Combining Texts

Ideas for 'works', 'Knowledge and the Philosophy of Number' and 'Ontological Categories'

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4 ideas

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets
23623 Predicativism says only predicated sets exist [Hossack]
     Full Idea: Predicativists doubt the existence of sets with no predicative definition.
     From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 02.3)
     A reaction: This would imply that sets which encounter paradoxes when they try to be predicative do not therefore exist. Surely you can have a set of random objects which don't fall under a single predicate?
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
23624 The iterative conception has to appropriate Replacement, to justify the ordinals [Hossack]
     Full Idea: The iterative conception justifies Power Set, but cannot justify a satisfactory theory of von Neumann ordinals, so ZFC appropriates Replacement from NBG set theory.
     From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 09.9)
     A reaction: The modern approach to axioms, where we want to prove something so we just add an axiom that does the job.
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
23625 Limitation of Size justifies Replacement, but then has to appropriate Power Set [Hossack]
     Full Idea: The limitation of size conception of sets justifies the axiom of Replacement, but cannot justify Power Set, so NBG set theory appropriates the Power Set axiom from ZFC.
     From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 09.9)
     A reaction: Which suggests that the Power Set axiom is not as indispensable as it at first appears to be.
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.