In this critical review, the utility of the infinite quantity, the aleph null, , which is the first cardinal of the Cantorian Algebra provides solutions to the Last Theorem of Fermat and to the Continuum Hypothesis of Georg Cantor (1845 – 1918). As inductive solutions using transfinte algebra, they confirm the earlier work of Nonstandard Analysis of Abraham Robinson (1918 – 1974), which gives his view of the bridge between mathematics and language, and hence provides a rule based syntax for both understanding and demonstrating the modern utility of the set theoretic calculus for the infinite quantity. A transfinite extension of the incompleteness theorem of Kurt Gödel (1906 - 1978), which illustrates the creative rather than the incomplete nature of axiomatic and postulational systems, further supports the construction of extended rule structures for the infinite quantity. In this way, a beginning may be established for the foundation of mathematics now free from contradiction within the formalist position of David Hilbert (1862 – 1943) following the transfinte set theoretic work of Georg Cantor.
Far East J. Math. Sci. (FJMS) 5(3) (2002), 323-331
Paul W. Dixon
University of Hawaii
An elliptical solution to the Last Theorem of Fermat is given by Andrew Wiles (Wiles, 1995; Taylor & Wiles, 1995; Ribet, 1990). Such a solution sets a geometrical frame for this result. There is also a solution termed the, "transfinite solution," to the Fermat problem. We may generalize from the transfinite solution showing that counting in a downward direction from the infinite permits an infinite number of solutions to the Last Theorem of Fermat (Dixon, 1981).
The expected solution to Fermat's problem where there are no rational integral solutions has been found to be consistent within the elliptical framework of Andrew Wiles' solution, yet it is not true in all cases. Let us employ the inverse of the method of Bernard Frenicle (1605-1675) - the method of infinite descent, yet here in reverse form. This method is as follows: “suppose that solution of the problem in question is possible in positive integers: then we show how to derive a solution in smaller positive integers, and so on. But since this process cannot go on indefinitely we reach a contradiction and show that no solution is possible" (Frenicle, 1986). In this instance, to paraphrase, we substitute an ascending series, and for no solution, n number of solutions.
How may we then reconcile these two solutions? Assistance to this difficulty may come from formal logic. Kripke in 1963 (Kripke, 1963) developed a model for propositional calculi, basing this notion on possible worlds. This concept is related to the notion of Hintikka's epistemically possible worlds (Hinntikka, 1969). These separate worlds may be identified with a maximally consistent set of propositions according to Peregrin (1993). We may conclude, therefore, these two solutions to Fermat's problem show independent validity within their respective mathematical frameworks (Shalit, 1987).
We may note the last theorem of Fermat asserts that, if n is a natural number greater than 2, then
xn +yn = zn
has no rational integral solution x,y,z, with xyz = 0 This problem may be extended so that we consider the transfinite number theory of Cantor (Cantor, 1985) which provides the solution (Dixon, 1981). Thus if each x,y,z is greater than 2, we note
x = y = z
And as noted in (Dixon, 1981) using the Cantorian algebra we have
x + y = z
which is then a solution.
The elliptical solution of the last theorem of Fermat shows no solutions for n, yet in the transfinite solution there are solutions to the Fermat problem.
We may note also
x-n + y-n = z-n
which is also a solution. The elliptical solution may, therefore, be subsumed under the transfinite solution using the denumerable aleph of the Cantorian algebra. The set of transfinite numbers - 1, - 2, - 3, ..., - n and the positive integers 1,2,3, ...,n are for each corresponding integer 1,2,3, ...n congruent for the same modulus (namely, O) thus illustrating the validity of this generalization of the transfinite solution for Fermat's last theorem.
Following Abraham Robinson (Robinson, 1996), let a be an infinite natural number, and let Da exist in relation to ~, which is the infinitesimal approximation. Then we may note that the numbers a - 1, a - 2, a - n, ...n, a number in N all exist since a is infinite and the numbers a, a +/- 1, a = +/- 2, ..., a = +/- n , n a number in N, all belong to Da since |(a = +/- n) - a| = n is finite.
Let the denumerable of the Cantorian algebra be placed in pairwise isomorphic relation to a, a- 1, a- 2, ...,a - n. These two progressions of descent from a and map the same descent within the natural numbers. This completes the illustration of what in this series is predecessor to .
It is clear from these examples that the choice of geometric form - such as elliptical, euclidean or perhaps hyperbolic makes a difference in the solution obtained therein. Covariant transformations without metricizing are possible between the hyperbolic and elliptical forms of the equations. These questions have been analyzed in the context of general relativity (Abrahams et al, 1995).
Such a result presents a fundamental riddle central to the concept of number. We note that the axiom of parallel lines extending to infinity is true in the Euclidean geometry but false in the elliptical, Riemannian geometry. So, likewise, we may note that those fundamental number theoretic conclusions within an Euclidean (e) and non-Euclidan (ne) framework show possible world configurations.
