1994 | What Required for Foundation for Maths? |
p.405-1 | p.405 | 17774 | Definitions make our intuitions mathematically useful |
p.405-1 | p.405 | 17776 | The ultimate principles and concepts of mathematics are presumed, or grasped directly |
p.405-1 | p.405 | 17775 | If proof and definition are central, then mathematics needs and possesses foundations |
p.405-2 | p.405 | 17777 | Foundations need concepts, definition rules, premises, and proof rules |
p.405-2 | p.405 | 17773 | Proof shows that it is true, but also why it must be true |
p.406-2 | p.406 | 17778 | Axiomatiation relies on isomorphic structures being essentially the same |
p.406-2 | p.406 | 17779 | 'Classificatory' axioms aim at revealing similarity in morphology of structures |
p.407-1 | p.407 | 17780 | 'Eliminatory' axioms get rid of traditional ideal and abstract objects |
p.407-2 | p.407 | 17782 | Greek quantities were concrete, and ratio and proportion were their science |
p.407-2 | p.407 | 17781 | Real numbers were invented, as objects, to simplify and generalise 'quantity' |
p.408-2 | p.408 | 17784 | Real numbers can be eliminated, by axiom systems for complete ordered fields |
p.408-2 | p.408 | 17785 | Real numbers as abstracted objects are now treated as complete ordered fields |
p.410-1 | p.410 | 17786 | The mainstream of modern logic sees it as a branch of mathematics |
p.410-2 | p.410 | 17787 | Big logic has one fixed domain, but standard logic has a domain for each interpretation |
p.411-2 | p.411 | 17788 | First-order logic only has its main theorems because it is so weak |
p.412-1 | p.412 | 17791 | Only second-order logic can capture mathematical structure up to isomorphism |
p.412-1 | p.412 | 17789 | No logic which can axiomatise arithmetic can be compact or complete |
p.412-1 | p.412 | 17790 | No Löwenheim-Skolem logic can axiomatise real analysis |
p.412-1 | p.412 | 17793 | It is only 2nd-order isomorphism which suggested first-order PA completeness |
p.412-1 | p.412 | 17792 | 1st-order PA is only interesting because of results which use 2nd-order PA |
p.412-2 | p.412 | 17794 | Set theory is not just first-order ZF, because that is inadequate for mathematics |
p.413-2 | p.413 | 17796 | There is a semi-categorical axiomatisation of set-theory |
p.413-2 | p.413 | 17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation |
p.414-2 | p.414 | 17800 | The misnamed Axiom of Infinity says the natural numbers are finite in size |
p.414-2 | p.414 | 17801 | The set hierarchy doesn't rely on the dubious notion of 'generating' them |
p.414-2 | p.414 | 17798 | Cantor presented the totality of natural numbers as finite, not infinite |
p.414-2 | p.414 | 17799 | Cantor's infinite is an absolute, of all the sets or all the ordinal numbers |
p.414-2 | p.414 | 17797 | Cantor extended the finite (rather than 'taming the infinite') |
p.415-1 | p.415 | 17802 | We don't translate mathematics into set theory, because it comes embodied in that way |
p.415-2 | p.415 | 17804 | Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms |
p.415-2 | p.415 | 17803 | Limitation of size is part of the very conception of a set |
p.416-1 | p.416 | 17805 | Set theory is not just another axiomatised part of mathematics |