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Single Idea 13677
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
]
Full Idea
By convention, the natural numbers are the finite ordinals, the integers are certain equivalence classes of pairs of finite ordinals, etc.
Gist of Idea
Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals
Source
Stewart Shapiro (Foundations without Foundationalism [1991], 9.3)
Book Ref
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.251
The
50 ideas
from 'Foundations without Foundationalism'
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15944
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Second-order logic is better than set theory, since it only adds relations and operations, and nothing else
[Shapiro, by Lavine]
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13629
|
Broad standard semantics, or Henkin semantics with a subclass, or many-sorted first-order semantics?
[Shapiro]
|
|
13626
|
Semantic consequence is ineffective in second-order logic
[Shapiro]
|
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13631
|
Are sets part of logic, or part of mathematics?
[Shapiro]
|
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13627
|
There is no 'correct' logic for natural languages
[Shapiro]
|
|
13624
|
The 'triumph' of first-order logic may be related to logicism and the Hilbert programme, which failed
[Shapiro]
|
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13628
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We can live well without completeness in logic
[Shapiro]
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13630
|
Non-compactness is a strength of second-order logic, enabling characterisation of infinite structures
[Shapiro]
|
|
13625
|
Mathematics and logic have no border, and logic must involve mathematics and its ontology
[Shapiro]
|
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13634
|
Satisfaction is 'truth in a model', which is a model of 'truth'
[Shapiro]
|
|
13632
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Finding the logical form of a sentence is difficult, and there are no criteria of correctness
[Shapiro]
|
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13633
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'Satisfaction' is a function from models, assignments, and formulas to {true,false}
[Shapiro]
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13635
|
'Weakly sound' if every theorem is a logical truth; 'sound' if every deduction is a semantic consequence
[Shapiro]
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13636
|
An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation
[Shapiro]
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|
13637
|
If a logic is incomplete, its semantic consequence relation is not effective
[Shapiro]
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13640
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Russell's paradox shows that there are classes which are not iterative sets
[Shapiro]
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13638
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Properties are often seen as intensional; equiangular and equilateral are different, despite identity of objects
[Shapiro]
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13641
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Complex numbers can be defined as reals, which are defined as rationals, then integers, then naturals
[Shapiro]
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13642
|
Logic is the ideal for learning new propositions on the basis of others
[Shapiro]
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13643
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Aristotelian logic is complete
[Shapiro]
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13644
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Semantics for models uses set-theory
[Shapiro]
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13650
|
Henkin semantics has separate variables ranging over the relations and over the functions
[Shapiro]
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|
13645
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In standard semantics for second-order logic, a single domain fixes the ranges for the variables
[Shapiro]
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13649
|
Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics
[Shapiro]
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|
13648
|
The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity
[Shapiro]
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13647
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Choice is essential for proving downward Löwenheim-Skolem
[Shapiro]
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13646
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Compactness is derived from soundness and completeness
[Shapiro]
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13651
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A set is 'transitive' if contains every member of each of its members
[Shapiro]
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13657
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First-order arithmetic can't even represent basic number theory
[Shapiro]
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13652
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The 'continuum' is the cardinality of the powerset of a denumerably infinite set
[Shapiro]
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13653
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'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element
[Shapiro]
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13654
|
It is central to the iterative conception that membership is well-founded, with no infinite descending chains
[Shapiro]
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13656
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Some sets of natural numbers are definable in set-theory but not in arithmetic
[Shapiro]
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13661
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A language is 'semantically effective' if its logical truths are recursively enumerable
[Shapiro]
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13660
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Maybe compactness, semantic effectiveness, and the Löwenheim-Skolem properties are desirable
[Shapiro]
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13658
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Downward Löwenheim-Skolem: each satisfiable countable set always has countable models
[Shapiro]
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13659
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Upward Löwenheim-Skolem: each infinite model has infinite models of all sizes
[Shapiro]
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13662
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First-order logic was an afterthought in the development of modern logic
[Shapiro]
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13666
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Iterative sets are not Boolean; the complement of an iterative set is not an iterative sets
[Shapiro]
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13664
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Logicism is distinctive in seeking a universal language, and denying that logic is a series of abstractions
[Shapiro]
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13663
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Some reject formal properties if they are not defined, or defined impredicatively
[Shapiro]
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13667
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Skolem and Gödel championed first-order, and Zermelo, Hilbert, and Bernays championed higher-order
[Shapiro]
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13668
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Bernays (1918) formulated and proved the completeness of propositional logic
[Shapiro]
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13669
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Can one develop set theory first, then derive numbers, or are numbers more basic?
[Shapiro]
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13670
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Categoricity can't be reached in a first-order language
[Shapiro]
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|
13673
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The notion of finitude is actually built into first-order languages
[Shapiro]
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13674
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We might reduce ontology by using truth of sentences and terms, instead of using objects satisfying models
[Shapiro]
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13675
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Substitutional semantics only has countably many terms, so Upward Löwenheim-Skolem trivially fails
[Shapiro]
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13676
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Only higher-order languages can specify that 0,1,2,... are all the natural numbers that there are
[Shapiro]
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13677
|
Natural numbers are the finite ordinals, and integers are equivalence classes of pairs of finite ordinals
[Shapiro]
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