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D. Gomez Ramirez:
"Mathematics: a meta-isomorphic version of classic mathematics based on proper classes";
Vortrag: Logic Colloquium Stockholm, Stockholm, Schweden (eingeladen); 14.08.2017.

An implicit working principle in Von Newmann-Bernays-Gödel Set Theory (NBG)
is that small classes (or `sets´) are more suitable objects to start and work with for
developing a general foundational framework for standard mathematics. On the other
hand, proper classes are just `too big´ and formally `too dangerous´ in order to be able
to ground any classic mathematical theory.
In this paper, we will mainly show that these classic quantitative considerations
about proper and small classes are just tangential facts regarding the consistency of
ZFC set theory. Effectively, we will construct a first-order logic theory D-ZFC (Dual
theory of ZFC set theory) strictly based on (a particular sub-collection of) proper
classes with a corresponding special membership relation, such that ZFC and D-ZFC
are meta-isomorphic frameworks. More specifically, for any standard formal definition,
axiom and theorem that can be described and deduced in ZFC set theory, there exists
a corresponding `dual´ version in D-ZFC and vice versa. In particular ZFC set theory
is consistent if and only if D-ZFC is consistent.
In addition, let us call modern Mathematics for all formal mathematical theories
which are grounded in ZFC set theory, for instance, Real and Complex Analysis, Ge-
ometry, Algebra, Number theory, Topology and Category Theory. So, we will name
Dathematics for the family of all dual versions of the (former) modern theories, where all
the subsequent concepts and theorems describing properties among them are expressed
and grounded by D-ZFC. Finally, we prove the meta-fact that (classic) mathematics
and dathematics are meta-isomorphic, i.e., for any concept, theory and conjecture in
(classic) mathematics there exists a symmetric d-concept, d-theory and d-conjecture
in dathematics with equivalent formal properties, and vice versa; e.g., a mathematical
conjecture C is true (resp. provable) if and only if the dual `dathematical´ conjec-
ture C+ is true (resp. provable). So, (standard) Mathematics and Dathematics are
equiconsistent and the last meta-framework has, strictly speaking, proper classes as
fundamental objects.