|
|
| Line 1: |
Line 1: |
| In [[mathematical logic]], '''positive set theory''' is the name for a class of alternative [[set theory|set theories]] in which the [[axiom of comprehension]]
| |
|
| |
|
| *"<math>\{x \mid \phi\}</math> exists"
| |
|
| |
|
| holds for at least the <strong>positive formulas</strong> <math>\phi</math> (the smallest class of formulas containing atomic membership and equality formulas and closed under conjunction, disjunction, existential and universal quantification).
| | Aleta is what's written referring to her birth certificate though she doesn't really like being called like it. Massachusetts is considered where he's always been living. Managing people today is what she is performing in her day job but she's always expected her own [http://www.Bbc.Co.uk/search/?q=business business]. To drive is something her sister doesn't really like but she does. She is running and sticking to a blog here: http://[http://Prometeu.org/ prometeu].net/<br><br>My homepage; [http://prometeu.net/ clash of Clans hack ipad Download] |
| | |
| Typically, the motivation for these theories is topological: the sets are the classes which are closed under a certain [[topology]]. The closure conditions for the various constructions allowed in building positive formulas are readily motivated (and one can further justify the use of universal quantifiers bounded in sets to get <strong>generalized positive comprehension</strong>): the justification of the existential quantifier seems to require that the topology be [[compact spaces|compact]].
| |
| | |
| The set theory <math>GPK^+_{\infty}</math> of Olivier Esser consists of the following axioms:
| |
| | |
| * The [[axiom of extensionality]]: <math>x=y \Leftrightarrow\forall a\, (a\in x \Leftrightarrow a\in y)</math>.
| |
| * The [[axiom of empty set]]: there exists a set <math>\emptyset</math> such that <math>\,\neg \exists x\; x\in\emptyset\,</math> (this axiom can be neatly dispensed with if a false formula <math>\perp</math> is included as a positive formula).
| |
| * The axiom of generalized positive [[axiom of comprehension|comprehension]]: if <math>\phi</math> is a formula in predicate logic using only <math>\vee</math>, <math>\wedge</math>, <math>\exists</math>, <math>\forall</math>, <math>=</math>, and <math>\in</math>, then the set of all <math>x</math> such that <math>\phi(x)</math> is also a set. Quantification (<math>\forall</math>, <math>\exists</math>) may be bounded.
| |
| ** Note that negation is specifically not permitted.
| |
| * The axiom of [[topological closure|closure]]: for every formula <math>\phi(x)</math>, a set exists which is the intersection of all sets which contain every ''x'' such that <math>\phi(x)</math>; this is called the '''closure of <math>\{x \mid \phi(x)\}</math>''' and is written in any of the various ways that topological closures can be presented. This can be put more briefly if class language is allowed (any condition on sets defining a class as in [[Von Neumann–Bernays–Gödel set theory|NBG]]): for any class ''C'' there is a set which is the intersection of all sets which contain ''C'' as a subclass. This is obviously a reasonable principle if the sets are understood as closed classes in a topology.
| |
| * The [[axiom of infinity]]: the [[von Neumann]] [[ordinal number|ordinal]] <math>\omega</math> exists. This is not an axiom of infinity in the usual sense; if Infinity does not hold, the closure of <math>\omega</math> exists and has itself as its sole additional member (it is certainly infinite); the point of this axiom is that <math>\omega</math> contains no additional elements at all, which boosts the theory from the strength of second order arithmetic to the strength of [[Morse–Kelley set theory]] with the proper class ordinal a [[weakly compact cardinal]].
| |
| | |
| | |
| == Interesting properties ==
| |
| | |
| * The [[universal set]] is a proper set in this theory.
| |
| * The sets of this theory are the collections of sets which are closed under a certain [[topology]] on the classes.
| |
| * The theory can interpret [[ZFC]] (by restricting oneself to the class of well-founded sets, which is not itself a set). It in fact interprets a stronger theory ([[Morse-Kelley set theory]] with the proper class ordinal a [[weakly compact cardinal]]).
| |
| <!--* ... many more -->
| |
| | |
| == Researchers ==
| |
| | |
| * [[Isaac Malitz]] originally introduced Positive Set Theory in his 1976 PhD Thesis at UCLA
| |
| * [[Alonzo Church]] was the chairman of the committee supervising the aforementioned thesis
| |
| * [[Olivier Esser]] seems to be the most active in this field.
| |
| | |
| == See also ==
| |
| * [[New Foundations]] by [[W._V._Quine|Quine]]
| |
| ==References==
| |
| *{{citation
| |
| |last=Esser|first= Olivier
| |
| |title=On the consistency of a positive theory.
| |
| |journal=MLQ Math. Log. Q.|volume= 45 |year=1999|issue= 1|pages= 105–116
| |
| |mr=1669902
| |
| |doi=10.1002/malq.19990450110}}
| |
| | |
| | |
| | |
| [[Category:Systems of set theory]]
| |
Aleta is what's written referring to her birth certificate though she doesn't really like being called like it. Massachusetts is considered where he's always been living. Managing people today is what she is performing in her day job but she's always expected her own business. To drive is something her sister doesn't really like but she does. She is running and sticking to a blog here: http://prometeu.net/
My homepage; clash of Clans hack ipad Download