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If the infinitesimal gives an unlimited precision as 1/∞ --> 0 Which can be thought of as the decimal 0.000000.....00000... then

Why can't the reals, which demands, simply, unlimited precision (this is probably where I'm wrong), be thought of as the infinite set of every possible sum of infinitesimals, where each point along the number line is defined as greater than the next?

If there are an infinite number of rational numbers between 0 and 1, why don't they encompass all the reals, despite being countable?

Why is countability so important? I can't wrap my head around what distinguishes the reals from the rationals, and one infinity from another (as posited by Cantor).

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What does $0.000000.....00000...$ mean and how is it different from $0$? –  RghtHndSd Apr 22 at 21:38
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Addressing "what distinguishes the reals from the rationals": $\sqrt2$ is real but not rational. The "infinite number" is a red herring; there are infinitely many even integers, but $1$ isn't even. –  Alex Becker Apr 22 at 21:46
    
The real numbers have no (nonzero) infinitessimals. Your question seems to be supposing the opposite, which may be where the confusion lies. As for why aren't all real numbers rational, have you seen the proof that $\sqrt{2}$ is irrational? –  RghtHndSd Apr 22 at 21:47
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Strictly speaking, infinitesimals aren't used in calculus anymore, except perhaps for intuition (unless you're doing so-called "nonstandard analysis", but that's is a whole different matter). Nowadays, calculus is based on the concepts of "real number" and "limit". –  Hans Lundmark Apr 22 at 22:03
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And there are no infinitesimals in the real number system (and hence not among the rationals either, since they are a subset of the reals). –  Hans Lundmark Apr 22 at 22:04

5 Answers 5

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Infinitesimals are not used in ordinary calculus. Instead, the ideas of limits are used.

Infinitesimals are used in nonstandard analysis, but that is another story.

Also, the concept of "next" does not apply to the reals. For any two distinct reals, there are an uncountable number of reals (and a countable infinity of rationals) between them.

As for countability, Cantor showed that the rationals are countable and the reals are not.

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Contrary to your unsourced claim, infinitesimals are being used in the classroom; see for example this recent article. –  user72694 Aug 3 at 9:19
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That is an interesting article. A problem of mine is that I am so comfortable with the usual concept of limits that I fail to perceive the problems that students may have. However, in my defense, I did not say that infinitesimals are being not used in the classroom; I said they are not being used in ordinary calculus. –  marty cohen Aug 3 at 23:33
    
Marty, post-Weierstrassian paraphrases of ordinary calculus is what is causing student difficulties, because of alternating quantifiers etc. Here is for example Cauchy's "ordinary calculus" definition of continuity: each infinitesimal increment $\alpha$ leads to infinitesimal change $f(x+\alpha)-f(x)$. The extraodinary paraphrase of this via four quantifiers you are surely familiar with. –  user72694 Aug 4 at 7:48

First, you need a well-defined notion of nonzero infinitesimals.

Nonstandard analysis gives us such a notion, and there is a construction of the real numbers using them. The basic idea is:

  • Construct the nonstandard version of the rational numbers.
  • Throw away every infinite nonstandard rational number. (that is, all $q$ such that $|q| > n$ for every standard natural number $n$)
  • Decree that two nonstandard rational numbers represent the same real number if and only if their difference is infinitesimal. (that is, if $|q_1-q_2| < 1/n$ for every positive standard natural number $n$)
  • Decree that every real number is represented by some nonstandard rational number

(the last two points are constructing the real numbers are a set modulo an equivalence relation, if you're familiar with that notion)

Some points regarding this construction:

  • You have to use nonstandard rational numbers that are not standard ones: no standard rational number is infinitesimally distant from an standard irrational number.
  • every standard real number has lots of nonstandard rationals representing it.
  • This only constructs the set of standard real numbers: if you want nonstandard real numbers too (e.g. so that you have nonzero infinitesimals), you have to do more work. (e.g. transfer principle)
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Thanks, Hurkyl. Note however that the term "transfinite" is not usually used in reference to the hyperreals. Usually one talks about "infinite" or "illimited" numbers. –  user72694 Aug 3 at 9:22

I think what you mean why can't we make irrationals with rationals given the fact that we can as you say get arbitrarily close using "infinite precision".

The problem is that irrationals have the property that their decimal part is a non terminating and non repeating sequence.

But all rationals must be represented as some kind of fraction, if the decimimal part behaves "irrationally" it can no longer be represented as a fraction.

But there are fractions that get arbitarily close, but these have the limitation of either terminating or repeating in some way. This is why the ratio als are dense in the teams.

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This was by far the most helpful to my very confused question! Though do you think you could you explain your last sentence? about the ratio als? thanks! :~) –  Rascalniikov Apr 24 at 5:43

The questions are a bit vague but let me try to answer my interpretation of one of them. You write that "infinitesimals give infinite precision", and furthermore "the real numbers require infinite precision", so why can't infinitesimals be used to construct the real numbers?

Now the usual construction of the reals passes via equivalence classes of sequences of rational numbers. This is usually attributed to Cantor even though historians know that Meray got there first.

What's interesting is that, if one throws in infinitesimals or more precisely works with the hyperrationals instead of the rationals, then one can bypass the sequences and construct the reals directly out of hyperrationals themselves. This is explained in Martin Davis's book from 1977. I can provide more details if you are interested.

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Leaving aside the definition and use of the term “infinitesimals”, you can see that real numbers can be constructed from rational numbers as the collection of all limits of convergent series made up by rationals. Many series, e.g. Euler’s limit of the sum of $1/n^2$ can be used to define pi as a limit. These reals can be constructed by a process defined by a formula. To define all the reals we must also include series which cannot be constructed by formulas, since all formulas can be ordered in a countable way.

There is some disagreement on how to present this issue as can be seen in the discussion behind Wikipedia’s definable real numbers. In my view, usually not accepted (and usually rewarded with minus votes), Cantor’s definition of real numbers, immediately lead to the question what rules do this new collection if numbers obey, and the handling of this “collection” (set) lead directly to set theory, not even as a separate step bur instantaneously, more or less embedded in the definition itself. (This is not how invention of set theory is usually described.)

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