forcing (set theory)
E446865
Forcing (set theory) is a powerful technique in mathematical logic, introduced by Paul Cohen, used to construct models of set theory and prove the independence of certain propositions from Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC).
All labels observed (4)
| Label | Occurrences |
|---|---|
| Laver forcing | 2 |
| Proper and Improper Forcing | 1 |
| forcing (set theory) canonical | 1 |
| proper forcing theory | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T4493079 — 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: forcing (set theory) Context triple: [Paul Cohen, notableConcept, forcing (set theory)]
-
A.
Fraenkel–Mostowski permutation models
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.
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B.
set theory
Set theory is a foundational branch of mathematical logic that studies collections of objects, called sets, and underpins much of modern mathematics.
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C.
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.
-
D.
constructible universe
The constructible universe is a class model of set theory introduced by Kurt Gödel that systematically builds sets in hierarchical stages and shows the relative consistency of the axiom of choice and the generalized continuum hypothesis with ZF.
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E.
Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
- 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: forcing (set theory) Target entity description: Forcing (set theory) is a powerful technique in mathematical logic, introduced by Paul Cohen, used to construct models of set theory and prove the independence of certain propositions from Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC).
-
A.
Fraenkel–Mostowski permutation models
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.
-
B.
set theory
Set theory is a foundational branch of mathematical logic that studies collections of objects, called sets, and underpins much of modern mathematics.
-
C.
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.
-
D.
constructible universe
The constructible universe is a class model of set theory introduced by Kurt Gödel that systematically builds sets in hierarchical stages and shows the relative consistency of the axiom of choice and the generalized continuum hypothesis with ZF.
-
E.
Zermelo–Fraenkel set theory
Zermelo–Fraenkel set theory is the standard axiomatic framework for modern set theory, designed to avoid paradoxes and provide a rigorous foundation for much of mathematics.
- F. None of above. chosen
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
method in set theory
ⓘ
model construction method ⓘ technique in mathematical logic ⓘ |
| appliesTo |
models of ZFC
ⓘ
transitive models of set theory ⓘ |
| basedOn |
Boolean-valued models
ⓘ
partial orders ⓘ |
| coreConcept |
dense subset
ⓘ
forcing condition ⓘ genericity over a ground model ⓘ names (forcing names) ⓘ |
| defines |
forcing extension
ⓘ
forcing notion ⓘ forcing relation ⓘ generic filter ⓘ |
| enables |
construction of models where the Continuum Hypothesis fails
ⓘ
construction of models where the Continuum Hypothesis holds ⓘ construction of models with different cardinal arithmetic ⓘ construction of models with or without certain combinatorial principles ⓘ construction of models with special subsets of the reals ⓘ |
| field |
mathematical logic
ⓘ
set theory ⓘ |
| goal |
control which new sets are added to a model
ⓘ
extend a ground model to a larger model ⓘ |
| hasVariant |
Martin’s Axiom related forcing
ⓘ
class forcing ⓘ forcing with side conditions ⓘ iterated forcing ⓘ proper forcing ⓘ semi-proper forcing ⓘ symmetric submodel forcing ⓘ |
| historicalPeriod | 1960s ⓘ |
| influenced |
modern set-theoretic research
ⓘ
philosophy of set-theoretic truth ⓘ |
| introducedBy | Paul Cohen ⓘ |
| notableResult |
independence of Suslin’s Hypothesis from ZFC
ⓘ
independence of the Axiom of Choice from some weaker systems ⓘ independence of the Continuum Hypothesis from ZFC ⓘ |
| relatedTo |
Boolean-valued models of set theory
ⓘ
descriptive set theory ⓘ inner model theory ⓘ large cardinal axioms ⓘ |
| standardReference |
Set Theory by Kenneth Kunen
ⓘ
Set Theory by Thomas Jech ⓘ |
| usedFor |
constructing models of set theory
ⓘ
proving independence results ⓘ proving the independence of Suslin’s Hypothesis ⓘ proving the independence of the Axiom of Choice variants ⓘ proving the independence of the Continuum Hypothesis ⓘ proving the independence of the existence of measurable cardinals from weaker theories ⓘ showing consistency of statements relative to ZFC ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: forcing (set theory) Description of subject: Forcing (set theory) is a powerful technique in mathematical logic, introduced by Paul Cohen, used to construct models of set theory and prove the independence of certain propositions from Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC).
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: forcing (set theory)
linked to: forcing (set theory)
linked to: forcing (set theory)
linked to: forcing (set theory)