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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 'Introduction to the Philosophy of Mathematics'

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