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Calculus as used normally uses infinitesimal quantity and nonstandard (infinitesimal) analysis to define limits, derivatives, integrations and so on.

So, is there any attempt to define calculus in term of ZFC or set theory?

Edit: So, using AC, epsilon-delta continuity/definition can be defined in term of set theory - but the problem is, let us say that there is some number 1.234 and 1.235. Between them, there would be uncountable number of real numbers - in order for epsilon-delta coninuity-and-set-theory equivalence to work, we have to expand the space to the sequence of $2^{\aleph_0} numbers (not real line, as real line cannot be well-ordered respecting its original order). Is this correct?

Edit: Simplifying the question: What is the way to express calculus in the standard model of ZFC?

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closed as not a real question by Andres Caicedo, Noah Snyder, Norbert, Jason DeVito, Thomas Oct 10 '12 at 15:17

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Limits aren't defined that way, you only need real numbers. –  Olivier Bégassat Oct 10 '12 at 2:26
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Calculus used normally does not use infinitesimals. That's why non-standard analysis is called "non-standard". –  EuYu Oct 10 '12 at 2:26
    
In response to your newest edit (What is the way to express calculus in the standard model of ZFC?), there is no problem here at all. All of the usual notions in calculus can be formulated in the usual model of ZFC and everything works just fine. –  KReiser Oct 10 '12 at 4:32

3 Answers 3

up vote 2 down vote accepted

Even if you want to use infinitesimals, you can use ZFC to construct such creatures, it is just not the usual way of doing things.

  • In ZFC start with some non-standard model of number theory. Note that in ZFC, one will be able to distinguish the standard part of such a model from the non-standard part.
  • Construct a non-standard model of the reals (hyperreals), in a manner similar to Hurkyl's answer.
  • In the hyperreals, you will have infinitesimals (the reciprocals of the non-standard natural numbers).
  • Develop calculus from here as in Keisler's Elementary Calculus: An Infinitesimal Approach
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So, no standard approach? –  W12 Oct 10 '12 at 3:45
    
@w12: I'm not certain if I understand your question. –  Arthur Fischer Oct 10 '12 at 3:53
    
This is the answer that I wanted - but I am just asking if there is a way to develop calculus from the standard model of ZFC. –  W12 Oct 10 '12 at 4:00
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@W12: What I wrote is all formalisable within ZFC, so it takes place in whatever model of set theory you want. That you begin the construction with non-standard natural numbers doesn't mean that the universe of sets is non-standard. It just means that instead of taking the construction to begin with, say, the family $\omega$ of finite ordinals (which satisfy the Peano Axioms), you start with some different set (which also satisfies the Peano Axioms, but which contains elements that cannot be reached from "zero" by finitely many applications of the successor operation.) (cont...) –  Arthur Fischer Oct 10 '12 at 4:23
    
(...inued) But what I wrote is an approach to non-standard analysis from set theory. As the common modern approach to analysis/calculus is via limits ($\forall \epsilon > 0 \exists \delta > 0 \ldots$), all you need is some construction of the (standard) real numbers, and these are more easily seen as done within the framework of ZFC. –  Arthur Fischer Oct 10 '12 at 4:23

Early on, calculus was interpreted (by many) in the context of infinitesimals. It was, however, pointed out by many that this led to problems. This forced the mathematical community to hammer down and clarify the mechanisms of calculus precisely so that we could avoid any reference to infinitesimal quantities. Intuitively, that's what it looks like, but looking at the definitions, that isn't at all what we're doing.

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This would have been a good answer before ca. 1960, when NSA came around. –  Ben Crowell Oct 10 '12 at 5:11
    
@BenCrowell: You'll (I hope) forgive me if I don't have my History of Mathematics text in front of me. What has the NSA to do with this? –  Cameron Buie Oct 10 '12 at 7:37
    
Non-standard analysis has so far had an extremely limited impact in calculus education. Apart from people who explicitly study NSA, mathematicians nearly universally treat calculus in the standard $\epsilon/\delta$ way. –  Carl Mummert Oct 10 '12 at 13:55
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Ah! I thought you were talking about the National Security Agency. XD –  Cameron Buie Oct 10 '12 at 17:11

So, is there any attempt to define calculus in term of ZFC or set theory?

Yes:

  • Justify naive set theory using ZFC
  • Construct the natural numbers out of ZFC-sets
  • Construct the real numbers out of ZFC-sets
  • Develop calculus in the usual fashion using natural/real numbers and naive set theory.

I'm guessing, though, the question you meant to ask is "what is the usual fashion?"

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How can you justify naive set thoery using ZFC? Isn't ZFC born due to limitation of naive set theory? Or you are saying that naive set theory works for calculus....? –  W12 Oct 10 '12 at 3:06
    
I was just trying to point out you only need a little tiny fragment of set theory for calculus; if you think of "naive set theory" referring to a specific theory, then pretend I said something else. :) –  Hurkyl Oct 10 '12 at 3:29

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