Combining Texts

Ideas for 'The Evolution of Logic', 'Preface to 'Principles of Philosophy'' and 'Review of Frege's 'Grundlagen''

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6 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
13463 There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD]
     Full Idea: Since we can map the transfinite ordinals one-one into the infinite cardinals, there are at least as many infinite cardinals as transfinite ordinals.
     From: William D. Hart (The Evolution of Logic [2010], 1)
13492 Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD]
     Full Idea: It is easier to generalize von Neumann's finite ordinals into the transfinite. All Zermelo's nonzero finite ordinals are singletons, but if ω were a singleton it is hard to see how if could fail to be the successor of its member and so not a limit.
     From: William D. Hart (The Evolution of Logic [2010], 3)
13459 The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD]
     Full Idea: We can show (using the axiom of choice) that the less-than relation, <, well-orders the ordinals, ...and that it partially orders the ordinals, ...and that it totally orders the ordinals.
     From: William D. Hart (The Evolution of Logic [2010], 1)
13491 The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD]
     Full Idea: The axiom of infinity with separation yields a least limit ordinal, which is called ω.
     From: William D. Hart (The Evolution of Logic [2010], 3)
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
13446 19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD]
     Full Idea: The real numbers were not isolated from geometry until the arithmetization of analysis during the nineteenth century.
     From: William D. Hart (The Evolution of Logic [2010], 1)
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
13509 We can establish truths about infinite numbers by means of induction [Hart,WD]
     Full Idea: Mathematical induction is a way to establish truths about the infinity of natural numbers by a finite proof.
     From: William D. Hart (The Evolution of Logic [2010], 5)
     A reaction: If there are truths about infinities, it is very tempting to infer that the infinities must therefore 'exist'. A nice, and large, question in philosophy is whether there can be truths without corresponding implications of existence.