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Ideas of John Mayberry, by Text
[American, fl. 1994, Professor of Mathematics at the University of the Pacific, California.]
1994

What Required for Foundation for Maths?

p.4051

p.405

17774

Definitions make our intuitions mathematically useful

p.4051

p.405

17775

If proof and definition are central, then mathematics needs and possesses foundations

p.4051

p.405

17776

The ultimate principles and concepts of mathematics are presumed, or grasped directly

p.4052

p.405

17777

Foundations need concepts, definition rules, premises, and proof rules

p.4052

p.405

17773

Proof shows that it is true, but also why it must be true

p.4062

p.406

17778

Axiomatiation relies on isomorphic structures being essentially the same

p.4062

p.406

17779

'Classificatory' axioms aim at revealing similarity in morphology of structures

p.4071

p.407

17780

'Eliminatory' axioms get rid of traditional ideal and abstract objects

p.4072

p.407

17782

Greek quantities were concrete, and ratio and proportion were their science

p.4072

p.407

17781

Real numbers were invented, as objects, to simplify and generalise 'quantity'

p.4082

p.408

17784

Real numbers can be eliminated, by axiom systems for complete ordered fields

p.4082

p.408

17785

Real numbers as abstracted objects are now treated as complete ordered fields

p.4101

p.410

17786

The mainstream of modern logic sees it as a branch of mathematics

p.4102

p.410

17787

Big logic has one fixed domain, but standard logic has a domain for each interpretation

p.4112

p.411

17788

Firstorder logic only has its main theorems because it is so weak

p.4121

p.412

17791

Only secondorder logic can capture mathematical structure up to isomorphism

p.4121

p.412

17790

No LöwenheimSkolem logic can axiomatise real analysis

p.4121

p.412

17789

No logic which can axiomatise arithmetic can be compact or complete

p.4121

p.412

17792

1storder PA is only interesting because of results which use 2ndorder PA

p.4121

p.412

17793

It is only 2ndorder isomorphism which suggested firstorder PA completeness

p.4122

p.412

17794

Set theory is not just firstorder ZF, because that is inadequate for mathematics

p.4132

p.413

17796

There is a semicategorical axiomatisation of settheory

p.4132

p.413

17795

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

p.4142

p.414

17800

The misnamed Axiom of Infinity says the natural numbers are finite in size

p.4142

p.414

17801

The set hierarchy doesn't rely on the dubious notion of 'generating' them

p.4142

p.414

17799

Cantor's infinite is an absolute, of all the sets or all the ordinal numbers

p.4142

p.414

17797

Cantor extended the finite (rather than 'taming the infinite')

p.4151

p.415

17802

We don't translate mathematics into set theory, because it comes embodied in that way

p.4152

p.415

17804

Axiom theories can't give foundations for mathematics  that's using axioms to explain axioms

p.4152

p.415

17803

Limitation of size is part of the very conception of a set

p.4161

p.416

17805

Set theory is not just another axiomatised part of mathematics
