display all the ideas for this combination of texts
20 ideas
| 18395 | Sets are mereological sums of the singletons of their members [Lewis, by Armstrong] |
| Full Idea: Lewis pointed out that many-membered classes are nothing more than the mereological wholes of the classes formed by taking the singleton of each member. | |
| From: report of David Lewis (Parts of Classes [1991]) by David M. Armstrong - Truth and Truthmakers 09.4 | |
| A reaction: You can't combine members to make the class, because the whole and the parts are of different type, but here the parts and whole are both sets, so they combine like waterdrops. |
| 15496 | We can build set theory on singletons: classes are then fusions of subclasses, membership is the singleton [Lewis] |
| Full Idea: The notion of a singleton, or unit set, can serve as the distinctive primitive of set theory. The rest is mereology: a class is the fusion of its singleton subclasses, something is a member of a class iff its singleton is part of that class. | |
| From: David Lewis (Parts of Classes [1991], Pref) | |
| A reaction: This is a gloriously bold proposal which I immediately like, because it cuts out the baffling empty set (which many people think 'exists'!), and gets mathematics back to being about the real world of entities (as the Greeks thought). |
| 15500 | Classes divide into subclasses in many ways, but into members in only one way [Lewis] |
| Full Idea: A class divides exhaustively into subclasses in many different ways; whereas a class divides exhaustively into members in only one way. | |
| From: David Lewis (Parts of Classes [1991], 1.2) |
| 15499 | A subclass of a subclass is itself a subclass; a member of a member is not in general a member [Lewis] |
| Full Idea: Just as a part of a part is itself a part, so a subclass of a subclass is itself a subclass; whereas a member of a member is not in general a member. | |
| From: David Lewis (Parts of Classes [1991], 1.2) | |
| A reaction: Lewis is showing the mereological character of sets, but this is a key distinction in basic set theory. When the members of members are themselves members, the set is said to be 'transitive'. |
| 15503 | We needn't accept this speck of nothingness, this black hole in the fabric of Reality! [Lewis] |
| Full Idea: Must we accept the null set as a most extraordinary individual, a little speck of sheer nothingness, a sort of black hole in the fabric of Reality itself? Not really. | |
| From: David Lewis (Parts of Classes [1991], 1.4) | |
| A reaction: We can only dream of reaching the level of confidence that Lewis reached, to make such beautiful fun of a highly counterintuitive idea that is rooted in the modern techniques of philosophy. |
| 15498 | We can accept the null set, but there is no null class of anything [Lewis] |
| Full Idea: There is no such class as the null class. I don't mind calling some memberless thing - some individual - the null 'set'. But that doesn't make it a memberless class. | |
| From: David Lewis (Parts of Classes [1991], 1.2) | |
| A reaction: The point is that set theory is a formal system which can do what it likes, but classes are classes 'of' things. Everyone assumes that sets are classes, reserving 'proper classes' for the tricky cases up at the far end. |
| 15502 | There are four main reasons for asserting that there is an empty set [Lewis] |
| Full Idea: The null set is a denotation of last resort for class-terms that fail to denote classes, an intersection of x and y where they have no members in common, the class of all self-members, and the real numbers such that x^2+1=0. This is all mere convenience. | |
| From: David Lewis (Parts of Classes [1991], 1.4) | |
| A reaction: A helpful catalogue of main motivations for the existence of the null set in set theory. Lewis aims to undermine these reasons, and dispense with the wretched thing. |
| 15506 | If we don't understand the singleton, then we don't understand classes [Lewis] |
| Full Idea: Our utter ignorance about the nature of the singletons amounts to sheer ignorance about the nature of classes generally. | |
| From: David Lewis (Parts of Classes [1991], 2.1) |
| 15497 | We can replace the membership relation with the member-singleton relation (plus mereology) [Lewis] |
| Full Idea: Given the theory of part and whole, the member-singleton relation may replace membership generally as the primitive notion of set theory. | |
| From: David Lewis (Parts of Classes [1991], Pref) | |
| A reaction: An obvious question is to ask what the member-singleton relation is if it isn't membership. |
| 15511 | If singleton membership is external, why is an object a member of one rather than another? [Lewis] |
| Full Idea: Suppose the relation of member to singleton is external. Why must Possum be a member of one singleton rather than another? Why isn't it contingent which singleton is his? | |
| From: David Lewis (Parts of Classes [1991], 2.2) | |
| A reaction: He cites Van Inwagen for raising this question, and answers it in terms of counterparts. So is the relation internal or external? I think of sets as pairs of curly brackets, not existing entities, so the question doesn't bother me. |
