Page last updated 25 April 2022
Note: for information on transfinite ordinal numbers rather than cardinal numbers, please see The Origins of Transfinite Numbers.
When one reads books on mathematics aimed at a popular readership and which eulogize the concept of transfinite numbers, a concept invented by Georg Cantor (see Cantor’s invented numbers) one might get the impression that a clear definition of “cardinal number ” is a simple straightforward matter. For a simple demonstration as to why this is not the case, one might observe the attempt at such a definition at Wikipedia’s formal definition of a Cardinal number where it is admitted that there is no straightforward definition of a cardinal number without relying on abstruse Platonist assumptions such as the Axiom of Choice and the notion of transfinite numbers. (Footnote: You can also read online an English translation of one of Cantor’s major works, Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Foundations of a general theory of sets), which lays out his philosophy on different sizes of infinity and cardinal-numbers.)
It should not be surprising that this is the case. For finite sets the definition is quite straightforward and the definition of the cardinal number for any finite set is given by a one-to-one correspondence with a subset of the natural numbers (see Definition D below), whereas for infinite sets, a transfinite cardinal number is given not by the existence of a one-to-one correspondence, but by the absence of a one-to-one correspondence to some other set. Unfortunately for conventional mathematics, this does not actually give a definitive cardinal number for many infinite sets, which means that ad hoc assumptions must be tagged onto this theory of transfinite numbers, see for example The Continuum Hypothesis.
The notion that different infinite sets can have different cardinal numbers leads directly to a contradiction - that a limitlessly large set can be smaller than some other limitlessly large set. Some people have tried to evade this contradiction by claiming that a cardinal number of an infinite set is not a measure of the quantity of its elements. But the cardinality of any finite set is a measure of the quantity of its elements and the measures of any two finite sets differ by some definitive measure (which may be zero). If it is not zero then, ipso facto, then one set has fewer elements than the other. And extending that concept to infinite sets necessarily requires that the cardinality of an infinite set is also a measure of the quantity of its elements. It is still the case that any two cardinal numbers differ by a definitive measure (which may be zero) and if it is not zero, then, ipso facto, then one set has fewer elements than the other.
The rather troubling inherent problem of how transfinite cardinal numbers should be defined, along with the aforementioned contradiction - is generated by a fundamental axiomatic assumption. That fundamental axiomatic assumption is within the definition:
Definition A: If and only if a one-to-one correspondence exists between two sets then both sets have the same numerical property (termed ‘cardinal number ’) and which satisfies the comparators ‘equals’, ‘less than’ and ‘greater than’. (Footnote: Note that this does not require that for different sets the property must always satisfy the same definition.) (Footnote: Note that there are different versions of this definition, but the fundamental assumption is always included in some way.)
Note B: Note that this definition does not define what the cardinal number for any particular set is, nor how it is calculated.
Note C: Note that by this definition, every set must have some such cardinal number, since for every set there exists a one-to-one correspondence between it and some other set. (Footnote: Note that the notion of an empty set is not a notion that is essential to any set theory. The term is an oxymoron, since any attempt to define a set as something that contains elements immediately precludes an empty set. When Zermelo and others were trying to reformulate set theory to remove the set paradoxes, there were considerable misgivings about the inclusion of the notion of an empty set for that reason, see: 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.” and an article that describes the emergence of the ‘empty’ set: Akihiro Kanamori, ‘The Empty Set, The Singleton, And The Ordered Pair’, The Bulletin of Symbolic Logic, 9.3, Sept 2003. )
The corollary of Definition A, together with Note C, is that if two sets cannot be put in a one-to-one correspondence then they both have cardinal numbers, and those cardinal numbers must be different. Furthermore, it follows from the Diagonal proof that since there is not a one-to-one correspondence between the set of real numbers and the set of natural numbers, then Definition A forces them to have different cardinal numbers. Hence Definition A gives rise to the bizarre contradiction that there can be limitlessly large sets that are smaller than other limitlessly large sets. Note that you will not see conventional mathematics using these terms since they make it so uncomfortably obvious how absurd the contradictory notion is. Instead they use euphemisms like “the set A has a smaller size than the set B ”.
Instead of the fundamental axiomatic Definition A we can simply define a definition without using “If and only if ”. Note that the use of the term “If and only if ” means that one condition (having a one-to-one correspondence) necessarily implies the other (having different cardinal numbers), and vice-versa. But if we simply use “if ” this gives us:
Definition D: If a one-to-one correspondence exists between two sets then both sets have the same numerical property (termed ‘cardinal number ’) and which satisfies the comparators ‘equals’, ‘less than’ and ‘greater than’. (Footnote: As for Definition A, this does not define what the cardinal number for any particular set is, nor how it is calculated; that must be defined separately.)
