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Before I get into details of the question, please note that what we know as the completeness of real numbers, Dedekind calls it continuity of real numbers. So I just stick with Dedekind terminology in asking the question. Please feel free to use either.

Dedekind tries to rigorously define the continuity of real numbers in his manuscript "Continuity and Irrational Numbers". To do so he first defines the continuity of straight line: "if all points of the straight line fall into two classes [sets] such that every point of the first class[set] lies to the left of every point of the second class[set], then there exists one and only one point which produces this division of all points ..."

I have three questions about this:

  1. Why not defining continuity simply as there exists a point in between any two points? In terms of real numbers this would be there exists a real number between any two real numbers.

  2. Is there a particular reason we have to void our intuition for defining continuity? I mean is the continuity of line really different from the intuition of continuity that we have?.

  3. Would it be possible to know how Dedekind arrived at the definition of continuity? what train of logical thoughts did he follow?

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    $\begingroup$ There always being a point in-between every number, doesn't imply that that there are no gaps in-between those numbers; for example all the rationals having a factor of two in the divisor have a member in-between every pair, but there are rationals without a $2$ in the divisor in-between those but not a member of them. $\endgroup$
    – it's a hire car baby
    Mar 22, 2017 at 9:57
  • $\begingroup$ In terms of voiding intuition, I think, as with many math matters, once one has worked with it in many ways, the intuition follows. The reason it isn't intuitive now is that there are details of the Real numbers which you haven't appreciated yet. $\endgroup$
    – it's a hire car baby
    Mar 22, 2017 at 9:59
  • $\begingroup$ On the final question, transcendental numbers are arrived at by the means of limit points and infinitessimals. As such I think it was inevitable that we would move towards a definition of the Real numbers which is capable of encapsulating those and as such defining numbers as limit points. $\endgroup$
    – it's a hire car baby
    Mar 22, 2017 at 10:02
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    $\begingroup$ Thanks a lot @RobertFrost excellent comment. $\endgroup$
    – abk
    Mar 22, 2017 at 22:24
  • $\begingroup$ @RobertFrost What confused me about the definition was Dedekind use of straight line from geometry to define continuity. Then I started basing my own intuition on geometry rather than numbers. It makes sense that our geometrical intuition does not apply to numbers. As you said there are more into real numbers than a simple geometry. Thanks again, it was really helpful. $\endgroup$
    – abk
    Mar 22, 2017 at 22:30

3 Answers 3

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  1. his idea was to identify a real number with the set of all rationals less than it, so square root of 2 is the set of all rational q such that q^2<2
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  1. The set of rational numbers has the property described in your first question , but as you know there are holes in the set, so you cannot define continuity like that .
  2. Dedekind's construction of the number continuum was inspired by the continuity of straight line, so he couldn't get rid of geometric intuition completely , although he does claim that " arithmetic shall be developed out of itself", his reason is the following :

    That such comparisons with non-arithmetic notions have furnished the immediate occasion for the extension of the number-concept may, in a general way, be granted (though this was certainly not the case in the introduction of complex numbers); but this surely is no sufficient ground for introducing these foreign notions into arithmetic, the science of numbers. Just as negative and fractional rational numbers are formed by a new creation, and as the laws of operating with these numbers must and can be reduced to the laws of operating with positive integers, so we must endeavor completely to define irrational numbers by means of the rational numbers alone. The question only remains how to do this.

I think this doesn't mean math based on geometry is non-rigorous , and there is no reason to get rid of geometry in analysis and stop your from using geometric intuition for inspiring foresight, things you should be aware of are geometric intuition is error-prone(learn more here, it may provide a satisfactory answer to your second question ), and algebraic method is more power than geometric methods, which caused shift of focus from geometry to arithmetic/algebra, learn more here

  1. Inspired by the line continuity axiom (as you mentioned it in your post), Dedekind thought a number continuum should have the property ( in his book Continuity and Irrational Numbers):

    the domain R possesses also continuity; i. e., the following theorem is true: If the system R of all real numbers breaks up into two classes A1 , A2 such that every number α1 of the class A1 is less than every number α2 of the class A2 then there exists one and only one number α by which this separation is produced.

But I don't think Dedekind really achieved his goal, since he distinguish a cut produced by no rational number from an irrational number, learn more here. Modern real analysis books just equate a cut produced by no rational number and an irrational number, so that avoid the problem Dedekind was questioned by Heinrich Weber , but which makes the real numbers a bit abstract and strange than you might think , so that Hermann Hankel made the comment (cite form here):

Every attempt to treat the irrational numbers formally and without the concept of [geometric] magnitude must lead to the most abstruse and troublesome artificialities, which, even if they can be carried through with complete rigor, as we have every right to doubt, do not have a higher scientific value.

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  • $\begingroup$ That past point about abstraction is true, but it does not appear to be a limitation Dedekind, but rather a limitation of his time. Mathematics at that time was not so much abstraction oriented as it is now. And with the work of Dedekind available, adding abstraction (ie saying that real numbers are Dedekind cuts) was a minor (but important) step. +1 for the nice answer with good references. $\endgroup$
    – Paramanand Singh ♦
    Nov 13, 2017 at 16:25
  • $\begingroup$ @ParamanandSingh Thanks ! (1) When you say $\sqrt2$ is a Dedekind cut, how do you mean by saying the length of a line segment is $\sqrt2$? I cannot find an explanation using the abstract definition of real numbers .(2) Why adding abstraction (ie saying that real numbers are Dedekind cuts) was a minor (but important) step? $\endgroup$
    – iMath
    Nov 14, 2017 at 1:27
  • $\begingroup$ The length of a line segment is a term which we define by associating a real number to each line segment. More precisely if $a, b$ are Dedekind cuts (real numbers) corresponding to end points of a line segment then $|a-b|$ is defined to be length of a line segment. $\endgroup$
    – Paramanand Singh ♦
    Nov 14, 2017 at 3:51
  • $\begingroup$ The abstraction here was minor because we did not have to do any hard work. We just had to say that a real number is the Dedekind cut itself. It was important because if this was not done, there was no way to create the real numbers out of rationals. The problem looks complicated if we don't do abstraction. You can try the simpler problem. How do we get integers out of natural numbers? $\endgroup$
    – Paramanand Singh ♦
    Nov 14, 2017 at 3:56
  • $\begingroup$ @ParamanandSingh I followed this approach in the book to get integers out of natural numbers, even rational numbers: The inverse operations, subtraction and division, are not always possible within the set of natural numbers; we cannot subtract 2 from 1 or divide 1 by 2 and stay within that set. To make these operations possible without restriction we are forced to extend the concept of number by inventing the number 0, the "negative" integers, and the fractions. The totality of all these numbers is called the class or set of rational nu $\endgroup$
    – iMath
    Nov 14, 2017 at 13:39
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  1. This is called density and is a property that the rationals have. But the rationals are not complete. So density does not capture completeness.
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