Fraenkel–Mostowski permutation models
E399414
Fraenkel–Mostowski permutation models are set-theoretic constructions using permutations of atoms to demonstrate the independence of certain choice principles from Zermelo–Fraenkel set theory.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Fraenkel set theory with atoms | 1 |
| Fraenkel–Mostowski permutation models canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3930985 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Fraenkel–Mostowski permutation models Context triple: [Abraham Fraenkel, knownFor, Fraenkel–Mostowski permutation models]
-
A.
Kripke–Platek set theory
Kripke–Platek set theory is a weaker, predicative subsystem of Zermelo–Fraenkel set theory focused on sets that are explicitly constructible and often used in the study of admissible sets and recursion theory.
-
B.
Foundations of Set Theory (with Andrey Kolmogorov)
"Foundations of Set Theory" is a classic 20th-century mathematical text co-authored by Pavel Alexandrov and Andrey Kolmogorov that systematically develops the basic concepts and axioms of set theory.
-
C.
New Foundations for Mathematical Logic
New Foundations for Mathematical Logic is W.V.O. Quine’s influential essay proposing an alternative set theory, known as "New Foundations," aimed at resolving paradoxes while preserving a broad, intuitive universe of sets.
-
D.
Morse–Kelley set theory by class–set distinction
Morse–Kelley set theory by class–set distinction is a foundational system that avoids certain set-theoretic paradoxes by rigorously distinguishing between sets and proper classes within a powerful axiomatic framework.
-
E.
von Neumann paradox in set theory
The von Neumann paradox in set theory is a foundational result showing that, under certain group-theoretic conditions, a set can be decomposed and reassembled into paradoxical subsets of equal “size,” illustrating the counterintuitive consequences of the axiom of choice.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Fraenkel–Mostowski permutation models Target entity description: Fraenkel–Mostowski permutation models are set-theoretic constructions using permutations of atoms to demonstrate the independence of certain choice principles from Zermelo–Fraenkel set theory.
-
A.
Kripke–Platek set theory
Kripke–Platek set theory is a weaker, predicative subsystem of Zermelo–Fraenkel set theory focused on sets that are explicitly constructible and often used in the study of admissible sets and recursion theory.
-
B.
Foundations of Set Theory (with Andrey Kolmogorov)
"Foundations of Set Theory" is a classic 20th-century mathematical text co-authored by Pavel Alexandrov and Andrey Kolmogorov that systematically develops the basic concepts and axioms of set theory.
-
C.
New Foundations for Mathematical Logic
New Foundations for Mathematical Logic is W.V.O. Quine’s influential essay proposing an alternative set theory, known as "New Foundations," aimed at resolving paradoxes while preserving a broad, intuitive universe of sets.
-
D.
Morse–Kelley set theory by class–set distinction
Morse–Kelley set theory by class–set distinction is a foundational system that avoids certain set-theoretic paradoxes by rigorously distinguishing between sets and proper classes within a powerful axiomatic framework.
-
E.
von Neumann paradox in set theory
The von Neumann paradox in set theory is a foundational result showing that, under certain group-theoretic conditions, a set can be decomposed and reassembled into paradoxical subsets of equal “size,” illustrating the counterintuitive consequences of the axiom of choice.
- F. None of above. chosen
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
model of set theory with atoms
ⓘ
permutation model ⓘ set-theoretic construction ⓘ |
| aimsToShow |
independence of choice principles
ⓘ
independence of the axiom of choice ⓘ independence of weaker forms of choice ⓘ |
| appliedIn |
study of cardinal arithmetic without choice
ⓘ
study of choice in analysis ⓘ study of partition properties without choice ⓘ |
| assumes | existence of a set of atoms disjoint from the pure sets ⓘ |
| basedOn |
Fraenkel–Mostowski permutation models
self-linksurface differs
ⓘ
surface form:
Fraenkel set theory with atoms
Mostowski’s permutation method ⓘ |
| clarifies | distinction between pure sets and atoms ⓘ |
| constructedIn | ZFA ⓘ |
| contrastsWith | forcing with generic filters ⓘ |
| defines | symmetric submodel of a universe with atoms ⓘ |
| developedBy |
Abraham Fraenkel
ⓘ
Andrzej Mostowski ⓘ |
| feature |
distinguished set of atoms
ⓘ
group of permutations of atoms ⓘ hereditarily symmetric sets ⓘ normal filter of subgroups ⓘ notion of support for sets ⓘ |
| formalSetting | first-order set theory ⓘ |
| historicalOrigin |
work of Fraenkel on set theory with atoms
ⓘ
work of Mostowski on permutation models ⓘ |
| influenced | development of independence proofs in set theory ⓘ |
| mathematicalArea | foundations of mathematics ⓘ |
| mathematicalDiscipline | set theory ⓘ |
| relatedConstruction | symmetric submodels of forcing extensions ⓘ |
| relatedToTheory |
Zermelo–Fraenkel set theory
ⓘ
set theory ⓘ
surface form:
set theory with urelements
|
| shows |
axiom of choice is independent of ZF
ⓘ
various weaker choice principles are independent of ZF ⓘ |
| usedFor |
analyzing the strength of choice principles
ⓘ
constructing models of ZF + ¬AC ⓘ constructing models of ZF with partial choice ⓘ demonstrating independence of combinatorial principles ⓘ |
| uses |
atoms
ⓘ
permutations of atoms ⓘ |
| yields |
models of ZF without the axiom of choice
ⓘ
models where choice holds for finite sets but fails for countable sets ⓘ models where every set of reals is Lebesgue measurable (in suitable extensions) ⓘ models where the axiom of choice fails for families of sets of reals ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Fraenkel–Mostowski permutation models Description of subject: Fraenkel–Mostowski permutation models are set-theoretic constructions using permutations of atoms to demonstrate the independence of certain choice principles from Zermelo–Fraenkel set theory.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
Fraenkel–Mostowski permutation models
→
basedOn
→
Fraenkel–Mostowski permutation models
self-linksurface differs
ⓘ
this entity surface form:
Fraenkel set theory with atoms