Combining Philosophers

Ideas for H.Putnam/P.Oppenheim, John Mayberry and Gregory Currie

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5 ideas

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets

17795	Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry]
	Full Idea: Set theory cannot be an axiomatic theory, because the very notion of an axiomatic theory makes no sense without it.
	From: John Mayberry (What Required for Foundation for Maths? [1994], p.413-2)
	A reaction: This will come as a surprise to Penelope Maddy, who battles with ways to accept the set theory axioms as the foundation of mathematics. Mayberry says that the basic set theory required is much more simple and intuitive.

17796	There is a semi-categorical axiomatisation of set-theory [Mayberry]
	Full Idea: We can give a semi-categorical axiomatisation of set-theory (all that remains undetermined is the size of the set of urelements and the length of the sequence of ordinals). The system is second-order in formalisation.
	From: John Mayberry (What Required for Foundation for Maths? [1994], p.413-2)
	A reaction: I gather this means the models may not be isomorphic to one another (because they differ in size), but can be shown to isomorphic to some third ingredient. I think. Mayberry says this shows there is no such thing as non-Cantorian set theory.

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]
	Full Idea: The (misnamed!) Axiom of Infinity expresses Cantor's fundamental assumption that the species of natural numbers is finite in size.
	From: John Mayberry (What Required for Foundation for Maths? [1994], p.414-2)

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]
	Full Idea: The idea of 'generating' sets is only a metaphor - the existence of the hierarchy is established without appealing to such dubious notions.
	From: John Mayberry (What Required for Foundation for Maths? [1994], p.414-2)
	A reaction: Presumably there can be a 'dependence' or 'determination' relation which does not involve actual generation.

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]
	Full Idea: Our very notion of a set is that of an extensional plurality limited in size.
	From: John Mayberry (What Required for Foundation for Maths? [1994], p.415-2)