I am beginning to read Set theory: an introduction by Vaught.
So far, I have already two questions:
- He writes:
Before we begin to study sets, a brief remark about logic will be made. The reader is probably already familiar with at least some of the following logical symbols or abbreviations:
[here is a list of these symbols including $\land, \lor, \exists, \iota, \forall, \exists ! \dots$]
In theoretical discussions about the language we use - for example in Chapters 6 and 9 - it is very useful to suppose, or pretend, that we write in a purely symbolic language. On the other hand, in actually doing mathematics it is desirable to write in pure English (or whatever ordinary language), not using any logical symbols (except = ).
Since I am a beginner, should I somehow try to convince me that all the usual statements formulated in mathematical English can be formalized with these symbols? How could I do that? Do I understand it correctly that, for example, the assertion "Let $m, n$ be two natural numbers. Then $A(m, n)$ and $B(m, n)$ are true." would be "$\forall m\forall n((m\in\mathbb N\land n\in\mathbb N)\to (A(m, n)\land B(m, n)))$" in the formal language?
- He also says:
Often we are in an extended discussion in which all sets A, B, C, etc., being considered are subsets of a particular set A'(a 'temporary universe'). For example, X might be the set of real numbers or some other 'space'.
Does "space" mean the same as "set" in this context? Checking wikipedia: "In mathematics, a space is a set (sometimes called a universe) with some added structure." This doesn't seem to help me. Why should a temporary universe have additional structure? I understand "space" to be a geometrical notion, but in this context "space" seems to mean something different.