THINKING MATHEMATICS SANS NOTHING

**
Foundational Natural Numbers**

The factors of the 9 primary natural numbers foundational complex,

^{(9!/}_{}
^{i)}^{,}

are 1, 784, and 82944, the natural numbers that, uniquely
among all natural numbers, are their integral
products (cf. D.G. Leahy, *Foundation: Matter the Body Itself* [Albany, 1996],
Section III.6; also, on this web,
Transdecimal Calculation of
Number Identity: A Note on *Integral
Product* & Related Terms). 82944 is the natural number
whose 9^{th} part
is, uniquely among all natural numbers other than 1, the integral product of its 3^{rd} part
(Leahy, *Foundation*, p. 524, n. 98).

**
Body Thinking v Set Theory**

_{0}
:
_{0}
::
Aleph-Null : Beth-Zero (Zero ≠ Null) :: infinite *set*
of the natural and rational numbers : infinite *body*
of the natural and rational numbers.
The important thing is that the analogy *immediately* breaks down. Unlike
the elements of sets and subsets, the inhabitants of a thinking now occurring
"place" do not "belong" to it nor does it "contain" or "include" them. The
"place" is a body *conveniently* constituted by its members. The "place"
or "non-set" that is a body freely embodies its elements. Likewise the power
body,
(see, on this web,
The Real Beyond the Void: the
Beginning/the Power Body).

**
The Cardinals**

Cantor's
_{0
}contains the infinite series of natural numbers
(together with the rational numbers) in which 0 (=
nothing) is the *placeholder* for a *potential *one of the 9 primary
natural numbers.

Let
_{0
}be the place of the infinite series of natural numbers
in which 0 (= not nothing) is the *place* *actually occupied* by one
or more of a finite number of 9's, to wit, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1+9, 2+9,
3+9, ... 2+9+9, ... 4+9+9+9+9, ... 6+9+9+9+9+9+9, ... 8+9+9+9+9+9+9+9+9,
... 1+9+9+9+9+9+9+9+9+9+9+9, ... 2+9+9+9+9+9+9+9+9+9+9+9+9+9+9+9+9+9+9+9+9+9+9
.... Where 1+9 = 10 the 0 = +9, where 2+9+9 = 20 the 0 = +9+9, where 40 =
4+9+9+9+9 the 0 = +9+9+9+9, etc., etc. 0 is not only "not nothing",
as it is purely in real trinary logic (RTL) (Leahy, *
Foundation*, Section III.1, et passim), here
it is always the place for a *finite number of 9's.* The place for an *
infinite number of 9's* is "not not nothing", not 0, i.e.,
. (cf.
The Real Beyond the Void: the
Beginning/the Power Body, for 0 : Finite ::
: Infinite.)

In
the infinite series of natural numbers so construed, in
_{0,
}there are an infinite number of each of the
9 primary natural numbers. Therefore in
_{0
}there are an infinite number of 1's and an infinite number of not
nothing 0's, i.e., an infinite number of places for a *finite* number of
9's. But there is no *place* for an infinite number of 0's, nor for an
infinite number of 1's. Most particularly there is no *place* for an *
infinite* number of 9's, i.e., no
.

In the nothing based decimal system assumed in
_{0 }the
number 10 cannot be construed as 1+0. In the decimal system not nothing
based assumed in
_{0}
10 is construed as 1+0 = 1+9. This fact—combined
with the notion that the elements of the latter’s power body
are
to those of the power set
as
members not belonging to a set are to those belonging to a set (ibid.)—ensures
that the power body
will
have a discrete place for each of the infinite numbers of the 9 primary natural
numbers including the 0’s found in
_{0}.

Since
,** ^{
}**the power body of

_{0}
< (=
_{01}
= ** _{1})**.

If
= *c* (the continuum),
=
*c _{d}* (the discontinuity of the continuum = an infinite number of
continua).

**
The Ordinals**

In set theory
ordinals begin with 0 = Ø (the null
set), and 1 = {Ø} = {0}, thus: Ø,
{Ø}, {Ø,{Ø}}, {Ø,{Ø},{Ø,{Ø}}} ...,
effectively, 0, {0}, {0,1}, {0,1,2}, {0,1,2,3}, ...,
{0,1,2,3,4,5,6,7,8,9,10,11}, ...,
or, 0, 1, 2, 3, 4, ..., 12, .... Here an ordinal = the number of its
predecessors. The first finite ordinal is 0 = Ø.
It has no predecessors. The first infinite ordinal,
ω_{0}, is likewise without
predecessors, including the infinite number of finite ordinals which belong to
it. In Badiou's words, between the former and the latter "there is an abyss
without mediation" (cf. A. Badiou, *Being and Event*, trans. O. Feltham
[London and New York, 2005], p. 159). Here, at the point where 'a place
fuses with beyond' (ibid., p. 157), order is imperfectly rational, number
relatively natural, not completely logical.

