| slovo | definícia |
set theory
(encz) | set theory, n: |
set theory
(wn) | set theory
n 1: the branch of pure mathematics that deals with the nature
and relations of sets |
set theory
(foldoc) | set theory
A mathematical formalisation of the theory of
"sets" (aggregates or collections) of objects ("elements" or
"members"). Many mathematicians use set theory as the basis
for all other mathematics.
Mathematicians began to realise toward the end of the 19th
century that just doing "the obvious thing" with sets led to
embarrassing paradoxes, the most famous being {Russell's
Paradox}. As a result, they acknowledged the need for a
suitable axiomatisation for talking about sets. Numerous
such axiomatisations exist; the most popular among ordinary
mathematicians is Zermelo Fränkel set theory.
{The beginnings of set theory
(http://www-groups.dcs.st-and.ac.uk/~history/HistoryTopics.html)}.
(1995-05-10)
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axiomatic set theory
(foldoc) | axiomatic set theory
One of several approaches to set theory, consisting
of a formal language for talking about sets and a collection
of axioms describing how they behave.
There are many different axiomatisations for set theory.
Each takes a slightly different approach to the problem of
finding a theory that captures as much as possible of the
intuitive idea of what a set is, while avoiding the
paradoxes that result from accepting all of it, the most
famous being Russell's paradox.
The main source of trouble in naive set theory is the idea
that you can specify a set by saying whether each object in
the universe is in the "set" or not. Accordingly, the most
important differences between different axiomatisations of set
theory concern the restrictions they place on this idea (known
as "comprehension").
Zermelo Fränkel set theory, the most commonly used
axiomatisation, gets round it by (in effect) saying that you can
only use this principle to define subsets of existing sets.
NBG (von Neumann-Bernays-Goedel) set theory sort of allows
comprehension for all formulae without restriction, but
distinguishes between two kinds of set, so that the sets
produced by applying comprehension are only second-class sets.
NBG is exactly as powerful as ZF, in the sense that any
statement that can be formalised in both theories is a theorem
of ZF if and only if it is a theorem of ZFC.
MK (Morse-Kelley) set theory is a strengthened version of NBG,
with a simpler axiom system. It is strictly stronger than
NBG, and it is possible that NBG might be consistent but MK
inconsistent.
NF (http://math.boisestate.edu/~holmes/holmes/nf.html) ("New
Foundations"), a theory developed by Willard Van Orman Quine,
places a very different restriction on comprehension: it only
works when the formula describing the membership condition for
your putative set is "stratified", which means that it could
be made to make sense if you worked in a system where every
set had a level attached to it, so that a level-n set could
only be a member of sets of level n+1. (This doesn't mean
that there are actually levels attached to sets in NF). NF is
very different from ZF; for instance, in NF the universe is a
set (which it isn't in ZF, because the whole point of ZF is
that it forbids sets that are "too large"), and it can be
proved that the Axiom of Choice is false in NF!
ML ("Modern Logic") is to NF as NBG is to ZF. (Its name
derives from the title of the book in which Quine introduced
an early, defective, form of it). It is stronger than ZF (it
can prove things that ZF can't), but if NF is consistent then
ML is too.
(2003-09-21)
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zermelo fränkel set theory
(foldoc) | Zermelo Fränkel set theory
A set theory with the axioms of {Zermelo set
theory} (Extensionality, Union, Pair-set, Foundation,
Restriction, Infinity, Power-set) plus the Replacement {axiom
schema}:
If F(x,y) is a formula such that for any x, there is a
unique y making F true, and X is a set, then
F x : x in X
is a set. In other words, if you do something to each element
of a set, the result is a set.
An important but controversial axiom which is NOT part of ZF
theory is the Axiom of Choice.
(1995-04-10)
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zermelo set theory
(foldoc) | Zermelo set theory
A set theory with the following set of
axioms:
Extensionality: two sets are equal if and only if they have
the same elements.
Union: If U is a set, so is the union of all its elements.
Pair-set: If a and b are sets, so is
a, b.
Foundation: Every set contains a set disjoint from itself.
Comprehension (or Restriction): If P is a formula with one
free variable and X a set then
x: x is in X and P(x).
is a set.
Infinity: There exists an infinite set.
Power-set: If X is a set, so is its power set.
Zermelo set theory avoids Russell's paradox by excluding
sets of elements with arbitrary properties - the Comprehension
axiom only allows a property to be used to select elements of
an existing set.
Zermelo Fränkel set theory adds the Replacement axiom.
[Other axioms?]
(1995-03-30)
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