2006 | What are Sets and What are they For? |
Intro | p.123 | 14199 | 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 | 14234 | 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 | 14237 | 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 | 14239 | 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 | 14240 | 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 | 14241 | 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 | 14242 | 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 | 14243 | 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 | 14245 | 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 | 14246 | 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 | 14247 | 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. |