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3 ideas
10537 | The ordered pairs <x,y> can be reduced to the class of sets of the form {{x},{x,y}} [Dummett] |
Full Idea: A classic reduction is the class of ordered pairs <x,y> being reduced to the class of sets of the form {{x},{x,y}}. | |
From: Michael Dummett (Frege Philosophy of Language (2nd ed) [1973], Ch.14) |
9879 | NF has no models, but just blocks the comprehension axiom, to avoid contradictions [Quine, by Dummett] |
Full Idea: Quine's New Foundations system of set theory, devised with no model in mind, but on the basis of a hunch that a purely formal restriction on the comprehension axiom would block all contradictions. | |
From: report of Willard Quine (New Foundations for Mathematical Logic [1937]) by Michael Dummett - Frege philosophy of mathematics Ch.18 | |
A reaction: The point is that Quine (who had an ontological preference for 'desert landscapes') attempted to do without an ontological commitment to objects (and their subsequent models), with a purely formal system. Quine's NF is not now highly regarded. |
10542 | To associate a cardinal with each set, we need the Axiom of Choice to find a representative [Dummett] |
Full Idea: We may suppose that with each set is associated an object as its cardinal number, but we have no systematic way, without appeal to the Axiom of Choice, of selecting a representative set of each cardinality. | |
From: Michael Dummett (Frege Philosophy of Language (2nd ed) [1973], Ch.14) |