more on this theme
|
more from this text
Single Idea 17757
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
]
Full Idea
Every member of an ordinal is itself an ordinal, and every ordinal is a transitive set (its members are also its subsets; a member of a member of an ordinal is also a member of the ordinal).
Gist of Idea
Members of ordinals are ordinals, and also subsets of ordinals
Source
Michal Walicki (Introduction to Mathematical Logic [2012], 2.3)
Book Ref
Walicki,Michal: 'Introduction to Mathematical Logic' [World Scientific 2012], p.89
The
19 ideas
from Michal Walicki
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]
|
17758
|
Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion
[Walicki]
|
17757
|
Members of ordinals are ordinals, and also subsets of ordinals
[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]
|
17759
|
Ordinals play the central role in set theory, providing the model of well-ordering
[Walicki]
|
17763
|
Axiomatic systems are purely syntactic, and do not presuppose any interpretation
[Walicki]
|
17761
|
A compact axiomatisation makes it possible to understand a field as a whole
[Walicki]
|
17762
|
In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate
[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]
|