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I've just read a chapter on axiomatic set theory, from Comprehensive Mathematics for Computer Scientists 1. It comes with basic notation on sets and some axioms:

  • Axiom 1 (Axiom of Empty Set)
  • Axiom 2 (Axiom of Equality)
  • Axiom 3 (Axiom of Union)
  • Axiom 4 (Axiom of Pairs)
  • Axiom 5 (Axiom of Subsets for Propositional Attributes)
  • Axiom 6 (Axiom of Powersets)
  • Axiom 7 (Axiom of Infinity)
  • Axiom 8 (Axiom of Choice)

I need to know what I should be able to do with this - in order to check if I've learned it decently.

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I'm not sure what your question is. –  Asaf Karagila Sep 4 '12 at 10:20
Many, many books and papers have been filled with the consequences of these axioms. Given the specific book you mentioned, just read on. You will see that way whether you understand them well enough for the purposes of the book. There are many excellent books on introductory set theory you can use to get a deeper understanding if you want so. For now, I think the question is not reasonably scoped and I'm going to vote to close as "too localized". –  Michael Greinecker Sep 4 '12 at 10:22
@MichaelGreinecker Suggest a better elaboration. –  Igäria Mnagarka Sep 4 '12 at 10:39
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closed as not a real question by Michael Greinecker, William, tomasz, rschwieb, J. M. Sep 19 '12 at 9:32

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

1 Answer

up vote 4 down vote accepted

One thing you could do is go through some mathematical arguments that are already known to you (preferably concerning sets and functions), and convince yourself that they can be formalized using the axioms you're given.

(Later: since functions and relations are only a later chapter, it is not clear to me that you're even supposed to be able to do anything interesting by yourself yet. That is some strange book you have gotten your hands on.)

It is possible that some of them can't, since you seem to be missing the Axiom of Replacement. However it is not needed for much of elementary mathematics.

Google found me a PDF of the book you refer to. It looks like you're also missing the Axiom of Extensionality (which the authors explicitly omit with a justification that makes no sense to me at all). You're also missing the Axiom of Foundation, which is not much of a problem because it has few real applications outside higher set theory.

In general, the entire idea of presenting an axiomatic set theory to a readership that the authors think will be helped by lots of cutesy drawings of bags-within-bags is either extraordinarily bold or raving mad. I tend towards "raving mad", especially because the chapter on "logic" that precedes it only treats propositional logic and doesn't even mention quantification.

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Can you suggest a better book on it? And, why strange? –  Igäria Mnagarka Sep 4 '12 at 16:51
@Gustavo: My first choice would be Introduction to Set Theory by Hrbacek & Jech; see the Amazon review by Michael Greinecker, who posts here. Judy Roitman’s Introduction to Modern Set Theory is pretty concise, but it has the great virtue of being available free in PDF form here. I’ve seen only a few pages of the book that you’re using, but on the basis of those pages I agree with Henning. In fact, almost any other set theory text would probably be better. –  Brian M. Scott Sep 4 '12 at 19:14
@GustavoBandeira I've read half of Just/Weese and I think it's very pleasant to read. –  Matt N. Sep 12 '12 at 7:49
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