Naive
set theory and paradoxes
Avoiding
paradoxes with type theory
Avoiding
paradoxes with axiomatic set theory
Naive set theory and paradoxes
Set
theory was created at the end of the
19-th century by Georg Cantor in order
to allow to work with infinite sets
consistently, and attempted to formalize
the set using a minimal collection of
independent axioms. Later it was named
"naive set theory".
In 1901 Bertrand Russell discovered
paradox, which was named Russell's
paradox. The paradox
arises within naive set theory by considering
the set of all sets that are not members
of themselves. Such a set appears to
be a member of itself if and only if
it is not a member of itself, hence
the paradox. After the discovery of
the paradox, it becomes clear that assuming
that one could perform any operations
on sets without restriction led to paradoxes,
and naive set theory must be replaced
by something in which the paradoxes
can't arise. Two solutions were proposed:
type theory and axiomatic
set theory.
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Avoiding
paradoxes with type theory
To
avoid Russell's paradox Russell himself,
together with Whitehead proposed a theory
of types in which sentences were arranged
hierarchically. At the lowest level
are sentences about individuals. At
the next level are sentences about sets
of individuals; at the next level, sentences
about sets of sets of individuals, and
so on. It is then possible to refer
to all objects for which a given condition
(or predicate) holds only if they are
all at the same level or of the same
"type". This avoids the possibility
of having to talk about the set of all
sets that are not members of themselves,
because the two parts of the sentence
are of different types - that is, at
different levels.
Theory of type have been criticized
for being too ad hoc to eliminate the
paradox successfully.
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Avoiding
paradoxes with axiomatic set theory
To
avoid Russell's paradox and other related
problems Ernst Zermelo proposed an axiomatic
set theory. Zermelo’s axioms were designed
to resolve Russell’s paradox by restricting
Cantor’s naive comprehension principle.
This theory determines what operations
were allowed and when. Today Zermelo-Fraenkel
set theory is the most common version
of the axiomatic set theory. Theory
does not assume that, for every property,
there is a set of all things satisfying
that property. It assumes that for any
given set and any definable property,
there is a subset of all elements of
the given set satisfying the property.
The set of all sets M cannot be constructed
like that and is not a set in this theory.
by
Tetyana Butler
