{"id":421,"date":"2007-05-16T19:02:10","date_gmt":"2007-05-16T19:02:10","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2007\/05\/16\/set-theory-some-basic-definitions\/"},"modified":"2007-05-16T19:02:10","modified_gmt":"2007-05-16T19:02:10","slug":"set-theory-some-basic-definitions","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2007\/05\/16\/set-theory-some-basic-definitions\/","title":{"rendered":"Set Theory – some basic definitions"},"content":{"rendered":"

So, what’s set theory really about?<\/p>\n

We’ll start off, for intuition’s sake, by talking a little bit about what’s now called naive set theory<\/em>, before moving into the formality of axiomatic set theory<\/em>. Most of this post might be a bit boring for a lot of you, but it’s worth
\nbeing a bit on the pedantic side to make sure that we’re starting from a clear basis.<\/p>\n

A set<\/em> is a collection of things. What it means to be a member of a set S is
\nthat there’s some predicate PS<\/sub> – that is, some way of describing things via logic – which is true only<\/em> for members of S. To be a tad more formal, that means that for any possible object x, PS<\/sub>(x) is true if and only if
\nx is a member of S. In general, I’ll just write S for both the set and the predicate that defines the set, unless there’s some reason that that would be confusing and\/or ambiguous. (Another way of saying that is that a set S is a collection of things that all share
\nsome property<\/em>, which is the defining property of the set. When you work through
\nthe formality of what a property means, that’s just another way of saying that there’s a
\npredicate.)<\/p>\n

<\/p>\n

Set theory – as you can see even from the first definition above – is incredibly closely intertwined with first order predicate logic (FOPL). In general, the two can form a nicely closed formal system: sets provide a semantic basis for the logic; and the logic provides a set of tools for talking about sets. That’s a big part of why set theory makes such a good basis for mathematics: it’s one of the simplest things that we can use to create a semantically meaningful complete logic.<\/p>\n

So I’m going to run through a quick reminder of the basic notations and concepts of FOPL; for more details, you can look at some posts about FOPL itself here<\/a>.<\/p>\n

In first order predicate logic, we talk about two kinds of things: predicates<\/em>, and objects<\/em>. Objects are the things that we can reason about using the logic; predicates are the things that we use<\/em> to reason about objects. <\/p>\n

A predicate<\/em> is a statement that says something about some object or objects. We’ll write predicates as either uppercase letters (“A”, “B”), or words starting with an uppercase letter (“Married”), and we’ll write objects in quotes. Every predicate is followed by a list of comma-separated objects (or variables representing objects). So, for a few examples:<\/p>\n