THEOREM 5.

THEOREM 6.

These Theorems demonstrate that any form, x, may be transmitted through an infinitely continuous universe and that an infinitely continuous universe may be divided as many times as we please.

The following Theorems are developed for use in demonstrating the existence of equations of higher degree in the algebra of distinctions for application to the physical sciences. Brief demonstrations of their validity are included to help the reader develop more familiarity with the algebra of distinctions.

THEOREM 7.

Demonstration:

by Ex. 3 (twice)

by Theorem 5

by Theorem 3

by Ex. 3.

THEOREM 8.

a

Demonstration:

by Theorem 7

by Ex. 3

by Theorem 5

by Ex. 2

by Theorem 7

by Theorem 5.

by Ex. 3.

by Theorem 7.

a by Theorem 3.

We have seen that a region defined by a distinction may be divided by further distinctions as many times as we please, but so far we have dealt only with finite expressions. All of our proofs and demonstrations consist of a finite number of steps. But suppose we divide a distinction, A, into two regions whose contents are described by the variables a and b: A observe that

A		AA by Ex. 2.
OR
A			by substitution.
			by Ex. 3.
			by Theorem 5.
			by Theorem 8.
			by Ex. 3.
			by definition of A
			by Ex. 2.
			by substitution

which by repetition of the first five steps above, implies

A

Since this procedure may be repeated over and over, we may represent A as:

A Ex. 9.

We see that this is a self-referential expression and the right hand side of Ex. 9 is not a finite expression. With a finite expression, we may determine unique values or solutions for every possible case by substitution of and for the variables. After substitution, a finite expression may be simplified in a finite number of steps. For example, is a finite expression and may be evaluated by considering all possible values of a and b:

CASE I a b

CASE II a , b

CASE III a , b

CASE IV a b

CASE I:

A

by substitution

by Ex. 3.

By Ex. 3.

CASE II:

A

by substitution

by Voidness of

by Ex. 3.

CASE III:

A

by substitution

by Voidness of

by Ex. 2.

By Ex. 3.

CASE IV:

A

by substitution

by Voidness of

By Ex. 3.

This demonstrates the statement that the finite expression has unique values for each possible case. However, the basic assumption of infinite continuity implies that the region A, in fact any region, is infinitely divisible. Therefore,

A Ex. 9.

must be viewed as truly representative of the infinite divisibility of region A. But this is an algebraic expression of a different degree of complexity than the expression .

One of the consequences of the basic assumption of infinite continuity is that an infinitely continuous universe is self-referential. This self-referentiality is clearly evident in Ex. 9 when we notice that the form of the first two, and every subsequent set of two regions is identical with the whole expression.

Ex. 9. may thus be transformed to

A Ex. 10.

reflecting self-referentiality. Solving Ex. 10 for every possible case of a, b, as before,

For CASE I:

a b :

Since, if A

Voidness of

by Ex. 3 (twice)

CASE II:

a , b :

Since, if A

Voidness of

by Ex. 2.

by Ex. 3.

CASE III :

a , b :

Since, if A

Voidness of

By Ex. 2.

By Ex. 3.

CASE IV :

a b : or

Since, if A

by Ex. 3.

and if A

by Ex. 3.

Therefore, in CASE IV, the solution of Ex. 10 is not unique. In fact, there are two solutions. Following the convention established for algebras based upon numerical arithmetics, we will call equations with one solution for each evaluation equations of the first degree. Equations with two solutions for any evaluation will be called equations of the second degree, and so forth.

We have only defined two distinct values. They are and . Is it possible that the arithmetic of distinctions may also have values other than and ? If so, then, since our calculus follows logically from the making of a distinction, these other values must correspond to specific forms existing within the universe of distinctions. The solutions of higher order equations would be of little mathematical interest and have limited application, if they could only be repetitions of these two values. However, development of the calculus of distinctions may produce values analogous to those of numerical mathematics, where in addition to the cardinal numbers: 1, 2, 3, ..., there are negative numbers, fractions, irrational numbers and imaginary numbers.

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