Sequence algebra arrives at its present status by adopting three extensions to first order arithmetic theory and first order number theory. These extensions are grafted on without affecting standard arithmetic rules nor the redefinition of number theoretic properties. Of the three extensions, the first defines a new arithmetical operator called Normc which is new and cannot be derived from standard arithmetic operators whilst the second defines the expression z/(z-1) axiomatically as representing a global set of natural numbers whilst the third defines mixed mode operations whereby boolean functions and arithmetic functions can be integrated and evaluated under the same algebraic formulation. The author cites findings from publications in symbolic logic and mathematical logic to back his claims [1].


1. Introduction

In this discussion, there is no need for readers to know in detail first order arithmetic theory and first order number theory. Other than mathematical logicians, ordinary users do not begin a mathematical course by learning first the underlying logic. In spite of insights provided by these theories, much progress in mathematics is made informally outside mathematical logic. Of late, the author discovers that some findings in mathematic logic do explain the increased expressivity of sequence algebra over conventional number theory.

First order sentences in number theory range over only the integer number objects in this domain of discourse. To make mathematical statements on sets of numbers we need to define second order quantifiers [1]. Thus far such second order quantifiers cannot be translated into true first-order sentences. The extensions provided in sequence algebra has changed this situation. Now we have a first-order quantifier which describes the global set of integer numbers in the form given by equation (1):

......................................z
Axiom:.....Natnum := ----- .........................(1).
....................................z - 1

Natnum is defined as an axiom to represent the global integer number set because when taylor series is expanded, the terms are ordered consecutively according to the natural number sequence as shown in equation (2). There are numerous first-order theories of arithmetic but in this paper we shall adopt an arithmetic theory with the axiomatized theory P where P stands for Peano Arithmetic [1]. It is recognised that taylor series expansions only rely on standard arithmetic operations and are well within first order arithmetic. What is new is the way we interprete the series as representing the natural number system.

..................................1........1........1........1........1........1.........1.........1...........1
Natnum := 1 + 1/z + ---- + ---- + ---- + ---- + ---- + ---- + ---- + ---- + O(---)...(2).
....................................2........3.........4........5........6........7.........8.........9..........10
..................................z........z.........z.........z.........z........z.........z.........z............z

If Natnum is the global model, then there are two versions of a theorem of compactness from mathematic logic which has an "easy direction" and a "hard direction" [1]. We give below the two versions translated for applications in the number theory domain.

(i) Easy direction: If Natum has a model, then every finite subset of Natnum has a model.

(ii) Hard direction: If every finite subset of Natnum has a model, then Natum itself has a model.

To simplify presentation, only intuitive proofs are given.

From sequence algebra, every finite subset of Natnum has the same algebraic structure as equation (3) where ub represents the upperbound of the number of terms in the taylor series.

Natnum := series(z/(z-1),z=infinity,ub);................(3).

Equation (3) ensures that all finite subsets of Natnum have models once the global model of z/(z-1) is defined according to equation (1).

The second version is harder to prove and yet is provable intuitively in sequence algebra as follows:

We start with Natnum_1= 1+1/z as representing the first two integer number set {0,1}. When the cartesian product of Natnum_1 is taken we get Natnum_2 which represents the integer number set {0,1,2} if we ignore the numerator coefficient values which has to do with duplicities of terms as shown in the second term in equation (4):

...................................1
Natnum2 := 1 + 2/z + ---- .........................(4).
......................................2
...................................z

We can continue to take successive cartesian products of the outputs which can be generalised to equation (5). Here it is necessary to invoke a new arithmetic operator called Normc which is defined as one of the extensions in sequence algebra. The function of Normc is to reduce all numerator coefficient in the taylor series to unities.

Natnum_2n:= Normc(expand(Natnum_n^2)); ...............(5).

This means that when n tends to infinity, we should get by induction the global natural number set

{0,1,2,..........,n,n+1,..........infinity}.............(6).

(i) Axiom: Let the taylor series expansions of z/(z-1) represent the global natural number set.

This axiom has already be discussed in the previous section. As far as arithmetic operations are concerned, those from first-order arithmetic theory will suffice without defining new ones.

(ii) Normc operator: This is defined as a normalising operator which reduces globally all numerator coefficients of the taylor series sequence to unities.

Seamless algebraic manipulations of number sequences within sequence algebraic formulations can only be effected if all number sequences are in normalised forms. This can be provided using the newly defined Normc arithmetic operator which simply reduces all numerator coefficients to unities. However, this operator cannot be derived from the existing ones. This is why a new one has to be defined. The very first application of Normc is in deriving Prime(z) as the arithmetic difference between Nat(z) and Normc(Comp(z)) as given by equation (7) derived using Macsyma 2.2.1.

series:taylor(sum(1/(z^i*(z^i-1)),i,2,40),z,inf,40);

Normc(series):=block([args: args(series)],apply( "+",args/map('numfactor,args)))$

Comp_z:Normc(series); Natnum:taylor(z/(z-1),z,inf,40)-1-1/z;

We thus obtain the Prime(z) sequence as the difference between Natnum and Comp_z:

Prime(z):=Natnum-Comp_z;

(1/(z^2)) + (1/(z^3)) + (1/(z^5)) + (1/(z^7)) + (1/(z^11)) + (1/(z^13)) + (1/(z^17)) + (1/(z^19)) + (1/(z^23)) + (1/(z^29)) + (1/(z^31)) + (1/(z^37)) + . . . ...........(7).

(iii) Mixed Mode Functions: The use of functions which include both boolean funtions and arithmetic functions but which are always reduced to arithmetic values in computations. These are described in section 3.

true + true
true + false
true/false and false/true
true^true and true^false
false^true and false^false

true-true = 0
false-false = 0
true/true = 1
false/false = 1
(true-false)/(true-false) = 1
(false-false)/(true-false) = 0
The next two lines follow from the previous two lines and is used to reverse the numeric output values:
(false-true)/(false-true) = 1
(true-true)/(false-true) = 0
We have not found any use for the next two lines but include these here for completeness:
true^0 = 1
false^0 = 1

.................................ub
.............................--------'
.............................'....| |
Conjunc_MMF :=....| | MMF(i)...........................(8).
..................................| |
..................................| |
................................i = 1

..............................ub
............................-----
.............................\
Disjunc_MMF := ) MMF(i).............................(9a).
............................./
............................-----
.............................i = 1

.......................................ub
.....................................-----
......................................\
......Xor_MMF :=(1/ub) ) MMF(i).............................(9b).
....................................../
.....................................-----
......................................i = 1

Primex(z):=sum(1/z^i*(isprime(i)-false)/(true-false),i=1..15);

......................1........1........1........1........1.......1
Primex(z) := ---- + ---- + ---- + ---- + --- + ---...................(10).
........................2........3........5........7.......11......13
......................z........z.........z.........z........z.......z

Comp(z):=sum(1/z^i*(isprime(i)-true)/(false-true),i=1..15);

..............................1........1........1........1.......1.......1......1......1
Comp(z) := 1/z + ---- + ---- + ---- + ---- + --- + --- + --- + ---.............(11).
................................4.........6........8........9......10....12.....14...15
..............................z.........z........z.........z.......z.......z.......z......z

6. Sequence Algebra - A Tutorial Paper - Huen Y.K. (Date Released 2/2/98, 46 Kbytes)