The corollary of Definition D is that if there is not a one-to-one correspondence between two sets R and N, those two sets can nevertheless have the same cardinal number; this applies if the sets are infinite, and so the cardinal number of both sets is infinite. Furthermore, there can be a one-to-one correspondence between such a set R and a set R2 where both have the same cardinal number, which is infinite (such as where R is the set of real numbers between 0 and 1 and R2 is the set of real numbers between 0 and 2). And there can be a one-to-one correspondence between such a set N and a set Q where both have the same cardinal number, which is infinite (such as the set N of natural numbers and the set Q of rational numbers).
Note that in this case, regardless of whatever cardinal number two sets have in common, in the same way as for Definition A, nothing in Definition D requires that the same definition for ‘cardinal number ’ applies for finite sets and for infinite sets. It only requires that if there is a one-to-one correspondence between two sets, then those two sets have the same numerical property (‘cardinal number’), which need not satisfy the same definition as a common numerical property of two other sets.
This means that we can define that if there exists a one-to-one correspondence between the elements of a set A and a specific proper subset B of the natural numbers (Footnote: For the purposes of this section, all references to natural numbers are to natural numbers excluding zero.) where that subset is the proper subset which includes all natural numbers less than its largest natural number, then we define the cardinal number for the set A to be the largest number in that set B. This defines ‘cardinal number’ for finite sets.
We can also define that if there does not exist a one-to-one correspondence between the elements of a set A and any subset B of the natural numbers where the subset has a largest natural number, then the cardinal number of that set A is infinitely larger than any natural number, and we can apply the term infinite for that cardinal number.
The above gives us only finite cardinal numbers and a single non-finite infinitely large cardinal number, and no contradictions, in contrast to the unfounded Definition A that necessarily implies that different infinite sets can have different cardinal numbers.
Absence of any proof
Besides the fact that the definition of cardinality does not prove that there can be different infinite sets that have different sizes, the fact is that there has never been any rigorous proof of that claim, see Proof of more Real numbers than Natural numbers.
Furthermore, it is a commonly accepted mathematical proof principle that if it can be demonstrated that an argument which entails certain assumptions leads to a contradiction, then one or more of those assumptions is untenable or there is one or more illogical steps in the argument. But for some bizarre reason, for the case of infinite sets mathematicians have turned this principle completely upside down by assuming that if two infinite sets have no one-to-one correspondence, then if they were the same size, then that would be a contradiction. They then proceed to “solve” this supposed problem by replacing it with a blatant contradiction - that one limitlessly large quantity can be smaller than another limitlessly large quantity.
The conventional definition of cardinality ignores a fundamental difference between how a one-to-one correspondence can be defined between finite sets and between infinite sets. For finite sets, a one-to-one correspondence can always be given without any reference to the properties of the elements of the sets, by simply creating an actual list (e.g. 1:A, 2:B, 3:C). But for infinite sets this is not possible, and a one-to-one correspondence can only be given by a definition where the definition defines a correspondence in terms of the properties of the elements of the sets. (Footnote: For example, the function 2x defines a one-to-one correspondence between the set of positive integers and the set of even positive integers.) It follows that the determination of whether a one-to-one correspondence can be defined between two infinite sets depends on the properties of the elements of the sets. This means that the impossibility of defining a one-to-one correspondence between two infinite sets would appear to imply that this impossibility depends on the properties of the elements of the sets rather than the quantity of the elements of infinite sets. And, in fact, the Power set proof actually depends on the definition of a subset which bestows on that subset a property which is specifically such that there can be no definition of a one-to-one correspondence between the two infinite sets involved.
As is the case for all mathematical assertions, if anyone wishes to claim that the impossibility of defining a one-to-one correspondence between two infinite sets is a result of differences in the quantities of the elements of the sets, rather than one or more properties of one or more elements, then that person is obliged to provide a proof of that hypothesis. Otherwise, it is logical to summarily reject that claim, particularly when it results in a blatant contradiction.
Why do people continue to defend the indefensible?
It is an intriguing question - why do people prefer to continue to believe something that is blatantly contradictory - that there can be limitlessly large sets that are smaller than other limitlessly large sets? Why would anyone decide to continue to believe it when, despite the claim being prevalent for well over a hundred years, there has never been anything even approaching a rigorous proof of the claim? Why is there an immediate knee-jerk reaction against any suggestion that one might be able to remove this contradiction by a full logical analysis of everything involved? See Why do people believe weird things? for some of the reasons for this strange attitude. Also see the appendix below.
