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I am beginning to read Set theory: an introduction by Vaught.

So far, I have already two questions:

  1. 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?

  1. 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.

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  • $\begingroup$ For (1) the author is encouraging the readers to explain things in English... for the example you give, make a distinction between $\forall$ and $\exists$. $\forall x \in N$ means for all natural numbers while $\exists x \in N$ means there exists a natural number 2) the wikipedia definition suffices... there are sets which really lack any "structure", you can construct many trivial examples... as your maths knowledge grows, you will understand what "structure" means $\endgroup$ – felasfa Oct 9 '16 at 23:30
  • $\begingroup$ In this context, "space" means "environment" where the sets we are interested in live. Due to the peculiarity of set theory (see : Russell's Paradox, Universal set) it is not always the case that these "environments" are sets in the "technical" sense. $\endgroup$ – Mauro ALLEGRANZA Oct 10 '16 at 11:36

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