Set Theory

(May 31)

Set theory starts with \(\in\), which is the only non-logical symbol. (The logical symbols are \(\neg,\wedge,\vee,\Rightarrow,\exists,\forall,(,),x,y,z,P,Q,R,\ldots\).)

Everything else, including all of mathematics, can be defined/encoded in terms of these symbols. How is that suppose to work?

For a first example we can define the subset relation from the elementship relation:

\(a \subseteq b \ \stackrel {\rm def} \Leftrightarrow \ \forall x\,.x\in a \Rightarrow x\in b\)

We can think of \(a\) as a property that the elements of \(b\) may, or may not, have.

For example

Proposition: The empty set is a subset of any set. In symbols \(\forall \)

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