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I will be leaving town Monday for a month; I will be taking printouts of everything pertinent and coming up with an ontology and a kludgesome on the plane. And I will keep trying to keep up.
OK. So. (1) A number can be singled out out of the mass of all numbers. It is a spisa.Real numbers are not atomic: you can always divide them into x < n/2 and x >= n/2. They are therefore stuff.
A bit of stuff can only be singled out of the mass of all the bits of stuff if it is physically separate. So the middle 1/9 of a glass of water is not a lo djacu}, because it is not physically distinct from its neighbours. However, a glassful of water is distinct from the Pacific Ocean, because it is a spisa.
For 3-D stuff, spisa is defined as I gave it in Ontology #3: a sectioning of 3D space, such that the lines of sectioning pass through non-stuff. The lines delimiting the glass pass through glass, not water. So the contents of the glass, the water, being encased by non-water, are a spisa of water. So it can be refered to as {lo djacu}. The middle 1/9 of a glass of water is not so encased. So it is not {lo djacu}.
For numbers, a spisa of numberhood is defined by the property = x, for some x. 5 is surrounded by numbers that are not 5.
Numbers have the peculiarity that every possible bit of a number is a spisa.3D objects have the peculiarity that every possible bit of a spisa are non-spisa.
As in, pa lo namcu cu se pagbu re lo namcu But pa lo djacu na se pagbu re lo djacu (2) Fractional quantification makes no sense for the set of real numbers.1 out of 2 real numbers *may* make sense, but all it means is that there is a bisection.
1 out of 2 x: P(x) means Ax:P(x) E!x': ~P(x) (For all x such that P, there exists a unique x such that non-P) If I use numerals, 1 out of 3 x: P(x) means Ax:P(x) E!2x' : ~P(x)However, because real numbers are infinitesimally subdividable and transfinitely many,
for *any* bisection of |R (set of real numbers), 1 out of 2 x will be within the bisection.
If we bisect |R at x=0, A(x):x<0 E!x' : x'>0 x' = -x If we bisect |R at x=1000, A(x):x<1000 E!x' : x'>1000 x' = -x + 2000This ends up meaning that pa fi'u ci loi namcu will end up meaing exactly the same as pi mu loi namcu:
1 out of 3 x : P(x) Ax: P(x) E!2x' : ~P(x) A(x):x<0 E!3x' : x > 0 x'1 = 1 - x x'2 = 1 / (1-x)So the set of all negative numbers is 1/2 of all Real numbers, *and* 1/3 of all Real numbers.
"But I wanted the bits counted by pimu to be equal", you might say. Sure they are. All three thirds of |R have the same cardinality: |R.
In fact, even with 3D objects, if you chop them into a left third and a right 2/3, you will always get any part in one mapping to a unique counterpart in the second. So you do get a one to one mapping between bits of the left third and bits of the right 2/3. So 1 out of 2 parts of the water are contained in the left third.
For say any bit of an individual of stuff is defined by a set of 3D points, at most infinite (but not transfinte), P1,P2,...Pn..., such that P1 = (x1/l, y1/w, z1/h), for l,w,h the length, width and height of the individual. (So the points are fractional --- the kind of thing you mean by "middle 1/9th".)
Bisect it at x=xi.Then any bit of the individual left of the bisection, delimited by points such that P1 = (x1/xi-l, y1/w, z1/h) has a corresponding bit in the individual right of the bisection, such that P1 = (x1/l-x1, y1/w, z1/h).
Bisect a loaf in half. The top 1/2 of the left bit of the loaf has a corresponding 1/2 of the right 1/2 of the loaf. The left 1/16 of the left bit has a corresponding left 1/16th of the right bit. The middle 1/37 sphere of the left has a corresponding 1/37 of the right bit.
Cut the loaf so that 1/3 of the loaf is to the left, 2/3 of the loaf is to the right. And guess what: for every bit as described in the left bit, there is still a corresponding such bit in the right bit. So 1/3 of the loaf still contains 1 in 2 bits of the substance.
"But I wanted the bits counted by pimu to be equal", you might say. Sure, and here, you have a 3-D specific notion: volume. Bits are equal, *not* because they have the same cardinality (which is still aleph-1 on both sides), but because they describe an equal volume. That's different from what you're doing with finite sets. And volume is inapplicable as a concept to real numbers.
So: for finite sets and collectives --- things of finite cardinality n --- pimu loi means a thing of cardinality n/2
For stuff of transfinite cardinality, we do *not* mean pimu loi is a thing of half that cardinality, because half of aleph-1 is aleph-1. Rather, we mean, for 3D objects, any portion occupying 1/2 the space of the piromei.
Where space is irrelevant, as in real numbers, then *all* fractional quantifications are the same, and the only distinction is between piro and na'ebo piro = pida'a.
So And, your 1 out of 2 is inapplicable to |R, and I'm right: pimu describes only volume for a 3D onject, not really quantifuing over bits.
over to you. -- **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** * Dr Nick Nicholas, French & Italian Studies nickn@hidden.email * University of Melbourne, Australia http://www.opoudjis.net * "Eschewing obfuscatory verbosity of locutional rendering, the * circumscriptional appelations are excised." --- W. Mann & S. Thompson, * _Rhetorical Structure Theory: A Theory of Text Organisation_, 1987. * **** **** **** **** **** **** **** **** **** **** **** **** **** **** ****