Set Theory 5:
Non-Natural Set Theory

Page last updated 14 Sep 2024

One might ask why, given that one can have a simple theory of sets, as described previously on the page Natural set theory, and which avoids Russell’s paradox and similar paradoxes, might we need a set theory with a complex collection of strange assumptions that have no real world basis at all? The story of the development of today’s set theories puts one in mind of the stereotypical absent-minded philosopher who has become so out of touch with reality that he has become convinced that something exists despite there being absolutely no evidence for its existence. It is the story of theorists that were so imbued with the pervasive conviction that a set theory must be able to have sets that can be elements of sets that they spent their entire time trying to remove the contradictions that this conviction creates, rather than questioning whether it might be that original conviction that might itself be the problem.

All of the set theories promoted today are Non-Natural set theories where sets may be elements of sets. The set theory that is most commonly favored is a theory called Zermelo-Fraenkel set theory, commonly called ZF set theory. There are also several other Non-Natural set theories, but they all share similar assumptions to those that underpin Zermelo-Fraenkel set theory. Unlike Natural set theory, none of them have origins based on empirical observation of the real world.

The origins of these theories can all be traced back to Georg Cantor, whose ideas were closely intertwined with his religious beliefs (see Cantor’s religious beliefs). He initiated the notion that a system of mathematics could be built upon the idea of sets being elements of sets, and since then, mathematicians became so enamored with the notion that that variety of set theory could be the underlying foundation of all of mathematics that a new concept of foundational mathematics began to take shape - the concept that it doesn’t matter if a mathematical theory has no underlying real-world basis at all.

Instead the notion arose that if a mathematical theory seemed to work and had fixes that appeared to prevent any contradictions then surely it couldn’t be wrong to say that statements that the theory came up with had to be ‘true’. It was Georg Cantor himself who started the ball rolling with his plea that: (Footnote: From Cantor, § 8 of “Grundlagen einer allgemeinen Mannigfaltigkeitslehre” (Foundations of a general theory of sets), Mathematische Annalen 21, Leipzig 1883, as in p 182, Gesammelte Abhandlungen: mathematischen und philosophischen Inhalts, ed Zermelo, Springer-Verlag, republished, 2013 (my translation).)

Mathematics is completely free in its development and is only bound to the self-evident consideration that its concepts are both free of contradictions and that they are in fixed relationships to proven concepts that have already been previously established.

And while he also remarked in the same paper that:

…every mathematical concept also carries the necessary corrective in itself; if it is sterile or inexpedient, it very soon shows it through its uselessness and it is then dropped because of lack of success.

the fact is that contradictions that arose from Non-Natural set theory have still not been eliminated despite over 100 years of effort - and yet Non-Natural set theory is still promoted, not only as valid mathematics, but as the only possible foundation for mathematics for all time - despite the fact that the theory allows the limitless proliferation of “objects” that are not themselves actual objects, and despite the fact that the theory has no usefulness at all in the real world, not in technology and not in science. As the mathematician Nik Weaver (b. 1969) has said: (Footnote: Nik Weaver, ‘The Concept Of A Set’, arXiv:0905.1677, 2009)

‘Virtually all modern mathematics outside set theory itself can be carried out in formal systems which are far weaker than Zermelo-Fraenkel set theory and which can be justified in very concrete terms without invoking any supernatural universe of sets … axiomatic set theory is not indispensable to mathematical practice, as most philosophers of mathematics have apparently assumed it to be. It is one arena in which mathematics can be formalized, but it is not the only one, nor even necessarily the best one.’

‘I am convinced that the Platonism which underlies Cantorian set theory is utterly unsatisfactory as a philosophy of our subject … Platonism is the medieval metaphysics of mathematics; surely we can do better.’

But the Platonist path that was set out by Cantor was seized on and followed by mathematicians of the time and also subsequently. Kurt Gödel was a prominent proponent of Platonism and Non-Natural set theory, and he claimed: (Footnote: Kurt Gödel, ‘What is Cantor’s continuum problem?’ American Mathematical Monthly, vol. 54 (1947).)

‘But, despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves on us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception, which induces us to build up physical theories and to expect that future sense perceptions will agree with them, and, moreover, to believe that a question not decidable now has meaning and may be decided in the future.’

But it is difficult to fathom how axioms might “force themselves” on anyone as being “true” when they result in the infinite proliferation of things that that have no analogue in the real world, and which result in limitlessly large sets that can somehow have fewer elements than other limitlessly large sets - and when they result in an irresolvable fundamental contradiction as we shall demonstrate below. Nevertheless, others followed Gödel’s footsteps along the same path, and among them was Paul Cohen (1934-2007), who pleaded that we should reject any scientific approach, and that although the axioms of set theory are not intuitive, we should instead grope in a search for truth by some sort of quasi-spiritual mysticism: (Footnote: Paul J. Cohen, Comments on the Foundations of Set Theory, in: Axiomatic Set Theory, ed. D. Scott, Proceedings of Symposia in Pure Mathematics, Vol. 13, Part 1, Amer. Math. Soc., Providence, RI, pp. 9‑15, p.15 (1971).)

‘Our intuition is probably not yet sufficiently developed or at least one cannot communicate it. Nevertheless I feel that this is a useful task, to develop our mystical feeling for which axioms should be accepted. Here of course, we must abandon the scientific program entirely and return to an almost instinctual level.’

