display all the ideas for this combination of philosophers
6 ideas
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) |
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) |
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) |
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) |
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. |