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Single Idea 17755
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
]
Full Idea
The collection of ordinals is defined inductively: Basis: the empty set is an ordinal; Ind: for an ordinal x, the union with its singleton is also an ordinal; and any arbitrary (possibly infinite) union of ordinals is an ordinal.
Gist of Idea
Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals
Source
Michal Walicki (Introduction to Mathematical Logic [2012], 2.3)
Book Ref
Walicki,Michal: 'Introduction to Mathematical Logic' [World Scientific 2012], p.88
A Reaction
[symbolism translated into English] Walicki says they are called 'ordinal numbers', but are in fact a set.
Related Idea
Idea 17756
The union of finite ordinals is the first 'limit ordinal'; 2ω is the second... [Walicki]
The
19 ideas
from Michal Walicki
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17742
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Scotus based modality on semantic consistency, instead of on what the future could allow
[Walicki]
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17747
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A 'model' of a theory specifies interpreting a language in a domain to make all theorems true
[Walicki]
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17748
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The L-S Theorem says no theory (even of reals) says more than a natural number theory
[Walicki]
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17749
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Post proved the consistency of propositional logic in 1921
[Walicki]
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17741
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To determine the patterns in logic, one must identify its 'building blocks'
[Walicki]
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17752
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The empty set is useful for defining sets by properties, when the members are not yet known
[Walicki]
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17753
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The empty set avoids having to take special precautions in case members vanish
[Walicki]
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17754
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Inductive proof depends on the choice of the ordering
[Walicki]
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17759
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Ordinals play the central role in set theory, providing the model of well-ordering
[Walicki]
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17758
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Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion
[Walicki]
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17755
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Ordinals are the empty set, union with the singleton, and any arbitrary union of ordinals
[Walicki]
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17756
|
The union of finite ordinals is the first 'limit ordinal'; 2ω is the second...
[Walicki]
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17760
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Two infinite ordinals can represent a single infinite cardinal
[Walicki]
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17757
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Members of ordinals are ordinals, and also subsets of ordinals
[Walicki]
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17762
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In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate
[Walicki]
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17761
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A compact axiomatisation makes it possible to understand a field as a whole
[Walicki]
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17763
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Axiomatic systems are purely syntactic, and do not presuppose any interpretation
[Walicki]
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17764
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Boolean connectives are interpreted as functions on the set {1,0}
[Walicki]
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17765
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Propositional language can only relate statements as the same or as different
[Walicki]
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