Set Theory

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

Type

Book

Intellectual base

Review or monograph

Year

2003

DOI

10.1007/3-540-44761-x

ISBN

9783540440857

Link

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

Cited from

mathematics

Cited by

13 citations across 13 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

• 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
  • … is written “`<κ`-closed”; Kunen's older convention calls a poset `κ`-closed when it has lower bounds for every descending sequence of length `<κ`. 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.
• 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.
• 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.
• 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

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