Hi Dlorde, you are right, we have been here before. We have seen these questions, and these answers, before. The trouble is, as I said to Pete earlier in this thread, answers tend to take the form: “this is the answer, whatever the question was”.

By that logic, anywhere on the road is the middle, which makes 'middle' meaningless in this context.

Great!

It is unclear if you are saying that you consider “middle” to be meaningless, or just that the logic I applied would make it meaningless. However, I would contend that middle, beginning or end; or indeed any position, in infinity is meaningless, but that’s probably a step further than we would be ready to go until we can progress beyond constantly returning to mathematical “infinities”.

Half of infinity is still infinity. You have two infinite roads.

You have two unbounded roads, but I argue that unbounded and infinite are not necessarily synonymous.

“Unbounded” can be quite different from “infinite”, because, although infinity can rightly be said to be boundless, all that is boundless is not necessarily infinite. For example, in an infinity of nothingness there could, in theory, exist a universe. The surrounding nothingness would place no limit on the amount of matter or energy that could be added to that universe, nor would there be any limit to the extent to which the universe could expand. Nevertheless, however much was added to the universe, or however great was its expansion, it would always be finite.

Its potential might be said to be infinite, as it has an infinity of nothingness into which it can expand, but even this is not strictly correct. It can never reach infinity; therefore it does not have the potential to become infinite. Nothing finite can become infinite.

Yes. An infinite extent can have a beginning.

Beginning and end are directional concepts. Turn round and your beginning becomes an end which, by definition, infinity cannot have.

You may say that the start is a specific point on the road, and the PIF, infinitely far away, are also at a specific point on the road, but these two points cannot be related by measurement;

Does it seem strange that something that constantly returns to mathematical definitions cannot be related by measurement? I am not disagreeing with you here; I too believe that points in infinity cannot be related by measurement; but take that logic a step further and it becomes: Two points in infinity cannot be distinguished from each other.

No, both infinities in this thought experiment are the same size. If you want to know about different 'sizes' or orders of infinity, check out Georg Cantor's Transfinite Numbers.

This is a recurring problem when trying to discuss infinity. Cantor’s infinities are valuable as mathematical tools, but let’s not forget that even Cantor had problems dealing with “absolute infinity”.

I don't know what you mean by it, but there are different orders of infinity. For example, there are an infinite number of natural numbers, but a larger infinity of real numbers.

This is another of those things that always comes up, but is never resolved. This, I think, is a shame, because Cantor’s work on infinities is a masterpiece, but it should not be expected to apply beyond the sphere of maths.