| 15513 | Maybe singletons have a structure, of a thing and a lasso? [Lewis] |
| Full Idea: Maybe the singleton of something x is not an atom, but consists of x plus a lasso. That gives a singleton an internal structure. ...But what do we know of the nature of the lasso, or how it fits? We are no better off. | |
| From: David Lewis (Parts of Classes [1991], 2.5) | |
| A reaction: [second bit on p.45] |
| 15507 | Set theory has some unofficial axioms, generalisations about how to understand it [Lewis] |
| Full Idea: Set theory has its unofficial axioms, traditional remarks about the nature of classes. They are never argued, but are passed heedlessly from one author to another. One of these says that the classes are nowhere: they are outside space and time. | |
| From: David Lewis (Parts of Classes [1991], 2.1) | |
| A reaction: Why don't the people who write formal books on set theory ever say things like this? |
| 10191 | Set theory reduces to a mereological theory with singletons as the only atoms [Lewis, by MacBride] |
| Full Idea: Lewis has shown that set theory may be reduced to a mereological theory in which singletons are the only atoms. | |
| From: report of David Lewis (Parts of Classes [1991]) by Fraser MacBride - Review of Chihara's 'Structural Acc of Maths' p.80 | |
| A reaction: Presumably the axiom of extensionality, that a set is no more than its members, translates into unrestricted composition, that any parts will make an object. Difficult territory, but I suspect that this is of great importance in metaphysics. |
| 15508 | If singletons are where their members are, then so are all sets [Lewis] |
| Full Idea: If every singleton was where its member was, then, in general, classes would be where there members were. | |
| From: David Lewis (Parts of Classes [1991], 2.1) | |
| A reaction: There seems to be a big dislocation of understanding of the nature of sets, between 'pure' set theory, and set theory with ur-elements. I take the pure to be just an 'abstraction' from the more located one. The empty set has a puzzling location. |
| 15514 | A huge part of Reality is only accepted as existing if you have accepted set theory [Lewis] |
| Full Idea: The preponderant part of Reality must consist of unfamiliar, unobserved things, whose existence would have gone unsuspected but for our acceptance of set theory. | |
| From: David Lewis (Parts of Classes [1991], 2.6) | |
| A reaction: He is referring to the enormous sets at the far end of set theory, of a size that had never been hitherto conceived. Excellent. Daft to believe in something entirely because you have accepted set theory, with no other basis. |
| 15523 | Set theory isn't innocent; it generates infinities from a single thing; but mathematics needs it [Lewis] |
| Full Idea: Set theory is not innocent. Its trouble is that when we have one thing, then somehow we have another wholly distinct thing, the singleton. And another, and another....ad infinitum. But that's the price for mathematical power. Pay it. | |
| From: David Lewis (Parts of Classes [1991], 3.6) |
| 23623 | Predicativism says only predicated sets exist [Hossack] |
| Full Idea: Predicativists doubt the existence of sets with no predicative definition. | |
| From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 02.3) | |
| A reaction: This would imply that sets which encounter paradoxes when they try to be predicative do not therefore exist. Surely you can have a set of random objects which don't fall under a single predicate? |
| 23624 | The iterative conception has to appropriate Replacement, to justify the ordinals [Hossack] |
| Full Idea: The iterative conception justifies Power Set, but cannot justify a satisfactory theory of von Neumann ordinals, so ZFC appropriates Replacement from NBG set theory. | |
| From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 09.9) | |
| A reaction: The modern approach to axioms, where we want to prove something so we just add an axiom that does the job. |
| 23625 | Limitation of Size justifies Replacement, but then has to appropriate Power Set [Hossack] |
| Full Idea: The limitation of size conception of sets justifies the axiom of Replacement, but cannot justify Power Set, so NBG set theory appropriates the Power Set axiom from ZFC. | |
| From: Keith Hossack (Knowledge and the Philosophy of Number [2020], 09.9) | |
| A reaction: Which suggests that the Power Set axiom is not as indispensable as it at first appears to be. |
| 6548 | Physicalism requires the naturalisation or rejection of set theory [Lycan] |
| Full Idea: Eventually set theory will have to be either naturalised or rejected, if a thoroughgoing physicalism is to be maintained. | |
| From: William Lycan (Consciousness [1987], 8.4) | |
| A reaction: Personally I regard Platonism as a form of naturalism (though a rather bold and dramatic one). The central issue seems to be the ability of the human main/brain to form 'abstract' notions about the physical world in which it lives. |