Set Theory

Jech, T. (2003). Set Theory: The Third Millennium Edition, Revised and Expanded. Springer.

Type

Book

Intellectual base

Review or monograph

Year

2003

Pages cited

1-3

DOI

10.1007/3-540-44761-x

ISBN

9783540440857

DOI registrant

Springer Nature

Link

https://doi.org/10.1007/3-540-44761-X

Cited from

mathematics

Cited by

23 citations across 23 artifacts.

Each citation links to the sentence it supports in the citing article.

Primes

• Well-Foundedness (Well-Ordering)
  • Well-ordering is the total-order version of well-foundedness; Zermelo (1904, restated 1908) proved the well-ordering theorem — equivalent to the axiom of choice — guaranteeing that every set admits a well-ordering, a result developed canonically in modern texts such as Jech (2003).

    This sourceCanonical reference for modern axiomatic set theory; develops well-foundedness, the well-ordering theorem, the cumulative hierarchy V_α, and ordinal/cardinal arithmetic.

Domain-specific

• Amoeba order
  • … demonstrates that the null ideal or amoeba forcing, carrier sets, measure bound, comparison relation, order direction, cardinal invariant and dependence on set-theoretic axioms are explicit. to An applied instance preserves the same invariant under a changed scale, notation, jurisdiction, dataset, or implementation..
• Axiom of infinity
  • The ZF axiom of infinity asserts that there is a set I such that the empty set belongs to I and, whenever x belongs to I, the successor \(x\cup\{x\}\) also belongs to I.
• Closed Preordered Set
  • Jech instead calls a forcing order `κ`-closed when the requirement extends through sequences of length `κ`, and uses `<κ`-closed for all smaller indexed closure properties.
• Club principle
  • … A-delta of each delta, arbitrary unbounded subset A, containment witness delta and comparison with club sets diamond and continuum-hypothesis implications and consistency results are explicit. to An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation..
• Continuous function (ordinal theory)
  • In ordinal theory, a continuous increasing function preserves suprema at limit stages: f(lambda) equals the supremum of f(alpha) for alpha below each limit lambda in its domain.
• Generic filter
  • … transfers from A canonical instance directly demonstrates that the filter obeys the order convention and meets each dense subset of the forcing notion that lies in the declared ground model. to An applied instance preserves the same invariant under a changed scale, notation, jurisdiction, or implementation..
• Knaster's condition
  • … constitutive identity and a convenient observable transfers from A Knaster forcing cannot contain an uncountable antichain because the required uncountable linked subset would contain compatible pairs. to A forcing proof distinguishes Knaster, precaliber-aleph-one and ccc, whose preservation properties differ..
• Kurepa tree
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Mouse (set theory)
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Normal function
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Normal measure
  • … closure, regressive functions on a U-large set and constant-on-U-large conclusion, ultrapower embedding and image of identity, concentration properties and equivalence of formulations are explicit. to An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation..
• Order type
  • … example, the distinction between constitutive identity and a convenient observable transfers from The natural numbers and the positive even integers in their usual order have the same order type omega. to A set theorist distinguishes order type from cardinality because equal-size sets can have nonisomorphic orders..
• Ordinal definable set
  • … rank, first-order formula and coding, finite ordinal parameters, uniqueness condition, absoluteness qualifications, class OD, relation to HOD and L and closure and forcing behavior are explicit. to An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation..
• Pairing function
  • The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.
• Proper forcing axiom
  • The proper forcing axiom states that every proper forcing P and sequence of ℵ1 dense subsets of P admit a filter intersecting each dense set.
• Set-Theoretic Code
  • The code is chosen so that this collapse is \(\operatorname{TC}(\{x\})\), the transitive closure of the singleton containing \(x\), with one distinguished element corresponding to \(x\).
• Square principle
  • … theory and cardinal or class, indexed club sequence, limit domains, order-type or width bounds, coherence equation, no-thread condition and consistency or implication claim are explicit. to An applied instance preserves the same invariant under changed scale, notation, dataset, jurisdiction, or implementation..
• Standard model (set theory)
  • It does not by itself say that \(M\) is transitive, a set, a proper class, well founded from outside, or a model of any particular axiom system; those are separate conditions.
• Strong measure zero set
  • Strong measure zero implies Lebesgue measure zero but is strictly stronger in standard models; whether every such subset of the real line is countable is Borel's conjecture and is independent of ZFC.
• Supercompact cardinal
  • … point, j(kappa) bound, sequence-closure requirement, or fine normal measure formulation, size conventions, equivalence proof, hierarchy implications, and consistency status are explicit. to An applied instance preserves the same invariant under a changed scale, notation, jurisdiction, dataset, or implementation..
• Transitive Set
  • the ambient membership relation — ordinary set membership, not an arbitrary relation labeled “contains.”
• Tree (Set Theory)
  • The definition permits incomparable nodes, so one history can split into many continuations, and it permits limit heights, where a node has predecessors at every earlier level without having an immediate predecessor.

    This sourceAuthoritative reference for trees, ordinal height and levels, Aronszajn and Suslin trees, the tree property, and its large-cardinal context.

Verification

Does it exist? Confirmed. This work's DOI resolves to a registered record, which fixes its identity. That is all it fixes.

Does it back the claim? Not recorded. Neither this nor any other of the 23 citations of this work carries a recorded support check.

Support is checked per citation rather than per work — the same source can be cited soundly in one article and wrongly in another. Per-citation recording began recently, so a citation with no recorded check is a gap in the record rather than evidence it went unchecked.

See how references were verified.

Registry ID ref:262645998302 · see in the full table