real numbers as points on a number line

[113]

Every point of a number line is assumed to correspond to a real number.

https://en.wikipedia.org/wiki/Number_line

Is it possible to find points corresponding to infinitesimals on a number line?  I mean finding an infinitesimal between two neighbouring points (between two real numbers).

I am assuming that every point is surrounded by neighbourhood. I got this idea of neighbouring points from John L . Bells' book A Primer of Infinitesimal Analyis (2008).

On page 6, he mentions the concept of ‘infinitesimal neighbourhood of 0’. But I think he would not consider his infinitesimals as points because on page 3 he writes

that  "Since an infinitesimal in the sense just described is a part of the continuum from which it has been extracted, it follows that it cannot be a point:

to emphasize this we shall call such infinitesimals nonpunctiform."

 

 

 

 

[studiot]

4 hours ago, 113 said:

Is it possible to find points corresponding to infinitesimals on a number line? 

No.

This is the question that exercised Cantor and Dedekind so much and boiled down to

What do you want the properties of 'point' and 'line' to be ?

An infinitesimal was regarded as a function, although the modern view of a function as a type of mapping was just then arriving.

This requires that one can (theoretically) separate off any point (or subset or aggregate of points) of a 'point set' and perform an oepration (function) on it (them).

Changing the court from clay to grass or even concrete has implications in tennis and changing the underlying ground in continuum maths has similar repercussions.

 

[Country Boy]

Pretty much by definition of "number line" every point on a number line corresponds to a real number.  "Infinitesmals" are not real numbers so are not on a number line.

 

[taeto]

On 2/13/2020 at 4:38 PM, Country Boy said:

Pretty much by definition of "number line" every point on a number line corresponds to a real number.  "Infinitesmals" are not real numbers so are not on a number line.

We could challenge that reasoning by saying that there is also a "number line" every point of which corresponds to a rational number, and another "number line" of algebraic numbers, and so on. So why not a "number line" with all infinitesimals? After all, a number line is meant to represent the total order of some kinds of numbers, and if you look at the sums of real numbers with infinitesimal numbers, then they are totally ordered too, so why not?

[studiot]

43 minutes ago, taeto said:

On 2/13/2020 at 3:38 PM, Country Boy said:

Pretty much by definition of "number line" every point on a number line corresponds to a real number.  "Infinitesmals" are not real numbers so are not on a number line.

We could challenge that reasoning by saying that there is also a "number line" every point of which corresponds to a rational number, and another "number line" of algebraic numbers, and so on. So why not a "number line" with all infinitesimals? After all, a number line is meant to represent the total order of some kinds of numbers, and if you look at the sums of real numbers with infinitesimal numbers, then they are totally ordered too, so why not?

Doesn't this boil down to what you man by "number line"  ?

 

What do regard as the essential characteristics of a number line ?

[taeto]

12 minutes ago, studiot said:

Doesn't this boil down to what you man by "number line"  ?

What do regard as the essential characteristics of a number line ?

The notion of a "number line" does not really come up in any serious mathematical context at all. It seems to be a teaching aid for illustrating the ordering and arithmetical properties of real numbers in particular. As such the properties which it demonstrates are no different from the properties of rational or algebraic numbers, that is, the total order and the Archimedean properties.

Wikipedia distinguishes between "basic mathematics" in which a number line represents the real numbers, and "advanced mathematics" in which only the "real number line" represents the real numbers, but others are possible. The "hyperreal number line" is the number line which includes the infinitesimals.

[Country Boy]

On 3/6/2020 at 11:24 AM, taeto said:

We could challenge that reasoning by saying that there is also a "number line" every point of which corresponds to a rational number, and another "number line" of algebraic numbers, and so on. So why not a "number line" with all infinitesimals? After all, a number line is meant to represent the total order of some kinds of numbers, and if you look at the sums of real numbers with infinitesimal numbers, then they are totally ordered too, so why not?

No, because such a "set of points" would not be connected so not a line in the geometric sense.

[taeto]

36 minutes ago, Country Boy said:

No, because such a "set of points" would not be connected so not a line in the geometric sense.

That seems to beg the question. To be "connected in geometric sense" ought to mean that any two points are contained in a common line. So if we declare that the set of all hyperreal numbers forms a single line, then this condition will be satisfied trivially. 

[studiot]

40 minutes ago, taeto said:

That seems to beg the question. To be "connected in geometric sense" ought to mean that any two points are contained in a common line. So if we declare that the set of all hyperreal numbers forms a single line, then this condition will be satisfied trivially. 

Isn't the problem with this that the ordering you get on a number line is not the well ordering of the (real or hyperreal) number system?

This begs the question where (next to which real number) would you place a hyperreal so that it lies between that real number and the neighbouring real number ?

[taeto]

16 minutes ago, studiot said:

Isn't the problem with this that the ordering you get on a number line is not the well ordering of the (real or hyperreal) number system?

This begs the question where (next to which real number) would you place a hyperreal so that it lies between that real number and the neighbouring real number ?

Now come on, a real number does not have "a neighbouring real number", right? It clearly has both a smaller and a larger neighbouring number 😆. Just kidding. But the total order is not a problem, since the lexicographic order is total, which is why you can fairly easily look up entries in a dictionary or telephone directory. In particular if \(dx\) is a fixed positive infinitesimal, then \(a+b\cdot dx\) is to the left of \(a'+b'\cdot dx\) if \(a < a'\) or ( \(a=a'\) and \(b< b'\) ). 

A well-ordering is not required anyway.