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Single Idea 17805

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory ]

Full Idea

The fons et origo of all confusion is the view that set theory is just another axiomatic theory and the universe of sets just another mathematical structure. ...The universe of sets ...is the world that all mathematical structures inhabit.

Clarification

'Fons et origo' is the original source

Gist of Idea

Set theory is not just another axiomatised part of mathematics

Source

John Mayberry (What Required for Foundation for Maths? [1994], p.416-1)

Book Ref

'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.416

Related Idea

Idea 17802 We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry]

The 31 ideas from John Mayberry

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]