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The power of mathematics relies on its logical representation and it depends on its strict disjunction between axiom and definition.

I'd learned lots of definitions while studying mathematics, but still however, axioms are always given only in the limited context, such as axioms postulated before learning set theory, and axioms postulated before learning analysis - not in a general context.

Is it my delusion to hope 'It would be better if there's a set of axioms that universally used in all mathematical branches.'?

I'd like to hear some advice on this issue.

snapper
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  • Most branches of math use axioms of logic and many use axioms of set theory. – Arthur Mar 06 '18 at 15:11
  • @Arthur You mean by 'axioms of set theory' is axioms postulated in ZFC? https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory – snapper Mar 06 '18 at 15:14
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    To build a textbook on a single axiom system is quite "unparctical". But you can see e.g.Ethan Bloch, The real numbers and real analysis, Springer (2011) for the construction of real analysis on top of the axioms for natural numbers, and you can find the proof of the said axioms from those of set theory in every set theory textbook. – Mauro ALLEGRANZA Mar 06 '18 at 15:18
  • @MauroALLEGRANZA Could you tell me slight more of your term "unpractical" in detail? Thx for your advice on the book – snapper Mar 06 '18 at 15:21
  • @delinco Those are the most common axioms, yes. But there are others, and most mathematicians who don't deal with set theory directly don't really care which axioms they're using, as long as they're allowed to use unions, power sets and so on. – Arthur Mar 06 '18 at 15:24
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    "Unpractical" because in a typical set-th textbook, the construction of naturals will be around page 50. In a book about the foundations of reals starting from naturals take about 100 page. Thus, when the student arrives at the initial def of continuity of a real function (around page 150) he is dead. – Mauro ALLEGRANZA Mar 06 '18 at 15:25
  • @Arthur I think with 8 or 9(including C in ZFC) axioms of sets, I can construct necessary definitions. But can't find any good summary of which axioms of logic widely accepted in mathematics. Could you give me some link of it? – snapper Mar 06 '18 at 15:26
  • @MauroALLEGRANZA You remind me of the freshman year.. thx. – snapper Mar 06 '18 at 15:27
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    You can see also Terence Tao, Analysis I, Springer (3rd ed 2016). – Mauro ALLEGRANZA Mar 06 '18 at 15:29

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