Going back to our previous real world example, a set of five motor cars, three motorbikes, and two vans, in Non-Natural set theory, the set whose subsets include a set of five motor cars, a set of three motor bikes, and a set of two vans is a different set to the set whose elements are a set of five motor cars, a set of three motor bikes, and a set of two vans. That is to say, in Non-Natural set theory, the set:

{car 1, car 2, car 3, car 4, car 5, bike 1, bike 2, bike 3, van 1, van 2}

which is the set whose subsets include a set of five motor cars, a set of three motor bikes, and a set of two vans is a different set to the set:

{ { car 1, car 2, car 3, car 4, car 5} , {bike 1, bike 2, bike 3} , {van 1, van 2} }

which is the set whose elements are a set of five motor cars, a set of three motor bikes, and a set of two vans.

And yet they refer to the one single real world scenario. Obviously this sort of Non-Natural set theory does not accurately describe the reality of the physical scenario. Because of the strange way that Non-Natural set theory refers to things, it seems rather odd to call Non-Natural set theory a theory about ‘sets’, where the natural language word ‘set’ is used in a way that is completely alien to its meaning in natural language. It is a theory about entities that are cannot be considered to be sets in the conventional sense of natural language.

But, despite the fact that Non-Natural set theory has never found any application to real world scenarios, its proponents point to instances where a sector of mathematics had been developed and at the time, no-one could envisage any real world application for it - and yet, years later, it was found to have a real world application. Set theorists postulate that could happen in the future for set theory. But that ignores the simple fact that the reason for real world applications being discovered for previously developed mathematics is very simple - all such cases occurred because that sector of mathematics had been developed from a basis that had its original foundations derived from empirical real world observation.

In real world science, theories fall into disuse when they no longer can compete with other theories to provide a model of the real world scenario being considered. And despite Non-Natural set theory being around for over 100 years, real world science has never found any use for Non-Natural set theory. The reason for this should be obvious – it is because the underlying assumptions behind Non-Natural set theory have no real world basis.

As remarked previously in the section on Russell’s paradox, mathematicians applied a Sticking Plaster fix to Cantor’s original ideas rather than attempting to discover the root cause of its inherent contradictions. But these fixes failed to eliminate a fundamental contradiction that lies within the very essence of Non-Natural set theory.

The Contradiction of Sets as Numbers

As already noted on the previous page The ‘Empty Set’, the notion of an empty set in Non-Natural set theories such as Zermelo-Fraenkel set theory forces every object in the theory to be a set and there are no entities that are not sets. This obfuscates the subtle distinction between single entities that are finite and collections that are infinite, and we will see that this results in a contradiction. Since everything is a set and there are no entities that are not sets, there is necessarily the inherent concomitant claim that all numbers are sets. One way that this is achieved in Non-Natural set theory is that the ‘empty set’ represents the number 0, the set that contains only the ‘empty set’ represents the number 1, the set that contains the ‘empty set’ as well as a set that contains the ‘empty set’ represents the number 2, etc, giving the following, where ∅ is a symbol for the ‘empty set’: (Footnote: Note that this is not the only possible way of that natural numbers might be represented in conventional set theory, but such methods are all based around some nesting of the ‘empty set’. Given such a method, with a bit of manipulation you can define sets that correspond to rational numbers. But to actually define an irrational number or a complex number in Non-Natural set theory would involve infinitely large sets that are all sets that are composed of elements that are all some sort of hierarchy of nested ‘empty sets’. And because defining an actual irrational number as a set in Non-Natural set theory is so horrendously difficult, nobody working in real world science or technology ever works with irrational numbers using Non-Natural set theory. We will talk a bit more about irrational numbers in Non-Natural set theory later in Myth: Non-Natural Set theory is all of Mathematics.)

0 is ∅ ,
1 is {∅},
2 is {∅, {∅}},
3 is {∅, {∅}, {∅, {∅}}},
4 is {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}}, 

and so on, where the number represented corresponds to a set containing that number of sets. For convenience in the following, we will refer to these sets as ‘natural number sets’. (Footnote: Note that the principle here is essentially the same basis as is used in the definition of the natural numbers as in Peano arithmetic where zero as the symbol 0 as a foundational base number, and a successor symbol s which gives other numbers: 0, s0, ss0, sss0, etc. This method does not require any sets to give a valid foundation for the natural numbers, only simple mathematical symbols which are manipulated by a simple system of rules.)

Now, natural numbers and real numbers are all finite - every natural number enumerates all natural numbers up to itself, that is, for every natural number, there is another natural number that is exactly one less than that number, and there are exactly that finite quantity of unit decrementations to reach zero, so there cannot be any infinite natural number - if there were then there would have to be a unit decrementation from an infinitely large natural number to some finitely large number, which is impossible. And for the same reason, there cannot be an infinitely large real number.

Now consider a fundamental set theoretical operation - the union of two or more sets; for example the union of the set 2 and the set 4 is the set {∅, {∅}, {∅, {∅}}, {∅, {∅}, {∅, {∅}}}}, which of course is the set that is the number 4, since in any union of sets, common elements are not duplicated, and every element of 2 is also an element of 4. (Footnote: Note that the concept of union of two or more sets is quite different to the notion of the summation of two or more natural numbers.)

And now consider the union of all natural number sets - and immediately we have a problem. That union cannot be finite, since that would imply that there is a largest finite natural number which of course is impossible. On the other hand, that union cannot be infinite either, since that would imply that among the natural number sets there exists at least one that is more than one greater than any other natural number, which again is impossible.

Either way there is a contradiction.

This contradiction demonstrates the crass absurdity of the common claim that Non-Natural set theory provides the best possible foundation for all of mathematics, a theory in which it is fundamental that numbers are sets, but Non-Natural set theory has a horrendous contradiction at its very core. On the next page Myth: Non-Natural Set theory is all of Mathematics we will examine further the risible claim that Non-Natural set theory is the best possible foundation for mathematics.