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13530
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An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive [Wolf,RS]
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Full Idea:
Less theoretically, an ordinal is an equivalence class of well-orderings. Formally, we say a set is 'transitive' if every member of it is a subset of it, and an ordinal is a transitive set, all of whose members are transitive.
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From:
Robert S. Wolf (A Tour through Mathematical Logic [2005], 2.4)
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A reaction:
He glosses 'transitive' as 'every member of a member of it is a member of it'. So it's membership all the way down. This is the von Neumann rather than the Zermelo approach (which is based on singletons).
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13518
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Modern mathematics has unified all of its objects within set theory [Wolf,RS]
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Full Idea:
One of the great achievements of modern mathematics has been the unification of its many types of objects. It began with showing geometric objects numerically or algebraically, and culminated with set theory representing all the normal objects.
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From:
Robert S. Wolf (A Tour through Mathematical Logic [2005], Pref)
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A reaction:
His use of the word 'object' begs all sorts of questions, if you are arriving from the street, where an object is something which can cause a bruise - but get used to it, because the word 'object' has been borrowed for new uses.
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16150
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One is, so numbers exist, so endless numbers exist, and each one must partake of being [Plato]
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Full Idea:
If one is, there must also necessarily be number - Necessarily - But if there is number, there would be many, and an unlimited multitude of beings. ..So if all partakes of being, each part of number would also partake of it.
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From:
Plato (Parmenides [c.364 BCE], 144a)
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A reaction:
This seems to commit to numbers having being, then to too many numbers, and hence to too much being - but without backing down and wondering whether numbers had being after all. Aristotle disagreed.
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