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real numbers as points on a number line

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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."

 

 

 

 

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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.

 

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