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Everything posted by Ripheus23
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Something I found was that in surreal arithmetic, there are numbers such as "omega minus one." These can fit to the reduction of the omega-metaindex, though anomalously: also, though, the relatively finite numbers can then be defined as the examples of the things like 1 and 2 and so on, anomalously, in the surrealm. In other words, relative 1 is larger than all actual finite numbers and less than omega, in this sense.
However, there also seem to be an infinite number of kinds of zero in the system. Assume any aleph with any metaindex, and assume its glyphdex is 0. If its glyphdex is 0, we will stipulate that all higher indices are 0. At any rate, we can imagine metaindexed alephs all of whose first-order indices are zero. By the principle of comparison, a zero-aleph with more indices is larger than one with less, so we would get the peculiar result that there are infinite relative-zero numbers in a sequence whereby each is larger than the predecessor. In other words, not only would there be infinite number of kinds of zeros, but there would be infinitely larger and larger forms of zero!
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Lol I guess the Veblen hierarchy involves the mechanics of the arithmetic I'm studying. Hmm...
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I feel like there must be different levels of xatrix functions. Like, in one case you infinitely compound the hypertower's two indexes by the HT's own form. That's maybe the smallest. I actually don't know the counterpart sigma-function case (aleph-zero ^(aleph-zero) aleph-zero ^(aleph-zero)...) for the predecessor to the counterpart xatrix-sigma arithmetic, so although I think the basic xatrix hypertower goes to (omega.1)aleph-zero, I don't know how much further the sigma-case goes, here.
But anyway, compounding the glyphdex and converting it into a series of xatrix hypertowers has to go farther, for the sake of the fact that the simplest aleph-case of a hyperglyph [((aleph-zero)aleph-zero)aleph-zero] already goes farther than the simplest xatrix hypertower (on my occurrent assumptions).
The next major question of form of levels, then, (that I have anyway...) is what happens when we have a xatrix with more than one kind of variable? Like, I've started with ones where all the variables are X, but we could have ones with X and Y, or ones with countably or uncountably many variables? In that event, taking the xatrix function of such a system seems as if it would have to go to some realm of continuously branching levels, even if each level is, inside of itself as such, countable? I don't know. Technically, I still don't have a clear enough presentation of transfinal arithmetic, which is what I'm working towards...



