Set Theory 3:
Natural Set Theory
Page last updated 06 Feb 2023
Although there are several different set theories promoted today, they are all of the Non-Natural type of set theory where sets may be elements of other sets, and where sets are treated as objects in the same way as the objects that the sets refer to. 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.
It is important to realize that the differences between a Natural set theory and a Non-Natural set theory are profound, and that they originate from the fact that Natural set theory has its foundations based on empirical observation of the real world, whereas Non-Natural set theory has no real world basis whatsoever. It is also important to realize that the term “naive” set theory as applied to Cantor’s original set theoretical ideas does not apply to Natural set theory - Cantor’s original ideas were not based on the natural everyday concept of a set, and his theory was a Non-Natural set theory (see Part 1: Different types of set theories for an overview of the different types of set theory).
This matters, because the mathematics that is used in commerce, science and technology was developed from fundamental concepts that derived from empirical observation of the real world. One may encounter claims that it is extraordinary how effective mathematics is in science and technology - but these claims conveniently ignore those parts of mathematics which are completely ineffective and useless for any application to real world scenarios. For more on this see Is Mathematics Unreasonably Effective?
For example, natural numbers were an obvious way to reflect the notches on a stick or the balls on an abacus. Starting from those natural numbers, logical manipulations with equations then gave rise to rational numbers, then irrational numbers (both algebraic and transcendental) and then complex numbers. By way of contrast, Cantor’s invented numbers never had any real world origins, and continue to have no innate connection to the real world, with the inevitable result that after over one hundred years they still have no real world applications. This is not surprising since those invented numbers had their origins in Cantor’s religious beliefs.
An example of real world sets
As an example of this fundamental difference between Natural and Non-Natural set theory, suppose that we have, for example, a set of vehicles that we call V, and which consists of five motor cars, three motorbikes, and two vans. We might think of all the vehicles as comprising one set, a set of vehicles. In a natural notion of sets, we can also consider that the set can be subdivided into further sets. In our example we could divide the set of vehicles into three sets: a set of motor cars, a set of motor bikes, and a set of vans. We could also sub-divide the set of vehicles differently, for example, into a set of all vehicles made prior to 1940, and a set of vehicles made in 1940 or later; or we could have a set of vehicles whose rear wheels are driven and a set of vehicles whose rear wheels are not driven. But conventionally, we would acknowledge that the overall set of vehicles remains the same set, regardless of how we subdivide it.
The important thing to recognize about a Natural set theory, is that while we can talk about the set of vehicles as being comprised of a set of motor cars, a set of motor bikes, and a set of vans, that we are not referring to any of these sets as elements of our set of vehicles. We do not think of the set of motor cars as an element of our set of vehicles, nor do we think of the set of motor bikes, or the set of vans as an element of our set of vehicles. We know that the elements of our set of vehicles are five motor cars, three motorbikes, and two vans. When we talk about the set of motor cars, the set of motor bikes, and the set of vans, we are, in fact, talking about these sets not as elements, but as subsets of our set of vehicles. (Footnote: A subset of a set is a set that includes some or all of the elements of a given set.) And conventionally in natural language, we would say that the set of elements that is the set of five motor cars, three motorbikes, and two vans is the same set as the set whose subsets include a set of five motor cars, a set of three motor bikes, and a set of two vans. The reason that we would not consider any of these sets to be elements is because we would not consider that a set, such as the set of five motor cars, is an object in its own right in the same way that each of the things that it refers to is an object.
Now suppose that we remove all the vehicles, while planning to replace them with other ones. In a Natural set theory, and in conventional language, we haven’t got any set of vehicles at all; we haven’t got a set of cars, we haven’t got a set of motorbikes, and we haven’t got a set of vans. In a Natural set theory there is no such thing as an ‘empty set’. This is a notion introduced by mathematicians trying to overcome the deficiencies in Cantor’s original theory; it is a notion that arises directly from all Non-Natural set theories, and which does not occur in natural language. We will have a detailed look at the notion of an ‘empty set’ later.
Sets as elements
The notion that sets can be elements of other sets is at the heart of all the problems in Non-Natural set theories. As demonstrated above, it can easily be seen that this notion is an invention that was not derived from the observation of real world cases. The notion that sets can be elements leads directly to problems because it is one of the ways that Non-Natural set theories can result in the limitless proliferation of “objects” that are not objects in themselves. There is no real-world scenario that is comparable to this concept.
One might encounter the critique that one should be able to talk about sets as mathematical objects, which would seem to imply that they can be elements of sets. But, returning to our example above, we can observe that we can divide the set of all vehicles into a set A that is of vehicles made prior to 1970, and a set B of all vehicles made in 1970 or later, and we can also divide the set of all vehicles into the set C of vehicles whose rear wheels are driven and a set D of vehicles whose rear wheels are not driven. Note that the set A is not the same set as set C, since it includes a Mini which was made prior to 1970 but has only the front wheels driven.
We cannot then say that we have simultaneously in our mathematical system four objects, A, B, C and D, and which is also an accurate isomorphism to the real world scenario, since the entities that it evokes are not in accordance with real-world scenarios - there aren’t four different objects that exist simultaneously and which are:
- a thing that contains vehicles made prior to 1970,
- a thing that contains vehicles made in 1970 or later,
- a thing that contains vehicles whose rear wheels are driven, and
- a thing that contains vehicles whose rear wheels are not driven.
Note that it might be thought that the basis of a Natural set theory is in principle similar to that of a type theory, such as Russell’s theory of types, (Footnote: See, for example Type Theory at Wikipedia or Type Theory at Stanford Encyclopedia.) since they preclude a set being an element of a set unless it is of a higher type. (Footnote: Note that in type theories sets of some levels may be called classes rather than sets, but this makes no difference in principle to the overall result.) However, there is a crucial difference; while superficially a type theory appears to solve the immediate problem that arises from sets being elements of themselves, it nevertheless allows sets to be elements of other sets, which still permits the limitless proliferation of “objects” that are not objects in themselves, which is something that is not reflected in real world scenarios. For example, in type theory, the Power set of a set A is a set of the next higher level, and where each element of this power set is a subset of the set A. Hence type theories are essentially simply another variety of Non-Natural set theory.
We look at Non-Natural set theories on the next page, Non-Natural set theory.