Logic and Language

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

# Set Theory 5:The ‘Empty Set’

Page last updated 14 Jan 2023

As demonstrated on the previous pages, an analysis shows that Non-Natural set theory is not a theory about things that, in the real world, we would normally consider to be sets. In normal language, when we talk about sets, we think of a set as a collection of things. Most Non-Natural set theories however, introduce the concept of an ‘empty set’. In normal language, we do not refer to the absence of anything as a set of any kind. Furthermore, the notion of an ‘empty set’ gives rise to other notions which are completely at odds with what we conventionally regard as sets in real world scenarios.

For suppose we have, as in our previous real world example, a set of five motor cars, three motorbikes, and two vans. We might think of all the vehicles as comprising a set of vehicles; or we might think of our collection of vehicles as being comprised of three sets, the set of motor cars, the set of motorbikes, and the set of vans. But now suppose that we remove all the vehicles, while planning to replace them with other ones. Now, since in Non-Natural set theory, we are allowing the notion of an empty set, it might appear that we now have three ‘empty sets’ in our ‘empty set’ of vehicles.

But in Non-Natural set theory, there only is one ‘empty set’; in the theory, you cannot have a set whose elements are three ‘empty sets’. So even if, for the moment, we suppose that we allow the notion of an ‘empty set’, Non-Natural set theory is still not an accurate description of what we would mean in everyday life if we were talking about our collection of vehicles. For if we allow the notion of an empty set, and if we temporarily remove our vehicles, we would expect to say that instead of a set that comprises a set of five cars, a set of three motorbikes, and a set of two vans, that we now would have a set of no cars, a set of no motorbikes, and a set of no vans. That is, we would expect to say that we now have a set of three ‘empty sets’. But, in Non-Natural set theory, we cannot have a set that contains more than one ‘empty set’. That means that in Non-Natural set theory, our ‘empty’ collection of vehicles would have to be either one single ‘empty set’ – or a set that contains just one single ‘empty set’.

The assumption of the existence of a thing called the ‘empty set’ is one of the cornerstones of conventional Non-Natural set theories. But when theorists were trying to rework Cantor’s set theory so that it would avoid Russell’s paradox, while on the one hand the notion of an ‘empty set’ seemed to solve some of the difficulties, there were also deep reservations about any use of a notion of an ‘empty set’. (Footnote: Ernst Zermelo, Letters to Abraham Fraenkel, as quoted in:
Heinz-Dieter Ebbinghaus & Volker Peckhaus, ‘Ernst Zermelo: An Approach to His Life and Work ’, Springer, 2007.
31 March 1921: ‘[The empty set] is not a genuine set and was introduced by me only for formal reasons.
9 May 1921: ‘I increasingly doubt the justifiability of the “null set.” Perhaps one can dispense with it by restricting the axiom of separation in a suitable way. Indeed, it serves only the purpose of formal simplification.)
Today, these reservations have, by and large, been forgotten. (Footnote: See Akihiro Kanamori, ‘The Empty Set, The Singleton, And The Ordered Pair ’, The Bulletin of Symbolic Logic, 9.3 Sept 2003, which describes the emergence of the ‘empty’ set.)

While at first glance the notion of an ‘empty’ set might appear unremarkable, when you stop to think about it, the notion of an ‘empty set’ creates a fundamental difficulty. This assumption of such a thing as an ‘empty set’ completely ignores the original basis for a theory of sets in the first instance – the notion that a set is something whose fundamental property is that it possesses elements. As Cantor said in a footnote to his Grundlagen, Section 1: “By a manifold or set, I generally understand every multiple that can be thought of as one, i.e. every set of certain elements, which can be connected into a whole by a rule.”

Once you claim that a set is something that can have the property that it does not possess any elements, then you have thrown away your original basis for the definition of a set. Once you say that there can be such a thing as an ‘empty set’, then you can’t define a set as something that has elements, since an ‘empty set’ does not have any elements. The rather inconvenient difficulty of what constitutes a set in modern set theory is conveniently put to one side in the hope and expectation that no-one will notice, or care.

It leads to the situation where, in a Non-Natural set theory, everything that the theory refers to has to be a set. You cannot define a Non-Natural set as something that has elements, since the ‘empty set’ is a Non-Natural set, but has no elements. As a result, you will find that Non-Natural set theory only actually defines two principal sets, one being the ‘empty set’, and the other is the infinite set, a set which is defined only in terms of ‘empty sets’, and its elements are the ‘empty set’, the set that contains only the ‘empty set’, the set that contains only the set that contains only the ‘empty set’, and so on – indicated by:

{ { } , {{ }} , {{{ }}} , …}.

From this infinite set, one can define other subsets of this set that contain some, but not all of the sets of this infinite set. And you can define a power set of this infinite set, which is a set that contains all of the subsets of the infinite set. And you can define the power set of that power set. And so on - all based on - absolutely nothing.

Another consequence of the introduction of the notion of the ‘empty set’ is that, since you cannot define sets to be entities that have elements, then every object in your system has to be a set - and this means that there cannot be any objects in your system that are not sets, hence every object in your system has to be a set, and hence every object in your system has to have elements. This creates a difficulty if one is attempting to refer in your system to an object such as a number in the format of Peano arithmetic, such as ffffff  0, since it has to have elements, and its elements have to have elements, and its elements have to have elements, and so on, ad infinitum… . Unless an object of the system is a set composed of hierarchies of sets that each reduce to the ‘empty set’, they are entities that have an limitless hierarchy of sets that have no definable reduction to fundamental elements. This crippling detail is conveniently ignored by the people who actually claim that they use Non-Natural set theory.

But despite all of these problems, proponents of today’s set theory promote the notion that it is a “Theory of everything” for mathematics. We look at this on the next page: Is Set Theory the Root of all Mathematics?

Footnotes:

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