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Ideas of Oliver,A/Smiley,T, by Text

[British, fl. 2006, Both at the University of Cambridge.]

2006 What are Sets and What are they For?
Intro p.123 Cantor's sets were just collections, but Dedekind's were containers
     Full Idea: Cantor's definition of a set was a collection of its members into a whole, but within a few years Dedekind had the idea of a set as a container, enclosing its members like a sack.
     From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], Intro)
     A reaction: As the article goes on to show, these two view don't seem significantly different until you start to ask about the status of the null set and of singletons. I intuitively vote for Dedekind. Set theory is the study of brackets.
Intro p.124 If you only refer to objects one at a time, you need sets in order to refer to a plurality
     Full Idea: A 'singularist', who refers to objects one at a time, must resort to the language of sets in order to replace plural reference to members ('Henry VIII's wives') by singular reference to a set ('the set of Henry VIII's wives').
     From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], Intro)
     A reaction: A simple and illuminating point about the motivation for plural reference. Null sets and singletons give me the creeps, so I would personally prefer to avoid set theory when dealing with ontology.
Intro p.125 We can use plural language to refer to the set theory domain, to avoid calling it a 'set'
     Full Idea: Plurals earn their keep in set theory, to answer Skolem's remark that 'in order to treat of 'sets', we must begin with 'domains' that are constituted in a certain way'. We can speak in the plural of 'the objects', not a 'domain' of objects.
     From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], Intro)
     A reaction: [Skolem 1922:291 in van Heijenoort] Zermelo has said that the domain cannot be a set, because every set belongs to it.
1.2 p.127 The empty set is usually derived from Separation, but it also seems to need Infinity
     Full Idea: The empty set is usually derived via Zermelo's axiom of separation. But the axiom of separation is conditional: it requires the existence of a set in order to generate others as subsets of it. The original set has to come from the axiom of infinity.
     From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2)
     A reaction: They charge that this leads to circularity, as Infinity depends on the empty set.
1.2 p.129 The empty set is something, not nothing!
     Full Idea: Some authors need to be told loud and clear: if there is an empty set, it is something, not nothing.
     From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2)
     A reaction: I'm inclined to think of a null set as a pair of brackets, so maybe that puts it into a metalanguage.
1.2 p.130 We don't need the empty set to express non-existence, as there are other ways to do that
     Full Idea: The empty set is said to be useful to express non-existence, but saying 'there are no Us', or ∃xUx are no less concise, and certainly less roundabout.
     From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2)
1.2 p.131 Maybe we can treat the empty set symbol as just meaning an empty term
     Full Idea: Suppose we introduce Ω not as a term standing for a supposed empty set, but as a paradigm of an empty term, not standing for anything.
     From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 1.2)
     A reaction: This proposal, which they go on to explore, seems to mean that Ω (i.e. the traditional empty set symbol) is no longer part of set theory but is part of semantics.
2.2 p.135 The unit set may be needed to express intersections that leave a single member
     Full Idea: Thomason says with no unit sets we couldn't call {1,2}∩{2,3} a set - but so what? Why shouldn't the intersection be the number 2? However, we then have to distinguish three different cases of intersection (common subset or member, or disjoint).
     From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 2.2)
5.1 p.145 Logical truths are true no matter what exists - but predicate calculus insists that something exists
     Full Idea: Logical truths should be true no matter what exists, so true even if nothing exists. The classical predicate calculus, however, makes it logically true that something exists.
     From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.1)
5.1 p.146 If mathematics purely concerned mathematical objects, there would be no applied mathematics
     Full Idea: If mathematics was purely concerned with mathematical objects, there would be no room for applied mathematics.
     From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.1)
     A reaction: Love it! Of course, they are using 'objects' in the rather Fregean sense of genuine abstract entities. I don't see why fictionalism shouldn't allow maths to be wholly 'pure', although we have invented fictions which actually have application.
5.2 p.147 Sets might either represent the numbers, or be the numbers, or replace the numbers
     Full Idea: Identifying numbers with sets may mean one of three quite different things: 1) the sets represent the numbers, or ii) they are the numbers, or iii) they replace the numbers.
     From: Oliver,A/Smiley,T (What are Sets and What are they For? [2006], 5.2)
     A reaction: Option one sounds the most plausible to me. I will take numbers to be patterns embedded in nature, and sets are one way of presenting them in shorthand form, in order to bring out what is repeated.