Logic and
Language

Copyright   James R Meyer    2012 - 2024 https://www.jamesrmeyer.com

Page last updated 14 Jan 2023

In 1888 Georg Cantor (1845-1918) wrote of his new theory of sets: (Footnote: Georg Cantor, letter of 21st June 1888 to Carl Friedrich Heman, professor of Theology at the University of Basel as quoted in:
‘Georg Cantor: His Mathematics and Philosophy of the Infinite’ by J W Dauben, Princeton University Press, 1990.)

‘My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things.’

Unfortunately for Cantor, about thirteen years after he proclaimed his convictions as in the above quotation, it was discovered that his theory of sets led directly to contradictions such as Russell’s paradox. This paradox was first publicly announced by Bertrand Russell in 1901, who observed that you can create a definition which is:

‘The set of all sets that are not elements of themselves.’

Now consider the question of whether the set that is itself defined by this definition – is it an element of itself or not? If we suppose that it is not an element of itself, then it satisfies the definition; and since it satisfies that definition, that means that it must be an element of itself.

But that is a contradiction, since the definition stipulates that elements of the set cannot be elements of themselves. On the other hand, if we suppose that it is an element of itself, then, as is the case for all sets, that element must satisfy the definition of the set. But the definition of the set says that each element of the set cannot be an element of itself. Again we have a contradiction, for if it satisfies the definition of itself, then it cannot be an element of itself.

Either way, there is a contradiction.

## Set theory’s sticking plaster

After Russell discovered this paradox, mathematicians and logicians spent many long years trying to patch up their current ideas on set theory (Footnote: These ideas were a form of Non-Natural Set theory as described in Part 1: Different types of set theories.) so that this paradox might be avoided. In these attempts to avoid the paradox, these set theorists could find no simple way to avoid the paradox; they found that they had to resort to attaching all sorts of assumptions onto their set theories that were not intuitively satisfactory, and whose function was to prohibit certain types of operations on sets. Unlike the assumptions that had previously been used in mathematics up to then, these were not assumptions that one would say were obvious, nor were they based on any empirical observations of real world scenarios, nor were they aesthetically pleasing. After Russell’s notification of the existence of his eponymous paradox, it took about 30 years, much discussion and many papers to come up with a fix that would become the first generally accepted fix to the problem, and which is known as Zermelo–Fraenkel set theory, defined by axioms known as the Zermelo–Fraenkel axioms.

Unfortunately, instead of delving down to the very roots of the problem, these axioms were invented simply as a sticking plaster, as a quick fix, a fix to avoid the problems arising from Russell’s paradox. All attempts at such fixes assumed that any mathematical set theory must be a Non-Natural type of set theory. Quite why that was the case is a mystery.

It is a mystery because, in a Natural Set Theory, the definition that is Russell’s paradox simply defines the set that contains every element. And that does not result in any contradiction in a Natural set theory - in Natural set theory Russell’s ‘paradox’ is not a paradox at all. Before going into any more detail, we first we need to recognize that the Russell’s paradox sentence could mean either:

(a) ‘the set of all elements that are sets that are not elements of themselves’

or

(b) ‘the set of all sets that are sets that are not elements of themselves’

In case (a) in a Natural set theory, since no sets can be elements, the sentence does not define a set at all - in fact, in Natural set theory, any sentence that begins with:

‘the set of all elements that are sets …’

does not define any set at all. Here we see a crucial difference between most Non-Natural set theories and Natural set theory – in most Non-Natural set theories, given any definition of a set, that set necessarily must exist, either as an ‘empty set’ with no elements, or as a set that has elements. On the other hand, in a Natural set theory, simply because there is a definition that asserts there is a set, that does not necessarily mean that it is a valid definition and that some set that satisfies that definition must exist. In a Natural set theory there is no such thing as an ‘empty set’, since in a Natural set theory, a set must always have at least one element. (Footnote: We will look at the notion of an ‘empty set’ in more detail in section 5 The ‘Empty Set’.) In Natural set theory, since elements cannot be sets, and sets cannot be elements, the definition ‘the set of all elements that are sets that are not elements of themselves’ does not define a set at all.

Of course, given a universal set that includes everything of a defined domain, there can be various sets that have only some of the elements of that set, and these are ‘subsets’. One such subset of any set is the set of every element of that set – the set itself. That doesn’t elicit any paradox.

Moving on to case (b) above, ‘the set of all sets that are sets that are not elements of themselves’, since every set cannot be an element of itself, then the sentence defines the set of all possible sets; that includes the set itself – but there is no paradox in a set being a subset of itself. (Footnote: A subset of a set is a set that includes some or all of the elements of a given set; a proper subset only includes some but not all of the elements of the given set. The subset that is the set itself is simply an identity.) Again, there is no contradiction, no paradox, and no problem – and the commonly cited ‘necessity’ for Non-Natural set theories with complex and strange axioms can be seen to not be a ‘necessity’ at all.

We can consider further questions, for example, what happens if we state Russell’s paradox as: ‘the set of all sets that are sets that are not subsets of themselves’. In a Natural set theory, every set is a subset of itself, so that this definition would not define a set at all in a Natural set theory. Again, there is no contradiction, there is no paradox. Our next page is an overview of Natural set theory.

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