Re > Rne
We are, therefore, met with a fundamental disproportion contrary to platonic idealism as to the ideal of number and also a universal anisotropic relationship between the euclidean and elliptical geometries.
Let us consider a circle of infinite circumference. Each segment of this circumference is of an elliptical geometric form. In this circumference, we may set each point in an isomorphic relationship to the aleph null, of Gregor Cantor. The transfinite solution to the last theorem of Fermat, therefore, may be derived from this general case. There is, therefore, also an elliptical geometric frame which permits n-number of solutions to the last theorem of Fermat.
Should we rotate this circle with infinite circumference through n-dimensional space, this sphere has the same properties now generalized to an n-dimensional rotation in n-dimensional space. Should each of these dimensional constructs have their own possible world propositional calculi, then it may be conjectured that there is an infinite number of subsets to the infinite set in R. In this way, the infinite quantity is parsed through separate geometries and hence qualities and with different essences of number as well.
A solution to the Continuum Hypothesis has been put forward in which the axiom 1 = is used to negate this hypothesis (Dixon, 1993). To postulate that the continuum is not an aleph, yet R is a subset of the continuum and is therefore in the sense of an ordinal relationship is denumerable, with the negation of the continuum hypothesis, infinitesimals may join any discontinuity in R, making R infinitely extensible (Kossak, 1996). This permits denumerable status to R.
As a bridge to the mapping between a and c of R, substitution of the infinitesimals by letting
a = , b = , n
a + b = c
a + b c
where b = n
Therefore by subtraction
a - b = c
any gap in R is accounted for by transfinite induction in a pair-wise isomorphic mapping to:
which is the continuum.
The Gödel numbers, G, which are set in correspondence to the axioms of arithmetic or more complex theoretical postulate systems in physics have by Gödel's incompleteness theorem, using an extended diagonal form of proof, shown that in any extended axiomatic system, new axioms may be added which cannot be proven to be either true or false in conjunction with the original postulate system (Gödel, 1931; Gödel, 1958).
This is, however, in error since numbers which are used to enumerate
objects with no underlying ordinal, interval or ratio scale property have only
nominal (naming) scale properties (Stevens, 1951). With this kind of scale,
no mathematical operations are permissible. To speak, therefore of being unable
to add additional axioms or postulates in an extended array is, alas, of at
best a tautological nature only. The pairwise isomorphic mapping used in the
incompletness theorem cannot be extended mathematically.
We note that in a nominal scale the function: X1 = f(x) indicates that the Gödel numbers when assigned as an index to the rules of arithmetic and symbolic logic permit an enumeration of these entities. Using a nominal scale, however, does not indicate any limit or any order to this isomorphic relationship. The nominal scale is defective in this sense so to use this scale within a formal mathematical proof, by its very use, provides for unlimited substitution and grouping into sets and subsets. Gödels Incompletness Theorem is, therefore incomplete due to the use a nominal scale as an essential step within its formal deductive logic.
With transfinite induction, using an analogous operation
so that at the limit of an isomorphic mapping G to , no G number can be added at the limit, yet again by transfinite induction
- G =
as previously indicated as G goes to limit of .
Thus by means of transfinite induction n number of postulates or axioms may be subtracted from an extended series of postulates without altering the posulational system. This forms an extension to Gödel's original demonstration. Through the method of transfinite induction, the reduction of a postulate system may then be carried out without this being shown to be true or false in terms of the original postulate system.
This conclusion indicates that the axioms or postulates of the scientific system are not absolute foundational constructs in an operational sense. In the realm of infinite possibilities, the final system of axioms or postulates for a given system may it be found that creation outweighs certitude in the sense of possible worlds configurations.
The original system of axioms given by Euclid for his geometry illustrates this. Thus the first lines of Euclid's book entitled Elements is as follows:
Thus Euclid can be shown to not rely upon Euclid. Axiomatic and postulational systems may therefore be thought of as organizing principles yet they lack foundational and constructive validity if they share the generality of these geometrical axioms (Vilenkin, 1995).
Thus it may be demonstrated under this example of possible worlds logical extension that R has different properties according to certain geometrical considerations of an Euclidean. Non-Euclidean. e.g., Riemannian, or other configuration.
Let us conclude with the thoughts of Hadamard writing in the Revue generale des Sciences where he suggests that the essence of progress in mathematics must rest upon flexibility and innovation where at every step in its history advances were made possible in mathematics by those who posited the utility or, indeed, the existence of ideas which mathematicians at the time refused to recognize (Borel, 1928).
ACKNOWLEGEMENT. Arthur A. Sagle and Sharon Nasuk Thompson have been of great assistance in these mathematical endeavors. Their contributions are greatly appreciated.