Category: Set Theory
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Sets, Pluralities, and the Axiom of Pair
In a thread from the old blog, resident nominalist gadfly 'Ockham'/'William' made the fascinating double-barreled claim that: . . . (a) there are such things as sets and (b) the axiom of pairs is false. Briefly, I claim that 'a set of x's' is just another way of saying 'those x's'. The fundamental error of set…
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A Cantorian Argument Why Possible Worlds Cannot be Maximally Consistent Sets of Propositions
A commenter in the 'Nothing' thread spoke of possible worlds as sets. What follows is a reposting from 1 March 2009 which opposes that notion. ……………. In a recent comment, Peter Lupu bids us construe possible worlds as maximally consistent sets of propositions. If this is right, then the actual world, which is of course one…
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On the Abstractness of Mathematical Sets
Let us agree that x is concrete iff x is causally/active passive and abstract otherwise. Many say that mathematical sets ('sets' hereafter: 'mathematical' as opposed to 'commonsense') are abstract objects, abstract entities, abstracta. Why? Argument One: In set theory there are singleton sets, e.g. {Quine}. Obviously, Quine is not identical to {Quine}. The second is…
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Another Example of a Necessary Being Depending for its Existence on a Necessary Being
The Father and the Son are both necessary beings. And yet the Father 'begets' the Son. Part, though not the whole, of the notion of begetting here must be this: if x begets y, then y depends for its existence on x. If that were not part of the meaning of 'begets'' in this context, I…
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Trinity and Set Theory
Let S and T be mathematical sets. Now consider the following two propositions: 1. S is a proper subset of T. 2. S and T have the same number of elements. Are (1) and (2) consistent? That is, can they both be true? If yes, explain how. If you think (1) and (2) are consistent,…
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A Cantorian Argument Why Possible Worlds Cannot be Maximally Consistent Sets of Propositions
In a recent comment, Peter Lupu bids us construe possible worlds as maximally consistent sets of propositions. If this is right, then the actual world, which is of course one of the possible worlds, is the maximally consistent set of true propositions. But Cantor's Theorem implies that there cannot be a set of all true propositions. Therefore,…
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Transitive Sets and the Distinctness of Sets From Their Members
Vlastimil asked for examples of transitive sets. A transitive set is a set every element of which is a subset of it. (Hrbacek and Jech, Introduction to Set Theory, p. 50) There is no lack of examples. The null set vacuously satisfies the condition 'if x is an element of S, then x is a subset of…
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On the Elusive Notion of a Set: Sets as Products of Collectings
In an important article, Max Black writes: Beginners are taught that a set having three members is a single thing, wholly constituted by its members but distinct from them. After this, the theological doctrine of the Trinity as "three in one" should be child's play. ("The Elusiveness of Sets," Review of Metaphysics, June 1971, p.…
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Non-Empty Thoughts About the Empty Set
1. The empty or null set is a strange animal. It is a set, but it has no members. This is of course not a contingent fact about it, but one bound up with its very identity: the null set is essentially null. Intuitively, however, one might have thought that a set is a group…