[FOM] One axiom for all sets.
Zuhair Abdul Ghafoor Al-Johar
zaljohar at yahoo.com
Sun Mar 1 21:49:46 EST 2015
Mr. M.Randall Holmes had put forth a proof on his home page of existence of
a set of all sets hereditarily smaller than a given set, for every set, in ZF.
The proof is independent of choice.
One consequence of such a result is that the following would be a theorem
of Morse-Kelley (MK) class theory (with any size limitation axiom sufficient
to prove replacement over sets)
Set(x) <-> Ez (Ayez(z<y) ^ Ayex(y<<TC(z)))
where "<" stand for "is subnumerouse to" defined in the usual manner
(non strict subnumeroustiy); "TC" stands for "transitive closure of"
defined as the minimal transitive superclass; and "<<" stands for
"hereditarily subnumerouse to".
The nice result is that adding that sentence to axioms of Extensionality,
Class comprehension and Pairing of MK results in a theory that proves
all other set axioms of a version of MK that is sufficient to interpret
ZFC. If we weaken the above axiom by just replacing "TC(z)" by "z", then
we'd get a theory that interpret Z.
PS:"e" stands for membership; "E" for the existential quantifier and
"A" for the universal quantifier.
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