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Modern mathematics work in a vast world of sets: the iterative hierarchy of set theory. Increasingly complex set theories describe increasingly rich universes of sets. These universe are laid out one after another in order of their complexity. Set theory is surprisingly concerned with the old Anselmian idea of God or the Absolute as "that than which no greater is possible". For mathematicians, greatness is quantified as complexity (cardinality). The theory of the universe of sets develops dialectically in a thesis-antithesis-synthesis pattern. The thesis is the assertion that some particular universe of sets is that than which no greater is possible (it is absolute). The antithesis is that the existence of some greater set is possible. The synthesis is the inclusion of this greater set into the universe of the thesis, thus making a new universe and the next thesis for the dialectical ascent.
THESIS-0: Let the set theory T0 be ZFC with the existence of the empty set as its only existence axiom. Let U0 be the iterative hierarchy of which T0 is true (the iterative model of T0). U0 is the finite universe. Thesis: The finite universe is that than which no greater is possible.
ANTITHESIS-0: The existence of an infinite set Aleph-0 is consistent with the existence of the finite universe U0; if the existence of Aleph-0 is added to T0, then the resulting transfinite universe is greater than U0; so, U0 is not that than which no greater is possible.
SYNTHESIS-0: Let the set theory T1 be T0 plus the existence of Aleph-0. Let U1 be the iterative hierarchy of which T1 is true (the iterative model of T1). U1 is the transfinite universe. Synthesis-0 absorbs both Thesis-0 and Antithesis-0.
THESIS-1: The transfinite universe U1 is that than which no greater is possible.
ANTITHESIS-1: The existence of an inaccessible set Kappa is consistent with the existence of the transfinite universe U1; if the existence of Kappa is added to T1, then the resulting inaccessible universe is greater than U1; so, U1 is not that than which no greater is possible.
SYNTHESIS-1: Let the set theory T2 be T1 plus the existence of Kappa. Let U2 be the iterative hierarchy of which T2 is true (the iterative model of T2). U2 is the inaccessible universe. Synthesis-1 absorbs both Thesis-1 and Antithesis-1.
THESIS-2: The inaccessible universe U2 is that than which no greater is possible.
ANTITHESIS-2: The existence of a regular fixed point for every normal function (call this statement "Axiom F") is consistent with the existence of the inaccessible universe U2; if Axiom F is added to T2, then the resulting regular fixed-point universe is greater than U2; so, U2 is not that than which no greater is possible.
SYNTHESIS-2: Let the set theory T3 be T2 plus Axiom F. Let U3 be the iterative hierarchy of which T3 is true (the iterative model of T3). U3 is the regular fixed-point universe. Synthesis-2 absorbs both Thesis-2 and Antithesis-2.
This dialectical process continues up through increasingly complex theories
and universes. It aims to converge to the Absolute, to that than which no
greater is possible.
William Paterson University Philosophy Department