In
mathematics, the Double extension set theory (DEST) is an
axiomatic set theory proposed by Andrzej Kisielewicz consisting of two separate membership relations on the universe of sets,
denoted here by
and
, and a set of axioms relating the two. The intention behind defining the two membership relations is to avoid the usual
paradoxes
A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
of
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, without substantially weakening the
axiom of unrestricted comprehension.
Intuitively, in DEST, comprehension is used to define the elements of a set under one membership relation using formulas that involve only the other membership relation. Let
be a first-order formula with free variable
in the language of DEST not involving the membership relation
. Then, the axioms of DEST posit a set
such that
. For instance,
is a formula involving only
, and thus DEST posits the
Russell set , where
. Observe that for
, we obtain
. Since the membership relations are different, we thus avoid the Russell's paradox.
The focus in DEST is on regular sets, which are sets whose extensions under the two membership relations coincide, i.e., sets
for which it holds that
. The preceding discussion suggests that the Russell set
cannot be regular, as otherwise it leads to the Russell's paradox.
References
Systems of set theory
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