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All the ideas for 'fragments/reports', 'works' and 'What Required for Foundation for Maths?'

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42 ideas

1. Philosophy / D. Nature of Philosophy / 3. Philosophy Defined
2666 Carneades' pinnacles of philosophy are the basis of knowledge (the criterion of truth) and the end of appetite (good) [Carneades, by Cicero]
2. Reason / D. Definition / 2. Aims of Definition
17774 Definitions make our intuitions mathematically useful [Mayberry]
2. Reason / E. Argument / 6. Conclusive Proof
17773 Proof shows that it is true, but also why it must be true [Mayberry]
3. Truth / B. Truthmakers / 10. Making Future Truths
21390 Future events are true if one day we will say 'this event is happening now' [Carneades]
21672 We say future things are true that will possess actuality at some following time [Carneades, by Cicero]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
17796 There is a semi-categorical axiomatisation of set-theory [Mayberry]
17795 Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
17800 The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
17801 The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
17803 Limitation of size is part of the very conception of a set [Mayberry]
5. Theory of Logic / A. Overview of Logic / 2. History of Logic
17786 The mainstream of modern logic sees it as a branch of mathematics [Mayberry]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
17788 First-order logic only has its main theorems because it is so weak [Mayberry]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
17791 Only second-order logic can capture mathematical structure up to isomorphism [Mayberry]
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
17787 Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
17790 No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
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]
5. Theory of Logic / K. Features of Logics / 6. Compactness
17789 No logic which can axiomatise arithmetic can be compact or complete [Mayberry]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
17784 Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity
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]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
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]
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
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]
17804 Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
17792 1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
17793 It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
17794 Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry]
17802 We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry]
17805 Set theory is not just another axiomatised part of mathematics [Mayberry]
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
17785 Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry]
9. Objects / F. Identity among Objects / 6. Identity between Objects
15825 Carneades denied the transitivity of identity [Carneades, by Chisholm]
10. Modality / A. Necessity / 3. Types of Necessity
21389 Carneades distinguished logical from causal necessity, when talking of future events [Long on Carneades]
16. Persons / F. Free Will / 2. Sources of Free Will
21671 Voluntary motion is intrinsically within our power, and this power is its cause [Carneades, by Cicero]
16. Persons / F. Free Will / 6. Determinism / a. Determinism
21391 Some actions are within our power; determinism needs prior causes for everything - so it is false [Carneades, by Cicero]
16. Persons / F. Free Will / 6. Determinism / b. Fate
21674 Even Apollo can only foretell the future when it is naturally necessary [Carneades, by Cicero]
22. Metaethics / B. Value / 2. Values / i. Self-interest
7398 Carneades said that after a shipwreck a wise man would seize the only plank by force [Carneades, by Tuck]
25. Social Practice / D. Justice / 1. Basis of justice
21392 People change laws for advantage; either there is no justice, or it is a form of self-injury [Carneades, by Lactantius]
29. Religion / D. Religious Issues / 3. Problem of Evil / a. Problem of Evil
20698 Irenaeus says evil is necessary for perfect human development [Irenaeus, by Davies,B]