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Single Idea 17792
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
]
Full Idea
The sole theoretical interest of first-order Peano arithmetic derives from the fact that it is a first-order reduct of a categorical second-order theory. Its axioms can be proved incomplete only because the second-order theory is categorical.
Gist of Idea
1st-order PA is only interesting because of results which use 2nd-order PA
Source
John Mayberry (What Required for Foundation for Maths? [1994], p.412-1)
Book Ref
'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.412
The
31 ideas
from John Mayberry
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17774
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Definitions make our intuitions mathematically useful
[Mayberry]
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17775
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If proof and definition are central, then mathematics needs and possesses foundations
[Mayberry]
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17776
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The ultimate principles and concepts of mathematics are presumed, or grasped directly
[Mayberry]
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17777
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Foundations need concepts, definition rules, premises, and proof rules
[Mayberry]
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17773
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Proof shows that it is true, but also why it must be true
[Mayberry]
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17779
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'Classificatory' axioms aim at revealing similarity in morphology of structures
[Mayberry]
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17778
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Axiomatiation relies on isomorphic structures being essentially the same
[Mayberry]
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17780
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'Eliminatory' axioms get rid of traditional ideal and abstract objects
[Mayberry]
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17781
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Real numbers were invented, as objects, to simplify and generalise 'quantity'
[Mayberry]
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17782
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Greek quantities were concrete, and ratio and proportion were their science
[Mayberry]
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17785
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Real numbers as abstracted objects are now treated as complete ordered fields
[Mayberry]
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17784
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Real numbers can be eliminated, by axiom systems for complete ordered fields
[Mayberry]
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17786
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The mainstream of modern logic sees it as a branch of mathematics
[Mayberry]
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17787
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Big logic has one fixed domain, but standard logic has a domain for each interpretation
[Mayberry]
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17788
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First-order logic only has its main theorems because it is so weak
[Mayberry]
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17791
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Only second-order logic can capture mathematical structure up to isomorphism
[Mayberry]
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17790
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No Löwenheim-Skolem logic can axiomatise real analysis
[Mayberry]
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17789
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No logic which can axiomatise arithmetic can be compact or complete
[Mayberry]
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17792
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1st-order PA is only interesting because of results which use 2nd-order PA
[Mayberry]
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17793
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It is only 2nd-order isomorphism which suggested first-order PA completeness
[Mayberry]
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17794
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Set theory is not just first-order ZF, because that is inadequate for mathematics
[Mayberry]
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17796
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There is a semi-categorical axiomatisation of set-theory
[Mayberry]
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17795
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Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation
[Mayberry]
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17801
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The set hierarchy doesn't rely on the dubious notion of 'generating' them
[Mayberry]
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17800
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The misnamed Axiom of Infinity says the natural numbers are finite in size
[Mayberry]
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17799
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Cantor's infinite is an absolute, of all the sets or all the ordinal numbers
[Mayberry]
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17797
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Cantor extended the finite (rather than 'taming the infinite')
[Mayberry]
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17802
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We don't translate mathematics into set theory, because it comes embodied in that way
[Mayberry]
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17804
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Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms
[Mayberry]
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17803
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Limitation of size is part of the very conception of a set
[Mayberry]
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17805
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Set theory is not just another axiomatised part of mathematics
[Mayberry]
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