1. Introduction

On unsolved number theoretic problems, Chaitin has the following to say [2]:

".... the fact that many mathematical problems have remained unsolved for hundreds and even thousands of years tends to support my contention. Mathematicians steadfastly assume that the failure to solve these problems lies strictly within themselves, but could the fault not lie in the incompleteness of their axioms? For example, the question of whether there are any perfect odd numbers has defied an answer since the time of the ancient Greeks ....".

Comments by Godel are just as relevant (Godel, 1964) [11]:

"..... Success here means fruitfulness in consequences, in particular in "verifiable" consequences, i.e., consequences demonstrable without the new axiom, whose proofs with the help of the new axiom, however, are considerably simpler and easier to discover, and make it possible to contract into one proof many different proofs. ........ Perhaps also the apparently insurmountable difficulties which some other mathematical problems have been presenting for many years are due to the fact that the necessary axioms have not yet been found. ... "

The above quotations are made to highlight their relevence in sequence algebra which came into being since 1994 arising from the introduction of a single axiom in Nat(z) [6]. For the uninitiated, the axiom simply defines the natural number sequence holistically by a generating function z/(z- 1) which is in fact a restatement of the order principle from number theory in sequence algebraic format. This holistic axiom makes it easy to prove or derive other holistic sequences such as Even(z), Odd(z), Comp(z) and Prime(z). Until 1994, conventional number theory was not able to predict globally Comp(z) and Prime(z). Once these two sequences were sucessfully derived, it becomes easy to derive higher secondary sequences such as Mersenneprime(z), Fermatprime(z) and Perfectnumb(z) [5 to 8]. The ease with which some unsolved number theoretic problems were solved justifies the adoption of Nat(z) as an axiom.

In the next section, we will define a canonical generating function called CGF(z), which is holistic and yet generates a single integer output for a single integer input. This type of function is what Emily Post described as a recursively enumerable function or (r.e. function) [12]. In sequence algebra, even a single integer is considered as a member of a global sequence since CGF(z) is a holistic function.

2. The CGF(z) function

This paper will begin by defining a canonical generating function called CGF(z) written in sequence algebraic format given by equation (1). I like to call it a Swiss-knife function. The program line is written in Maple V R 3 syntax.

>CGF(Z):=1/(z^f(i)-1)-1/(z^f(i)*(z^f(i)-1));

..........................1...........................1
CGF(z) :=..------------..--.. ------------------ ..........................(1)
.....................f(i).....................f(i)...f(i)
..................(z.... -....1) ........z......(z....-....1)

For a fixed value of f(i), both the first and second expressions on the RHS of equation (1) are impulse sequences with uniform intervals of f(i) units. The only difference between the two is that the second sequence is shifted forward by an interval of f(i) units with respect to the first sequence. This explains why for fixed f(i), the CGF(z) will generate only a single, isolated impulse signal which in sequence algebra is taken to represent a single integer number also called f(i). You can only get this result using the sequence algebraic format but not the Euler's format. For more information on the differences between the sequence algebraic format and the Euler's format, please read references 5 and 9.

If we make f(i) a random function, i.e., f(i) = rand(0..infinity) where random integers are generated in the range from 0 to infinity, then we expect that CGF(z) will generate all the integers with equal probability given infinite time. Mathematically speaking, true random number generators have not been developed although it exists in nature such as the emission of alpha particles from a radioactive source. For experimental purposes, we shall have to make do with various pseudo-random functions provided by symbolic softwares or compiled languages. There is no proof that a true random number generator cannot be developed. Perhaps it is just that axioms in number theory itself do not have anything to say about randomness. Chaitin's remark that one cannot prove a 20 pound theorem using a 10 pound axiom is an information theoretic interpretation of Godel's incompleteness theorem [1,2]. To demonstrate the versatilities of CGF(z), examples will be given in this paper to backed up the claims.

(i)Proof of Unsolvability Of Halting Problem

The unsolvability of the halting problem is equivalent to the assertion that there is a recursively enumerable set (an r.e. set) that is not recursive. Interestingly CGF(z) fits this description of an r.e.set perfectly, i.e., it is a computable function which have a natural number as value for each natural number input and the running time is computational complexity. CGF(z) can be recursive or nonrecursive being dependent on the properties of the input function f(i). If f(i) is defined as an input function which generates integers randomly, CGF(z) will only output a single integer with a random interval which exactly duplicates the magnitude of the input integer. Would you call CGF(z) a random function? The author thinks so provided f(i)s are random since these are embedded in CGF(z). If the halting criterion of this function or program is defined as that when normality is attained amongst the ouput integers, how could the program ever halt since perfect random function generators do not exist? This is the simplest proof of the Halting Problem. But how can anyone be so sure that a perfect random number generator will not appear in future? To skirt around this problem, consider testing CGF(z) with increasing lengths of natural number sequences starting from two consecutive integers. Assuming the availability of a hypothetical perfect random generator then for a finite integer sequence, CGF(z) must halt since normality is assured. However, as the length of the natural number sequence grows, there will come a time when we reach computational complexity and the program will never halt. Theoretically CGF(z) must halt but in practice one will wait forever. This is an alternative proof of the unsolvability of the halting problem.

(ii) Nat(z) can be defined as a random sequence

Originally Nat(z) was defined based on the axiom of orderly principle defined in classical number theory [4]. In other words, the natural number sequence is defined as an orderly set. In this section, it will be proved that Nat(z) can be generated using either the orderly model or the random model. The objective is to prove that the same CGF(z) can be used to generate the same Nat(z) using contradictory axioms. If this is true, does it imply that Godel's incompleteness theorem is itself incomplete? This is what we will try to find out.

The CGF(z) function can only accept ordinary algebraic functions via its embedded f(i)- exponents. There is nothing to stop one from defining f(i) as an orderly algebraic function or a random function. Two examples are given in table 1 where example 1 shows how the natural number sequence Nat(z) can be generated by the linear recursive relation f(i) = f(i) + 1 in equation (1) whilst example 2 shows a small problem where integers in the range from 1 to 9 can be generated randomly by defining f(i) = rand(1..10) which are iterated and summed 10000 times.

Table 1 - Nat(z) Based On Two Contradictory Axioms