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 9th part is, uniquely among all natural numbers other than 1, the integral product of its 3rd 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, has a place for each infinite number of each of the 9 primary natural numbers, it has the place for an infinite number of 9's, i.e., , as well as the place for an infinite number of not nothing 0's (the place for an infinite number of places for a finite number of 9's), and the place for an infinite number of 1's.  As 0 < ,

0 < (= 01 = 1).

If = c (the continuum), = cd (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, ω0, 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/x3 = 8x3/x4 [= (√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 ω0, 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.

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