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Ideas for 'works', 'Naturalism in Mathematics' and 'Symbolic Reasoning'

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7 ideas

4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
18194 'Forcing' can produce new models of ZFC from old models [Maddy]
     Full Idea: Cohen's method of 'forcing' produces a new model of ZFC from an old model by appending a carefully chosen 'generic' set.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.4)
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
21548 The null class is the class with all the non-existents as its members [MacColl, by Lackey]
     Full Idea: In 1905 the Scottish logician Hugh MacColl published a paper in which he argued that the null class in logic should be taken as the class with all the non-existents as its members.
     From: report of Hugh MacColl (Symbolic Reasoning [1905]) by Douglas Lackey - Intros to Russell's 'Essays in Analysis' p.95
     A reaction: For the null object (zero) Frege just chose one sample concept with an empty extension. MacColl's set seems to have a lot of members, given that it is 'null'. How many, I wonder? Russell responded to this paper.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
18195 A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy [Maddy]
     Full Idea: A possible axiom is the Large Cardinal Axiom, which asserts that there are more and more stages in the cumulative hierarchy. Infinity can be seen as the first of these stages, and Replacement pushes further in this direction.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.5)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
18191 Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy]
     Full Idea: The axiom of infinity: that there are infinite sets is to claim that completed infinite collections can be treated mathematically. In its standard contemporary form, the axioms assert the existence of the set of all finite ordinals.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.3)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
18193 The Axiom of Foundation says every set exists at a level in the set hierarchy [Maddy]
     Full Idea: In the presence of other axioms, the Axiom of Foundation is equivalent to the claim that every set is a member of some Vα.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.3)
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
18169 Axiom of Reducibility: propositional functions are extensionally predicative [Maddy]
     Full Idea: The Axiom of Reducibility states that every propositional function is extensionally equivalent to some predicative proposition function.
     From: Penelope Maddy (Naturalism in Mathematics [1997], I.1)
4. Formal Logic / G. Formal Mereology / 1. Mereology
13282 Aristotle relativises the notion of wholeness to different measures [Aristotle, by Koslicki]
     Full Idea: Aristotle proposes to relativise unity and plurality, so that a single object can be both one (indivisible) and many (divisible) simultaneously, without contradiction, relative to different measures. Wholeness has degrees, with the strength of the unity.
     From: report of Aristotle (works [c.330 BCE]) by Kathrin Koslicki - The Structure of Objects 7.2.12
     A reaction: [see Koslicki's account of Aristotle for details] As always, the Aristotelian approach looks by far the most promising. Simplistic mechanical accounts of how parts make wholes aren't going to work. We must include the conventional and conceptual bit.