We want a linearly ordered field extending the reals, where we can take arbitrary/countable infinite sums of positive summands (which should be independent of the order of the summands).
This is somewhat equivalent to that every increasing sequence (for the countable case) has a limit.
This post is due to time limits a bit a stub. But I see 3 possibilites to check
- surreal/Conway numbers
- hyperreals (non-standard analysis)
- home made construction
bo198214 Wrote:This is somewhat equivalent to that every increasing sequence (for the countable case) has a limit.
As was just shown by
G.A. Edgar this dream displodes.
If the (increasing) sequence
of natural numbers has a limit
then also the sequence
has the limit
by continuity of addition is then
. As we are in a field and have a subtraction it follows contradictively
.
Well, that is if we assume that commutativity holds for transfinite L. But without commutativity, we wouldn't have many of the nice properties that we wish the resulting vector space to have.
Actually, something else just occurred to me. Would it be possible in this case to use Ramanujan summation? (We'd have to distinguish between finite cases and infinite cases, of course. But it gives us a way to obtain a value out of an infinite summation that perhaps may have the properties we want.)