Distribution of natural numbers inward in a line segment

Question

Is there a way one can distribute natural numbers inwards and between two given points? For example, imagine a line segment starting with 0 and ending 1. Now I want to distribute all the rest of natural numbers between these two numbers. I think I'm asking if there's an order for natural number so that it distributes them this way. Any help is much appreciated.

PS: I'm curious because I need to know that if there is such a distribution, the line segment seems to be dense, and I don't know if the next number (due to this new order) can be defined.

Answer 1

There is a simple distribution, which for say for integers up to $16$ might look like

0  9  5  10  3  11  6  12  2  13  7  14  4  15  8  16  1          

which you might see as as one line of

0                                                      1          
0                          2                           1
0            3             2             4             1          
0     5      3      6      2      7      4      8      1          
0  9  5  10  3  11  6  12  2  13  7  14  4  15  8  16  1          

and clearly you can go on inserting natural numbers in this way indefinitely

If you want an algorithm for the position of $n \gt 1$ then write $2(n-1)$ in binary then reverse signs of the powers of $2$ in that expression. For example, to find the position of eleven, write $2(n-1)$ i.e. twenty in binary as $$20_{\text{ten}} = 10100_{\text{two}} = 2^4+2^2$$ and then take $2^{-4}+2^{-2} = 0.3125$ as the position for eleven in the unit interval. The position of each integer is a specified dyadic rational with all the positions together being dense on the unit interval

These positions are totally ordered in the sense that it is possible to find the positions of any two integers and compare them (for example, you can see that eleven is positioned after five but before four) but not well-ordered in that you cannot find the number in the next position (six comes after eleven, but twenty-two comes halfway between them, and forty-three comes closer to eleven, and eighty-five even closer, and so on infinitely). Any dense distribution (of a set with more than one element) precludes a total well-ordering