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Single Idea 10044

[filed under theme 4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory ]

Full Idea

Russell adduces two reasons against the extensional view of classes, namely the existence of the null class (which cannot very well be a collection), and the unit classes (which would have to be identical with their single elements).

Gist of Idea

Russell denies extensional sets, because the null can't be a collection, and the singleton is just its element

Source

report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Stewart Shapiro - Structure and Ontology p.459

Book Ref

'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.459

A Reaction

Gödel believes in the reality of classes. I have great sympathy with Russell, when people start to claim that sets are not just conveniences to help us think about things, but actual abstract entities. Is the singleton of my pencil is on this table?

The 23 ideas with the same theme [objections to the whole idea of set theory]:

8745 Classes are logical fictions, and are not part of the ultimate furniture of the world [Russell]
6436 I gradually replaced classes with properties, and they ended as a symbolic convenience [Russell]
10044 Russell denies extensional sets, because the null can't be a collection, and the singleton is just its element [Russell/Whitehead, by Shapiro]
18208 We regard classes as mere symbolic or linguistic conveniences [Russell/Whitehead]
11064 Classes can be reduced to propositional functions [Russell, by Hanna]
7548 Classes, grouped by a convenient property, are logical constructions [Russell]
13536 Skolem did not believe in the existence of uncountable sets [Skolem]
10304 Very few things in set theory remain valid in intuitionist mathematics [Bernays]
3355 Von Neumann wanted mathematical functions to replace sets [Neumann, by Benardete,JA]
9920 Two objects can apparently make up quite distinct arrangements in sets [Goodman, by Burgess/Rosen]
3336 Two things can never entail three things [Quine, by Benardete,JA]
10699 Does a bowl of Cheerios contain all its sets and subsets? [Boolos]
8972 What in the real world could ground the distinction between the sets {A,{A,B}} and {B,{A,B}}? [Inwagen]
9570 In Field's Platonist view, set theory is false because it asserts existence for non-existent things [Field,H, by Chihara]
6548 Physicalism requires the naturalisation or rejection of set theory [Lycan]
13481 Maybe sets should be rethought in terms of the even more basic categories [Hart,WD]
7035 God does not create the world, and then add the classes [Heil]
10207 Anti-realists reject set theory [Shapiro]
8758 We could talk of open sentences, instead of sets [Chihara, by Shapiro]
18151 Could we replace sets by the open sentences that define them? [Chihara, by Bostock]
9563 A pack of wolves doesn't cease when one member dies [Chihara]
10108 As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman]
10687 Maybe we reduce sets to ordinals, rather than the other way round [Hossack]