display all the ideas for this combination of texts
6 ideas
18114 | There is no single agreed structure for set theory [Bostock] |
Full Idea: There is so far no agreed set of axioms for set theory which is categorical, i.e. which does pick just one structure. | |
From: David Bostock (Philosophy of Mathematics [2009], 6.4) | |
A reaction: This contrasts with Peano Arithmetic, which is categorical in its second-order version. |
18107 | A 'proper class' cannot be a member of anything [Bostock] |
Full Idea: A 'proper class' cannot be a member of anything, neither of a set nor of another proper class. | |
From: David Bostock (Philosophy of Mathematics [2009], 5.4) |
18115 | We could add axioms to make sets either as small or as large as possible [Bostock] |
Full Idea: We could add the axiom that all sets are constructible (V = L), making the universe of sets as small as possible, or add the axiom that there is a supercompact cardinal (SC), making the universe as large as we no know how to. | |
From: David Bostock (Philosophy of Mathematics [2009], 6.4) | |
A reaction: Bostock says most mathematicians reject the first option, and are undecided about the second option. |
18139 | The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock] |
Full Idea: The usual accounts of ZF are not restricted to subsets that we can describe, and that is what justifies the axiom of choice. | |
From: David Bostock (Philosophy of Mathematics [2009], 8.4 n36) | |
A reaction: This contrasts interestingly with predicativism, which says we can only discuss things which we can describe or define. Something like verificationism hovers in the background. |
18170 | The Axiom of Reducibility is self-effacing: if true, it isn't needed [Quine] |
Full Idea: The Axiom of Reducibility is self-effacing: if it is true, the ramification it is meant to cope with was pointless to begin with. | |
From: Willard Quine (Introduction to Russell's Theory of Types [1967], p.152), quoted by Penelope Maddy - Naturalism in Mathematics I.1 | |
A reaction: Maddy says the rejection of Reducibility collapsed the ramified theory of types into the simple theory. |
18105 | Replacement enforces a 'limitation of size' test for the existence of sets [Bostock] |
Full Idea: The Axiom of Replacement (or the Axiom of Subsets, 'Aussonderung', Fraenkel 1922) in effect enforces the idea that 'limitation of size' is a crucial factor when deciding whether a proposed set or does not not exist. | |
From: David Bostock (Philosophy of Mathematics [2009], 5.4) |