more on this theme     |     more from this text

Single Idea 13530

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers ]

Full Idea

Less theoretically, an ordinal is an equivalence class of well-orderings. Formally, we say a set is 'transitive' if every member of it is a subset of it, and an ordinal is a transitive set, all of whose members are transitive.

Gist of Idea

An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive

Source

Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.4)

Book Ref

Wolf,Robert S.: 'A Tour Through Mathematical Logic' [Carus Maths Monographs 2005], p.77

A Reaction

He glosses 'transitive' as 'every member of a member of it is a member of it'. So it's membership all the way down. This is the von Neumann rather than the Zermelo approach (which is based on singletons).

The 19 ideas from Robert S. Wolf

13519 Model theory uses sets to show that mathematical deduction fits mathematical truth [Wolf,RS]
13518 Modern mathematics has unified all of its objects within set theory [Wolf,RS]
13520 A 'tautology' must include connectives [Wolf,RS]
13524 Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof [Wolf,RS]
13522 Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) [Wolf,RS]
13521 Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance [Wolf,RS]
13523 Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P [Wolf,RS]
13525 Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens [Wolf,RS]
13526 Comprehension Axiom: if a collection is clearly specified, it is a set [Wolf,RS]
13529 Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists [Wolf,RS]
13530 An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS]
13531 Model theory reveals the structures of mathematics [Wolf,RS]
13532 Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' [Wolf,RS]
13533 First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem [Wolf,RS]
13534 In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide [Wolf,RS]
13535 First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation [Wolf,RS]
13537 An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS]
13538 If a theory is complete, only a more powerful language can strengthen it [Wolf,RS]
13539 The LST Theorem is a serious limitation of first-order logic [Wolf,RS]