2025/12/26 by Lisica, Ju. T.
#FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2512.22314
The basic one in this work is the axiomatic set theory NBG (von Neumann-Bernays-Gödel), which is a first-order theory with its own axioms, including in particular the axiom of choice \bf AC and the axiom of regularity \bf RA. The universal class \bf V of all sets in this theory exactly coincides with the class of all founded sets, i.e., such X∈\bf V that \it does not exist an infinitely descending ∈-sequence X\ni X1\ni X2\ni...\ni Xn\ni... of sets Xn, n=1,2,3,... . In the first part of the paper, a new concept of \it skand is introduced -- a random aggregate, or \grqq decreasing\grqq tuple composed of founded sets, e.g., X=\1,\2,\3,\... ...\\\\, and the theory of NBG-=NBG-\bf RA, i.e., the theory of NBG without the axiom of regularity \bf RA, to which is added the new axiom \bf SEA of the existence of infinite-length skands and the pseudo-founding axiom \bf PFA. These new axioms are a negation of the axiom of regularity and are thus less restrictive than the axiom of regularity \bf RA in the sense that they admit the existence of non-founded sets, and the axiom of regularity excludes the existence of such sets. At the same time of course the axiom of extensionality \bf EA is replaced by a more accurate axiom of extensionality \bf EEA, since it takes into account the equality of new objects. In the second part of the paper, a new concept of \it coskand is introduced, which is dual to a notion of skand and is a random aggregate, or \grqq increasing\grqq tuple composed of founded sets and the theory of NBG and actually is a theory NBG[\cal U] with individuals as limiting coskands, e.g., X=...\3,\2,\1,\0\\\\... .