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Single Idea 17754
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
]
Full Idea
Inductive proof is not guaranteed to work in all cases and, particularly, it depends heavily on the choice of the ordering.
Gist of Idea
Inductive proof depends on the choice of the ordering
Source
Michal Walicki (Introduction to Mathematical Logic [2012], 2.1.1)
Book Ref
Walicki,Michal: 'Introduction to Mathematical Logic' [World Scientific 2012], p.69
A Reaction
There has to be an well-founded ordering for inductive proofs to be possible.
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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17757
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Members of ordinals are ordinals, and also subsets of ordinals
[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
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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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17762
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In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate
[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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17761
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A compact axiomatisation makes it possible to understand a field as a whole
[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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