Footnotes:
Cardinal Numbers
Page last updated 2 Jun 2023
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” in transfinite set theory is a simple straightforward matter.
It should not be surprising that this is the case. For finite sets the definition is quite straightforward and there are various ways to define cardinal numbers for finite sets. Prior to transfinite set theory, the number of things in any collection was either, for finite sets, simply that number, or otherwise the number of things was simply infinite. But in transfinite cardinal number theory there are infinite sets for which the theory cannot actually assign a definitive cardinal number to them - which means that ad hoc assumptions must be tagged onto the theory, see for example The Continuum Hypothesis.
The key to understanding the determination of a cardinal number lies within the concept of a one-to-one correspondence between two sets, where each element of one set is paired up uniquely with one element of the other set.
One-to-one Correspondences and Properties
For finite sets that have the same number of elements, 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). Replacing any element by some other element (not already in the set) cannot change the cardinal number of the set, so that the cardinal number of any finite set is a property that is independent of the properties of its elements.
For infinite sets it is not possible to create an actual list (since such a list would be infinitely long). A one-to-one correspondence can only be given by a definition of a function that defines that one-to-one 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.)
While the impossibility of defining a one-to-one correspondence between two finite sets implies that there is a difference in a property of the sets, where that property is independent of the actual details of the elements of the sets, the impossibility of defining a one-to-one correspondence between two infinite sets implies that there is a difference in the properties of the elements of the sets, rather than a difference in a property of the sets that is independent of the actual details of the elements of the sets. The same applies to the impossibility of a surjection between two infinite sets, this is discussed in more detail on the page Surjections and Cardinal Numbers.
But transfinite cardinality theory turns a blind eye to this fundamental difference between finite sets and between infinite sets in relation to the presence or absence of a one-to-one correspondence between two sets. But we can note that the Diagonal proof is an instance of a proof that there is no one-to-one correspondence where the proof depends on the properties of the elements of the set. And the Power set proof 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.
Note regarding terminology
Some commenters have taken issue with some of the words I have used, such as limitlessly large, size of a set, smaller, quantity of the elements of a set, fewer elements, etc, complaining that the meaning of those words is not clear. But any ambiguity regarding such terms arises only from the illogical assumptions which ignore the fact that the impossibility of defining a one-to-one correspondence or a surjection between two sets may simply imply certain properties of the elements of the sets, rather than any property of the sets in themselves.
By rejecting such illogical assumptions, the meanings of such terms are easily defined, and are perfectly clear and unambiguous and completely in accordance what they intend in ordinary language, as is done further below.
It might be noted that there are people who try to evade the contradictions of transfinite cardinals by claiming that sets with different transfinite cardinal numbers do not imply that those sets have different quantities of elements, that there is some sort of subtlety involved - which is never explained - and which everyone fails to understand other than the people making such claims. There is a striking similarly of the mindset of such people to the mindset of fundamentalist religious fanatics, who will resort to any absurdity in attempts to defend their position.
The Origins of Comparisons of Cardinal Numbers
The details of the notion of transfinite cardinal numbers originated in Cantor’s 1895 paper “Beiträge zur Begründung der transfiniten Mengenlehre”, English translation online at Contributions to the Foundations of the Theory Of Transfinite Numbers. (Footnote: Georg Cantor, PDF "Beiträge zur Begründung der transfiniten Mengenlehre”, Mathematische Annalen 46.4, 1895, pp 481-512. Translated by Philip Jourdain as “Contributions to the Foundations of the Theory Of Transfinite Numbers”, Open Court Publishing, Chicago, 1915.) The concepts detailed in Cantor’s paper have been followed essentially unchanged since then. Section 1 of Cantor’s paper talks about “cardinal number” being a general concept that can arise by considering of a set in terms of an abstraction of its elements, where the individual properties of the elements are unimportant (in the paper, the cardinal number of a set M is represented by M̿ ):
“Since every single element m, if we abstract from its nature, becomes a “unit”, the cardinal number M̿ is a definite aggregate composed of units, and this number has existence in our mind as an intellectual image or projection of the given aggregate M.”
In Section 2 Cantor talks about how cardinal numbers can differ, and here he uses the symbols =, <, and > without making any reference to the fact that these symbols are already in use for natural numbers (and real numbers), so that a reader at this point of the paper might question whether there is an implied association of cardinal numbers to natural numbers and real numbers, and whether Cantor could have instead simply used completely new symbols.