For further reading, see also the Diagonal proof, the secondary argument of the Diagonal proof, and A List with no Diagonal number and Proof of more Real numbers than Natural numbers, and the papers On Considerations of Language in the Diagonal Proof and On the Reality of the Continuum and Russell’s Moment of Candour.
Appendix: A typical defence of different sizes of infinity
The multi-talented mathematician Eugenia Cheng, who also is a lecturer and pianist, has written a book aimed at the popular market entitled Beyond Infinity. (Footnote: Eugenia Cheng, Beyond infinity: An expedition to the outer limits of mathematics, Basic Books, 2017.) In the book she says that, having found that there is no one-to-one correspondence of natural numbers to real numbers, that:
Intuitively this means that there must be “more” real numbers than natural numbers, but what could this possibly mean if they’re both infinite? Some infinities are bigger than others – how is that possible, seeing as infinity is already infinitely big? Isn’t it the biggest thing that there is? How can anything be bigger than it?
Cheng answers her own question by continuing:
Just like questions of the soul, everlasting life, and whether or not I’m fat, this comes down to definitions. What is the definition of “fat”? In the case of infinity the question is: What is the definition of “big”?
She then proclaims that the definition of big for sets is to be called cardinality, and says:
The cardinality of a set of things is a measure of how many things there are in it.
If the set only contains a finite number of things, then its cardinality is simply the number of things in it. If it contains an infinite number of things, it’s more complicated.
…the smallest possible cardinality is 0. After that there are all the finite possibilities: a set with 1 object, a set with 2 objects, a set with n objects for any finite n.
It turns out that the natural numbers are the smallest possible infinite set.
But she has already established that the natural numbers have a limitless quantity of elements. So, she has claimed that a definition will remove the contradiction that arises from the notion that some sets with a limitless quantity of elements can have more elements than another set with a limitless quantity of elements, and then conveniently claimed that while for finite sets cardinality is the number of elements in the set, it means something different for larger sets, but Cheng never manages to tell us what that actually is - although she admits that for the infinite set of natural numbers, the cardinality of that set is the quantity of elements in that set.
Cheng tells us that while the set of natural numbers is the “smallest possible infinite set ”:
…we have seen only one set that is genuinely bigger: the set of real numbers. The question is: Is this the next infinity up? This is a very difficult issue….
It’s not a difficult issue. According to conventional set theory, it is not decidable whether the size of the set of real numbers is the next biggest infinity after natural numbers. So the question, according to conventional set theory, can only be settled by an arbitrary choice. One person can choose, without any difficulty at all, that the set of real numbers is the next biggest infinity after natural numbers, another person can choose that the set of real numbers is not the next biggest infinity after natural numbers. And another person can decide that the contradiction that arises from the notion that some sets with a limitless quantity of elements can have more elements than another set with a limitless quantity of elements is illogical and invalidates the notion that infinite sets can be of different sizes - but the set theory police don’t allow that.
Later Cheng says of cardinal arithmetic:
This is called “cardinal arithmetic,” because cardinality is just about how many things there are.
What she is saying is literally insane. Having previously asserted that the solution to the contradiction that arises from the notion that some sets with a limitless quantity of elements can have more elements than another set with a limitless quantity of elements “comes down to definitions ”, the only assertion that she has made approaching a definition is that:
The cardinality of a set of things is a measure of how many things there are in it.
and blithely continues to refer to infinite sets in terms such as “cardinality is just about how many things there are”.
It is worth noting that Cheng has also written a book about logic The Art of Logic (Footnote: Eugenia Cheng, The Art of Logic: How to Make Sense in a World that Doesn’t, Profile Books, 2018.) where she says:
Logic is to mathematics as evidence is to science. That is to say that the role of logic in mathematics is analogous to the role of evidence in science, but logic and evidence are fundamentally different. Unlike evidence, logic tells us when something has to be true, not by cause and effect, not by probability, not by observation, but by something inherent that will never ever change.
But her solution to the contradiction that arises from the notion that some sets with a limitless quantity of elements can have more elements than another set with a limitless quantity of elements, by her own admission is to try and sweep it under the carpet by claiming that it can be solved by applying an arbitrary definition. Cheng fails to see the irony that her mathematical notion of different sizes of infinity isn’t an edifice built upon a solid foundation - it is an edifice built upon the sand of an arbitrary definition. It isn’t inherently ‘true’, and it isn’t immune to change. It can be changed simply by changing the definition. Perhaps one day mathematicians will acknowledge the folly of building an edifice on sand.
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