In the body
thinking now occurring for the
first time there is no 0 = Ø.
Ordinals begin
with the RTL foundational unity 0
= 1,
and 1 = 0 = 1, thus: 0,
0, 00,
000,
0000,
..., or, 1, 1, 1 1, 1 1 1, 1 1 1 1, effectively, 0,
0, 0
1, 0 12, 0
123, ..., 0 12 123456789,
..., or, 1, 1, 1 1, 1 12, 1 123,
1 1234, 1 12345, 1
123456, 1 1234567, 1 12345678, 1 123456789, 1
1 123456789, 1 12
123456789, ..., 1 12345678 123456789, 1 123456789 123456789,
1 1 123456789 123456789, 1
12 123456789 123456789, ...,
or, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, ... (cf. Leahy, *Foundation*, pp. 299ff.,
for a similar, but importantly different, repetition, 1, 1, at the beginning of
the Fibonacci sequence).

Thus in the
thinking now occurring the first transfinite ordinal consists of an infinitely
cumulative beginning and ending of 1-otherwise-than-prefix-of an infinitely
cumulative beginning and ending of the 9 primary natural numbers.
(Cf., on this web, The
Deep Epidermal Surface: The Cornerstone Construction Order, Minimum Order
Tetrahedron Hypercube, & Absolute Dead Center Hypercube,
for the definition of the uniquely hypercubic
absolute dead center cube, [*x*/6]^{4}/*x*^{3 }
= 8*x*^{3}/*x*^{4 }
[= (√2×9)^{-1}].)

Universally nature *is* beginning to exist. Here
an ordinal = 1 + its immediate predecessor. The first natural ordinal is
1. Its immediate predecessor is RTL 1, whose immediate predecessor in turn
is 1,
whose immediate predecessor is 0
= 1, whose immediate predecessor in turn is 1,
etc., etc., *ad infinitum* (cf. Leahy, *Foundation*, pp. 266ff., "the
infinite lattice of logical digits," especially as matrix of the 100-square of
the unum-founded Fibonacci sequence, ibid., pp. 301ff.; cf. also, on this web,
**
Real Trinary Logic Geometric
Series Matrix of the Numeric Geometric Series & the Series of Perfect Numbers**).

Here the first finite ordinal is the sum of the infinite
series that is the infinite repetition of the three RTL digits, 01
= **1**, + 0, i.e., 01+0
= **1**+1 = 1, and the first ordinal corresponding to the natural number 1 is the
sum of this 1 and the infinite sum of its predecessors **1**, that is,* *
again, **1**+1
= 1. Then, consequently, **1**+1 = 1+1 = 2, **1**+2 = 1+1+1 = 3, **1**+3
= 1+1+1+1 = 4, etc., etc.
Here there is neither Ø nor ω_{0},
no ordinal, finite or infinite, without predecessor (cf. Leahy, *Foundation*,
pp. 255f., 303f., and 311, for
= "since," as the
fundamental RTL operator).Here,
beyond beyond finite, counting *ex abysso* (cf. Leahy, *Foundation*,
pp. 478ff., et passim), order is absolutely rational, number perfectly natural,
ontology identical with logic. (Cf., on this web,
Note to *Faith and
Philosophy* Further to the Ontology of Real Trinary Logic
and Beyond the Good:
Not Hither the Good [& Not Hither Beyond the Good].)

**Real Trinary Logic: Mathematical Analogue**

Arrange the elements in the place for an infinite number of 9's, into the number 999999..., likewise arrange the elements in the place for an infinite number of places for a finite number of 9's, into the number 999999..., adding 1 to differentiate it from the first 999999..., just so, 1+999999....

Where .999999... = 1 (http://en.wikipedia.org/wiki/0.999...),
and *xxxxxx*.../(1+*xxxxxx*...) = .999999... = 1, let RTL
(not
not nothing), the
place for an infinite number of 9's, =
999999..., let RTL 0 (not nothing), the
place for an infinite number of places for a
finite number of 9's, = 1+999999..., and let RTL 1, the
place for an infinite number of 1's, =
.
Then,

999999.../(1+999999...) = .999999... =
=
/(1)
= /0
= 1.