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Single Idea 17796

[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets ]

Full Idea

We can give a semi-categorical axiomatisation of set-theory (all that remains undetermined is the size of the set of urelements and the length of the sequence of ordinals). The system is second-order in formalisation.

Gist of Idea

There is a semi-categorical axiomatisation of set-theory

Source

John Mayberry (What Required for Foundation for Maths? [1994], p.413-2)

Book Ref

'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.413

A Reaction

I gather this means the models may not be isomorphic to one another (because they differ in size), but can be shown to isomorphic to some third ingredient. I think. Mayberry says this shows there is no such thing as non-Cantorian set theory.

Related Idea

Idea 17795 Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry]

The 31 ideas from 'What Required for Foundation for Maths?'

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