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Single Idea 17760

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

Full Idea

There may be several ordinals for the same cardinality. ...Two ordinals can represent different ways of well-ordering the same number (aleph-0) of elements.

Gist of Idea

Two infinite ordinals can represent a single infinite cardinal

Source

Michal Walicki (Introduction to Mathematical Logic [2012], 2.3)

Book Ref

Walicki,Michal: 'Introduction to Mathematical Logic' [World Scientific 2012], p.89

A Reaction

This only applies to infinite ordinals and cardinals. For the finite, the two coincide. In infinite arithmetic the rules are different.

The 19 ideas from 'Introduction to Mathematical Logic'

17742 Scotus based modality on semantic consistency, instead of on what the future could allow [Walicki]
17747 A 'model' of a theory specifies interpreting a language in a domain to make all theorems true [Walicki]
17748 The L-S Theorem says no theory (even of reals) says more than a natural number theory [Walicki]
17749 Post proved the consistency of propositional logic in 1921 [Walicki]
17741 To determine the patterns in logic, one must identify its 'building blocks' [Walicki]
17752 The empty set is useful for defining sets by properties, when the members are not yet known [Walicki]
17753 The empty set avoids having to take special precautions in case members vanish [Walicki]
17754 Inductive proof depends on the choice of the ordering [Walicki]
17759 Ordinals play the central role in set theory, providing the model of well-ordering [Walicki]
17758 Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion [Walicki]
17755 Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals [Walicki]
17756 The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki]
17760 Two infinite ordinals can represent a single infinite cardinal [Walicki]
17757 Members of ordinals are ordinals, and also subsets of ordinals [Walicki]
17762 In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate [Walicki]
17761 A compact axiomatisation makes it possible to understand a field as a whole [Walicki]
17763 Axiomatic systems are purely syntactic, and do not presuppose any interpretation [Walicki]
17764 Boolean connectives are interpreted as functions on the set {1,0} [Walicki]
17765 Propositional language can only relate statements as the same or as different [Walicki]