more on this theme     |     more from this thinker

Single Idea 17795

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

Full Idea

Set theory cannot be an axiomatic theory, because the very notion of an axiomatic theory makes no sense without it.

Gist of Idea

Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation

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

This will come as a surprise to Penelope Maddy, who battles with ways to accept the set theory axioms as the foundation of mathematics. Mayberry says that the basic set theory required is much more simple and intuitive.

Related Idea

Idea 17796 There is a semi-categorical axiomatisation of set-theory [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]