Set Theory 6:
Is Set Theory the Root of all Mathematics?
Page last updated 30 Mar 2023
Since the arrival of Quantum Mechanics and Einstein’s theory of General Relativity in the first part of the 20th century, physicists have hoped that they might discover a ‘Theory of Everything’ that would explain all of the physical world. As yet, no-one knows if we will ever be able to come up with such a theory. On the other hand, mathematicians claim that today’s Non-Natural set theory really is a ‘Theory of Everything’ for mathematics; they claim that it provides the core foundation for all of mathematics for all time. But Non-Natural set theory is flawed at its very core, and the notion that it should be the foundation stone of all mathematics is preposterous.
And today, 100 years after Ernst Zermelo and Abraham Fraenkel set out their axioms to try to circumvent the contradictions of non-natural set theory, mathematicians are still arguing whether that collection of unwieldy axioms is the fundamentally correct basis for all mathematics for all time, see, for example To Settle Infinity Dispute, a New Law of Logic or How Many Numbers Exist? Somehow, they fail to see the irony of the claim that their non-natural set theory must be the fundamental basis of mathematics for all time while at the same time 100 years of effort has failed to provide a rock solid basis for the claim.
An irrational number is often defined by reference to a “Cauchy sequence”, which defines an irrational number by referring to a limitlessly large set of ever-decreasing rational numbers. For example, Pi can be defined as the limiting value of the infinite series 4 - 4⁄3 + 4⁄5 - 4⁄7 + 4⁄9 + … (Liebniz’s series), and as the limiting value of the series 3 + 4⁄24 - 4⁄120 + 4⁄336 - 4⁄720 … (Nilakantha’s Series). In terms of Non-Natural set theory, one might define the irrational number to be the set that is the summation of all those ever-decreasing rational numbers. Of course, a real number is commonly asserted to be just this, for example, in the decimal system, a real number is usually asserted to be something such as 89.238467612… where the real meaning of the dots is conveniently ignored.
However, such a definition as an actual summation of a limitless quantity of numbers leads us straight into contradictions, such as in Sums of infinitely many fractions and The Courant & Robbins Contradiction.
On the other hand, without using Non-Natural set theory, we could define the irrational number as the limiting value of all finite sums of such rationals, and we don’t arrive at these contradictions.
One might believe that the difficulty might be evaded by defining a set A as the set of all finite sums of all the ever-decreasing rational numbers, and that from this set one can claim that there is a “supremum” of this set A, which is a value that is greater than every element of A.
However, there are problems with this within Non-Natural set theory, one problem being that, in Non-Natural set theory, since everything is a set, this supremum must be a set, yet it has been defined without any reference to what its elements might be. This is a problem for Non-Natural set theory; since there are infinitely many such supremum sets, we now have infinitely many sets for which there is no way of determining what the elements of such supremum sets are.
With such a Non-Natural set theory it can be seen that, for example, the number Pi arises not as a set defined by direct reference to the properties of its elements, but by a definition that refers to infintely many sets that are deemed to represent ever-decreasing rational numbers. But how does a real world mathematician refer to the number Pi? By a definition that refers to ever-decreasing rational numbers, in terms of the limit value of all finite sums of such rationals, without any need to refer to complex infinite sets.
So, despite the persistent claim that Non-Natural set theory encompasses all of mathematics, in fact it doesn’t add anything of any use to the mathematics that is used in real world science and technology. But the unrelenting contention continues that Non-Natural set theory is the foundation for all mathematics for all time, that it is indispensable to the foundations of mathematics. All that has happened is that Non-Natural set theory attached its definitions onto the definitions that were already defined for ordinary real world mathematics. There’s nothing there to indicate that Non-Natural set theory has some extra essential basis of mathematics that ordinary real world mathematics doesn’t have. As the mathematician William Thurston (1946-2012) remarked: (Footnote: William Thurston, quoted in article, ‘The Death of Proof ’, by John Horgan, Scientific American, October 1993.)
‘Set theory is based on polite lies, things we agree on even though we know they’re not true. In some ways, the foundations of mathematics have an air of unreality’.
Numbers and set theory
In Non-Natural set theory, numbers are sets that that can be elements of other sets, but do we want - or do we even need - to be able to refer to numbers as sets that can themselves be elements of sets? The notion that numbers should be sets only arose following Cantor’s ideas - prior to Cantor the notion that numbers should be sets did not exist, and yet mathematics had been developed before that without any need for the notion, and indeed, continues to do so. There is no logical imperative that dictates that the best mathematical theory is one where numbers should be sets. Peano arithmetic provides a simple starting point that does not require any sets, see for example: The Peano Axioms at Wolfram.
It would appear that in real world applications there is no need for any requirement that sets can be elements - that all the mathematics that is used in real world applications has no need of that concept. Of course, if numbers are not sets, and if there are no sets that are numbers, then the whole edifice of transfinite sets as transfinite numbers within Non-Natural set theory has its foundations of sand pulled out from under it - and this is no surprise since, as demonstrated elsewhere on this site, the notion of transfinite numbers is replete with contradictions that would not be tolerated in any science. (Footnote: Of course, a set may have a property that corresponds to the quantity of its elements. But it is the notion that that property itself also is a set that results in all the problems that arise from the notion that numbers should be sets, since it allows the generation of an endless hierarchy of “objects” where every such “object” is generated from the concept in Non-Natural set theory that there exists an “object” that is a “set” that contains absolutely nothing.) On the other hand, in a Natural set theory, where elements cannot be sets, every number is finite (as in the case for real world numbers), while there is no difficulty in defining sets that have no limit to the quantity of their elements, while not in any way implying that there “exists” any sort of simultaneous collection of those elements. For example, any element of the set of natural numbers is finite, while the set of natural numbers is limitless. This subtle distinction between finite entities and infinite collections is obliterated by conventional set theory since it treats sets and the elements of sets in exactly the same way. (Footnote: Note that in Natural set theory, one can have a concept represented by a symbol such as ∞, but this is not itself a real number, and hence cannot be an element of a set of real numbers; but the concept can be used as in what is called the extended real number line. Rules can be defined where ∞ can be treated as a number where some additional rules apply to that symbol, but this does not imply that ∞ actually is a real number.)
At this point it is important to realize that when a set theorist talks about a Non-Natural set as a number, although it is commonly assumed that such a “set~number” is exactly the same, in every respect, as a number that we refer to in normal English terminology and in real world mathematics, the reality is that the properties of these “set~numbers” are not identical to those of numbers used in real world mathematics. And since that is the case, it is an illogical leap of faith to simply assume that such “set~numbers” will always behave like the real numbers of real world mathematics. They don’t, the use of the same term “number” for two different definitions does not make those definitions equivalent.