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Single Idea 17748
[filed under theme 5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
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Full Idea
The L-S Theorem is ...a shocking result, since it implies that any consistent formal theory of everything - even about biology, physics, sets or the real numbers - can just as well be understood as being about natural numbers. It says nothing more.
Gist of Idea
The L-S Theorem says no theory (even of reals) says more than a natural number theory
Source
Michal Walicki (Introduction to Mathematical Logic [2012], History E.2)
Book Ref
Walicki,Michal: 'Introduction to Mathematical Logic' [World Scientific 2012], p.34
A Reaction
Illuminating. Particularly the point that no theory about the real numbers can say anything more than a theory about the natural numbers. So the natural numbers contain all the truths we can ever express? Eh?????
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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17757
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Members of ordinals are ordinals, and also subsets of ordinals
[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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17758
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Ordinals are transitive sets of transitive sets; or transitive sets totally ordered by inclusion
[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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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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17762
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In non-Euclidean geometry, all Euclidean theorems are valid that avoid the fifth postulate
[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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