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Copyright   James R Meyer    2012 - 2023 https://www.jamesrmeyer.com

Set Theory 4: Non-Natural Set Theory


 

 


 

Page last updated 14 Jan 2023

 

One might ask why, given that one can have a simple theory of sets, as described on the previous 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.

And the renowned mathematician and philosopher Solomon Feferman (1928-2016) remarked that: (Footnote: Solomon Feferman: ‘Infinity in Mathematics: Is Cantor Necessary?’ in “Infinity in Science”, Instituto dello Enciclopedia Italiana (1987), pp.151‑209, also in the book: In the Light of Logic, OUP on Demand, 1998.)

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. Nevertheless, others followed Gödel’s footsteps along the same path, 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.

 

Numbers in Non-Natural set theory

In Non-Natural set theory, the claim is that all numbers are sets. The ‘empty set’ represents the number 0, the set that contains only the ‘empty set’ represents the number 1, and so on, so that the claim is that: (Footnote: Note that this is not the only possible way of that natural numbers might be represented in conventional set theory, for example, another method is:
0 is { },
1 is {{ }, {{ }}},
2 is {{ }, {{ }}, {{ }, {{ }}}},
3 is {{ }, {{ }}, {{ }, {{ }}}, {{ }, {{ }}, {{ }, {{ }}}}},

but such methods are all based around some nesting of the ‘empty set’.)

{ } represents 0,

{{ }} represents 1,

{{{ }}} represents 2,

and so on, where the number represented corresponds to the degree of nesting of the ‘empty set’. (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.) This covers the natural numbers, and 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.

 

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, mathematicians applied a Sticking Plaster fix to Cantor’s original ideas rather than attempting to discover the root cause of its inherent contradictions. Among these fixes was the notion of an ‘empty set’ which we deal with on our next page: The ‘Empty Set’.

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Copyright   James R Meyer   2012 - 2023
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