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Single Idea 17930
[filed under theme 5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
]
Full Idea
A set of axioms is said to be 'categorical' if all models of the axioms in question are isomorphic.
Gist of Idea
Axioms are 'categorical' if all of their models are isomorphic
Source
Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2)
Book Ref
Colyvan,Mark: 'An Introduction to the Philosophy of Mathematics' [CUP 2012], p.25
A Reaction
The best example is the Peano Axioms, which are 'true up to isomorphism'. Set theory axioms are only 'quasi-isomorphic'.
The
21 ideas
from Mark Colyvan
17922
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Reducing real numbers to rationals suggested arithmetic as the foundation of maths
[Colyvan]
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17923
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Intuitionists only accept a few safe infinities
[Colyvan]
|
17925
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Showing a disproof is impossible is not a proof, so don't eliminate double negation
[Colyvan]
|
17926
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Rejecting double negation elimination undermines reductio proofs
[Colyvan]
|
17924
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Excluded middle says P or not-P; bivalence says P is either true or false
[Colyvan]
|
17928
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Ordinal numbers represent order relations
[Colyvan]
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17929
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Löwenheim proved his result for a first-order sentence, and Skolem generalised it
[Colyvan]
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17930
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Axioms are 'categorical' if all of their models are isomorphic
[Colyvan]
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17931
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Structuralism say only 'up to isomorphism' matters because that is all there is to it
[Colyvan]
|
17932
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If 'in re' structures relies on the world, does the world contain rich enough structures?
[Colyvan]
|
17934
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Proof by cases (by 'exhaustion') is said to be unexplanatory
[Colyvan]
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17933
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Reductio proofs do not seem to be very explanatory
[Colyvan]
|
17935
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If inductive proofs hold because of the structure of natural numbers, they may explain theorems
[Colyvan]
|
17936
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Transfinite induction moves from all cases, up to the limit ordinal
[Colyvan]
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17937
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Mathematical generalisation is by extending a system, or by abstracting away from it
[Colyvan]
|
17938
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Mathematics can show why some surprising events have to occur
[Colyvan]
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17939
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Mathematics can reveal structural similarities in diverse systems
[Colyvan]
|
17940
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Most mathematical proofs are using set theory, but without saying so
[Colyvan]
|
17941
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Infinitesimals were sometimes zero, and sometimes close to zero
[Colyvan]
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17942
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Can a proof that no one understands (of the four-colour theorem) really be a proof?
[Colyvan]
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17943
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Probability supports Bayesianism better as degrees of belief than as ratios of frequencies
[Colyvan]
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