It is only when we get to Section 5 (Finite Cardinal Numbers) that Cantor actually gives his first definition that can give an actual value for a cardinal number - he defines the cardinal number 1 - that it corresponds to a set with only one element, and that the number 2 corresponds to a set with 2 elements, and so on. So it is only at this point that we discover that Cantor’s intention is that the usage of the symbols =, <, and > for cardinal numbers are to correspond to the use of these symbols for real numbers for equality, less than and greater than. We see now that when he uses the term > (“greater than”), the usage precisely corresponds to the prior notion of > (“greater than”) for real numbers, which that is, that if a > b, then there exists some number r such that a = b + r. (Footnote: While the term ∞ is, strictly speaking, not a number, with certain caveats it can be used in a similar manner to represent the notion of approaching a limitlessly large quantity in certain expressions, such as limit expressions.)
The key point here is that Cantor defines the addition of two cardinal numbers as corresponding to the union of two abstract sets where the elements are all abstract elements that are simply units stripped of any individual distinction - in other words, Cantor is saying that the increase of a cardinal number of a set corresponds to the addition of abstract elements to an abstract set, where the precise nature of the elements are unimportant. Recalling Cantor’s assertion,
“Since every single element m, if we abstract from its nature, becomes a “unit”, the cardinal number M̿ is a definite aggregate composed of units, and this number has existence in our mind as an intellectual image or projection of the given aggregate M.”
it is clear that Cantor did not mean that such “units” would all be identical, since that would create a difficulty in talking about a set of multiple elements that are all identical, but the salient point here is that his intention was that the actual specific properties of the elements are irrelevant in the determination of the cardinal number of a set. In fact he states (in his Section 1):
“…the cardinal number M̿ remains unaltered if in the place of each of one or many or even all elements m of M other things are substituted.”
The logical implication is that if a set X has a greater cardinal number than a set Y, then simply by augmenting the set Y with sufficiently many elements a set can be created that has the same cardinal number as the set X. Similarly, the logical implication is that if the set of reals has a greater cardinal number than the set of natural numbers, then simply by augmenting the set of natural numbers with sufficiently many elements a set can be created that has the same cardinal number as the real numbers.
Cantor’s decision to use the absence of any one-to-one correspondence to indicate a difference of cardinal number was rather unfortunate, since for certain sets the absence of a one-to-one correspondence is the result of the elements of the sets having certain properties that prevent a one-to-one correspondence, rather than there being any actual difference in the properties of the sets that are independent of the properties of the elements.
That Cantor and others did not notice those facts regarding one-to-one correspondences can be attributed to a limited understanding at a time when the consideration of such matters was at an early and developing stage - but now that these facts are readily apparent, it is illogical to simply turn a blind eye to those facts and continue as though these facts are irrelevant. Why does conventional mathematics continue to rely on an 150 year old assumption when it obliterates the fundamental difference between the two completely different types of a one-to-one correspondence? The acknowledgment of the two different types of a one-to-one correspondence implies that the cardinal number of a set is simply a unique property of the set (and which is not the property that it is a set) which is independent of the properties of the elements of the set (other than the property of each element that it is an element).
Contradictions
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 number 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 (a smaller number of 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 number of its elements, and it is still the case that the cardinal numbers of two sets must differ by a definitive measure, which may be zero - if it is not zero, then, ipso facto, then one set has fewer elements than the other. And so, if the cardinal numbers of two infinite sets that both have limitlessly many elements are different, then we have the result that one of these two sets, where both have no limit to the number of elements that they contain, must have fewer elements than the other set - a blatant contradiction.
The Axiom of Transfinite Cardinals
The rather troubling inherent problem (as previously mentioned) 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:
Axiom 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 numerical comparators ‘equals’, ‘less than’ and ‘greater than’. (Footnote: This axiom was devised by Georg Cantor in Section 1 of his 1895 paper PDF Beiträge zur Begründung der transfiniten Mengenlehre, there is an English translation viewable online at Contributions to the Foundations of the Theory Of Transfinite Numbers - (trans: P. Jourdain). It should be noted that at this point in his translation, for some unfathomable reason Jourdain has inserted the word theorem, giving his translation as: “Of fundamental importance is the theorem that two aggregates M and N have the same cardinal number if, and only if, they are equivalent” whereas Cantor’s original German text makes no mention of a theorem: “Von fundamentaler Bedeutung ist es, dass zwei Mengen M und N dann und nur dann dieselbe Cardinalzahl haben, wenn sie äquivalent sind” A better translation would be: “It is of fundamental importance that two sets M and N have the same cardinal number if and only if they are equivalent” and which is an axiomatic assertion that assumes that all sets, whether finite or infinite, behave in exactly the same way with regard to one-to-one correspondences.) (Footnote: Note that there are different versions of this axiom, but the fundamental assumption is always included in some way.)
