Set Theory 1:
Different types of set theories
Page last updated 14 Jan 2023
It’s not uncommon to come across a criticism of conventional set theory - but unless the criticism also suggests a viable alternative its value is questionable. This overview of set theory is different in that it does provide a viable alternative. These pages are an overview of conventional set theories and an alternative to such conventional set theories
Fallacy: Natural set theory is inconsistent
The most important fact regarding set theories is that there are two fundamental types of set theory:
Type 1: Sets can be elements
Type 2: Sets cannot be elements
In conventional mathematical literature you will not find any reference to this distinction between these two fundamentally different types of set theory. Conventional mathematics assumes that the only acceptable form of a mathematical theory of sets must be a theory in which a set may be an element of a set. However, this naive assumption is the root of all the difficulties that are intrinsic to conventional set theory.
Natural set theory
Natural set theory is the term I will use for the type of type of set theory in which sets and elements are completely different types, so that a set cannot be an element of any set. I call it Natural set theory since the way it deals with sets is essentially the way that we deal with the notion of sets in the real world using everyday natural language such as English. Conventionally, in natural language, given any set, we don’t think of that set as being an element. We think of sets and elements as being completely different kinds of things. We don’t think of sets as being objects in themselves, but as a convenient way of referring to multiple objects.
In fact, in English, the actual meaning of the word ‘element’ is that an element is something that doesn’t have any constituent parts, and cannot be divided into other things. The use of the word ‘element’ indicates that it isn’t a set, since in natural language, we generally do think of a set as being a collection of things. (Footnote: In natural language, one wouldn’t usually talk about a set that contains only one thing. However, if you were creating a collection, you might start with just one item, and in this case you would refer to your new collection as having just one item, even though normally a collection would have at least two items. The key point here is that in Natural set theory, a set of one thing doesn’t cause any contradictions.)
We will discuss Natural set theory in more detail in the following pages; if you want to jump ahead, the specific page is Natural Set Theory.
Non-Natural set theory
We will refer to the other fundamental type of set theory, in which sets can be elements of other sets, as ‘Non-Natural’ set theory, since it deals with sets in a manner which is quite different to the way we deal with the notion of sets in everyday natural language. It was Georg Cantor who first made any serious attempt to formulate a mathematical theory of sets. Unfortunately, the theory of sets that Cantor devised is not a Natural set theory, but a Non-Natural set theory in which sets can be elements of other sets, and where sets are treated as objects in exactly the same way as the actual objects that they refer to.
Cantor’s original set theory is not used today in its original form since it was found to lead to severe contradictions such as Russell’s paradox (see below). These contradictions arise directly from the notion that a set can be an element of a set.
We will discuss Non-Natural set theory in more detail in the following pages.
‘Naive’ set theory
You will often see the term ‘naive set theory’. It is important to understand that the term ‘naive set theory’ refers to Non-Natural set theories in which sets may be elements of other sets, in which sets are to be treated in exactly the same way as the objects they refer to. Without additional prohibitions, such theories are contradictory.
Cantor’s ideas regarding sets were a naive set theory, and those ideas have been the basis of all mainstream attempts at set theory since that time, where various rules have been added to try to prevent contradictions arising from those naive ideas.
One should be aware that there is a common erroneous assumption that ‘naive set theory’ was derived from notions of sets that we use in everyday natural language. This is quite simply incorrect, as will be demonstrated on the following pages.
Contradictions in Set Theories
As noted above, the origins of today’s set theories can all be traced back to the ideas of Georg Cantor, who started to develop his ideas regarding sets in the 1870’s.
Unfortunately, as his ideas were developed, some contradictions started to become apparent in the application of these ideas, culminating in a shattering blow that would forever change the face of mathematics. That hammer blow is the subject of the next part of this overview of set theory, and is now known as Russell’s paradox.