pimu of real numbers?

[Nick Nicholas <opoudjis@optushome.com.au>]

This is unimaginably behind the times (i.e. outflow from yesterday), 
but then, that's the nature of this. When I think how many 
definitions and redefs have flowed in the past 5 days...

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 + 2000

This 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.


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* Dr Nick Nicholas, French & Italian Studies nickn@unimelb.edu.au *
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. *
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