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I need some help.

Quoted from Elliott Mendelson's book, "Real numbers will have to be defined in such a way that, not only are the ordinary arithmetic operations of addition, subtraction, multiplication, and division (by a nonzero number) performable, but also such that every distance of a point on the line from the origin corresponds to some real number, and, vice versa, every real number corresponds to some distance. (Positive real numbers are to correspond to distances from the origin of points to the right of the origin, and negative real numbers to distances from the origin of points to the left of the origin.)"

The author then gives the definition of a cut in an ordered field, as an ordered pair (A,B) of subsets A and B of F such that, 1) A and B are nonempty; 2) A ∪ B = F; 3) x∈A and y∈B implies x < y.

Then the author gave the definition that an ordered field is complete when there are no gaps, that is for every cut (A,B), either A has a maximum element or B has a minimum element.

My question is, by creating a complete ordered field, how is it that I'll be able to do the following, "every distance of a point on the line from the origin corresponds to some real number, and, vice versa, every real number corresponds to some distance ..." as mentioned by the author using the complete ordered field?

Little Rookie
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  • The cut point (A max or B min) corresponds to a real number. This approach, called Dedekind cut, is how real numbers get defined, starting with rationals. – herb steinberg Feb 13 '19 at 02:57
  • @herbsteinberg Yes I know that, but how does this answer my question ツ – Little Rookie Feb 13 '19 at 03:20
  • I believe that you need to set up an ordered field for distances on the line as well as for the reals, then set up the correspondence between cut points of the two fields. – herb steinberg Feb 13 '19 at 04:08
  • More basically. What is the definition of distance other than with a real number? – herb steinberg Feb 13 '19 at 04:21
  • @herbsteinberg The author did not elaborate more on that matter. It's a book about Number system. – Little Rookie Feb 13 '19 at 04:28
  • It is possible to construct the equivalence to the real numbers as distances along a line (horizontal) using straight edge and compass. First mark $0$ and $1$. Integers can then be marked off with the compass. Next rational numbers can be marked off with straight edge and compass and finally real numbers using Dedekind cut idea. – herb steinberg Feb 13 '19 at 17:19

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It is very complicated to define this function, what the most authors do is define a real's line, so by definition, a real line is the real set.

To be more formal, you need to construct a more complex idea. First, you do not have the definition of a point, so a line is a set of undefined things between two points. The line is a set of infinity points. Cool! Even we don´t know what is a point, we know what is a line.

The infinity set that has an order has a cardinal number even if the set is infinity, by another complex construction you can verify that the set of real numbers is ordered. So to represent the real numbers you need a straight line, because the straight line is an ordered set of points.

Now to verify if there is a bijection function between these two sets is just verify if the sets have the same cardinal number.

Now is the most difficult part, you need to prove the Continuum hypothesis! It is the first Hibert's problem. The Continuum hypothesis affirms:

"There is no set whose cardinality is strictly between that of the integers and the real numbers."

The CH is an open problem. If it is true, you can use the Dedekind cut to prove that some real number can be represented in a straight line, so all real can be represented in all points of a straight line. If not it should have some real number that can't "be" a pointer if the cardinal number of a straight line set is less than a cardinal number of real numbers set.

Guilherme Namen
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