- #1

- 471

- 63

https://upload.wikimedia.org/wikipe...led.svg/487px-Omega-exp-omega-labeled.svg.png

For example, some people might have seen similar pictures for ω

^{2}. So it seems interestingly there is a fairly reasonable correspondence between the kind of pictures posted above and what I describe below:

Place the number 1/2 at 0

Place the number 1/4 at 1

Place the number 1/8 at 2

and so on.....

We can also add the assumption that the number placed on each position must be strictly positive.

But we want to distinguish between "number placed" and "number assigned".Now we can define the "number assigned" for each position as:

The running sum of numbers placed at all lower positions.

So the (unique) "numbers assigned" in the above case are:

0 is assigned 0

1 is assigned 1/2

2 is assigned 1/2+1/4

3 is assigned 1/2+1/4+1/8

.... and so on.

1 is assigned to ω

But we can go beyond easily here of course. For example, going a bit further to see this:

Place the number 1/4 at ω

Place the number 1/8 at ω+1

Place the number 1/16 at ω+2

Place the number 1/32 at ω+3

.....

So now we have unique numbers assigned as:

1 is assigned to ω

1+1/4 is assigned to ω+1

1+1/4+1/8 is assigned to ω+2

1+1/4+1/8+1/16 is assigned to ω+3

.... and so on

1.5 is assigned to ω.2

Now indeed we can generalize this suitably to assign unique rational numbers to all elements less than ω

^{2}. For example, following the pattern, we would have 1.75 assigned to ω.3 and 1.875 to ω.4.

The main point here is that the "numbers placed" must be such that the running sum doesn't go unbounded. Now what is interesting in this kind of assignment is that if r1 is the unique rational number assigned to ordinal t1 and r2 is the unique rational number assigned to t2, we have:

(r1<r2) only if (t1<t2)

======

One point here seems to be that we must have running sum as convergent for each limit. And furthermore, it seems that if the convergence value is not a rational number on some limit, we might have some further worries to attend to. I don't know about the generic conditions when the convergence value of a convergent rational sequence is itself supposed to be rational.