓H␈↓1)␈↓ α8  ␈↓↓∀x.(x ε a ≡ P(x))␈↓?

␈↓ ↓H␈↓␈↓ α_2.␈αFor␈α
which␈α␈↓↓P(x)␈↓␈α
is␈α(1)␈α
consistent␈αwith␈α
any␈αconsistent␈α
extension␈αof␈α
ZF?␈α I␈α
guess␈αthis␈αmust␈α
be
␈↓ ↓H␈↓the same as the answer to the previous question.

␈↓ ↓H␈↓␈↓ α_3. Which sets of ␈↓↓P(x)␈↓ are consistent with ZF?

␈↓ ↓H␈↓␈↓ α_4. Which sets of ␈↓↓P(x)␈↓ are consistent with just extensionality?

␈↓ ↓H␈↓␈↓ α_5.␈α∞Is␈α∂there␈α∞a␈α∞nice␈α∂way␈α∞of␈α∞extending␈α∂the␈α∞comprehension␈α∞axiom␈α∂that␈α∞obviates␈α∞the␈α∂need␈α∞for
␈↓ ↓H␈↓␈↓↓pairing,␈↓ ␈↓↓unions␈↓ and ␈↓↓power set␈↓?

␈↓ ↓H␈↓␈↓ α_6.␈α∩Is␈α⊃there␈α∩any␈α∩nice␈α⊃maximal␈α∩set␈α∩of␈α⊃comprehension␈α∩terms,␈α∩such␈α⊃that␈α∩adding␈α∩more␈α⊃will
␈↓ ↓H␈↓produce␈αinconsistency?␈α My␈αguess␈αis␈αthat␈αthere␈αare␈αsuch␈αmaximal␈αsets,␈αand␈αthey␈αcan␈αbe␈αprofitably
␈↓ ↓H␈↓studied, but such a maximal set is not recursively enumerable.

␈↓ ↓H␈↓This is SET.NOT[W78,JMC] and was pubbed on March 15, 1978 at 15:34.