Note 1: Note that this axiom does not define what this property ‘cardinal number’ for any particular set is, nor how it is calculated.
Note 2: Note that by this axiom, 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 - other than the “empty 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.
For more on the ‘empty set’, see the page: The ‘Empty Set’.)
The corollary of Axiom A, together with Note 2, 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 Axiom A forces them to have different cardinal numbers. Hence Axiom 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 X has a smaller size than the set Y.”
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 would give us:
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 numerical comparators ‘equals’, ‘less than’ and ‘greater than’. (Footnote: As for Axiom A, this does not define what the cardinal number for any particular set is, nor how it is calculated; that must be defined separately.)
In this way, we can make a logical consideration of the fact that it may not be possible to define a one-to-one correspondence between two infinite sets simply on account of certain properties of the elements of the sets (as noted above in One-to-one Correspondences and Properties). Taking this into account, we now introduce some definitions.
Definitions
We now introduce some definitions that reject the illogical assumptions referred to above and which result in contradictions.
The Cardinal Number of a set, when defined without illogical assumptions regarding one-to-one correspondences between infinite sets, is defined as follows:
The Cardinal Number c of a set A is the largest n, such that for all m ≤ n there is a one-to-one correspondence of each m to an element of A, if there is a largest n (n, m ∈ N, n, m > 0). If there is no largest n, the cardinal number of the set is said to be infinite, and this may be represented by the symbol ∞, where ∞ > n for all n. Hence a cardinal number c is such that c ∈ N or c = ∞, c > 0. (Footnote: In this definition, there is no cardinal number that is zero, since the notion of an “empty set” is inherently problematic, see The ‘Empty Set’.)
Limitlessly large set / Infinite set:
A set A is assigned the term “limitlessly large” or “infinite” if there is a subset of the set for which there is a one-to-one correspondence with the set of all natural numbers, i.e: there is a function f (n) that enumerates some or all of the elements of that set (where n ∈ N, n > 0), and for every n, there is an element of the set A, and there is no largest n in that enumeration, that is, for every n where f (n) ∈ A there exists another m = n + 1 (where m ∈ N), and f (m) ∈ A where f (n) ≠ f (m). (Footnote: Note that the existence of so-called “indefinable” numbers has never been afforded any logical proof, see the Diagonal Proof - the secondary argument and Indefinable numbers.)
A set that is not limitlessly large is assigned the term “finite”.
Quantity of elements of a set / Number of elements of a set / Size of a set:
This is simply the Cardinal Number c of a set as defined above: c ∈ N or c = ∞, c > 0.
If a set is finitely large, then c ∈ N, c > 0. Otherwise, it is a limitlessly large set, and its size may be said to be limitlessly large / infinitely large, and it may be said to have infinitely many/limitlessly many elements, and c = ∞.
Smaller / Having fewer elements:
A set A is said to be smaller than a set B, and having fewer elements than B if the cardinal number (as defined above) of A is smaller than that of the set B, i.e: if a is the cardinal number of A, and b is the cardinal number of B, A is smaller than B if a < b, and there exists a cardinal number c where a + c = b. (Footnote: This corresponds to: the union of the set A and a set C that has the cardinal number c, if they have no elements in common, will result in a set that has the same cardinal number as the cardinal number b of the set B.) One commenter complained that the definition should be by the use of surjections, but this has the same problem with infinite sets as for one-to-one correspondences since the absence of a surjection depends on the particular properties of the elements rather than of the sets themselves; I deal with this in more detail on a separate page, see Surjections and Cardinal Numbers.
Now, when we take account of the fact that it may not be possible to define a one-to-one correspondence between two infinite sets because of certain properties of the elements of the sets, then, although there cannot be 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 (e.g: where R is the set of real numbers between 0 and 1 and R2 is the set of real numbers between 0 and 2, or between the set N of natural numbers and the set Q of rational numbers).
So, by omitting the problematic Axiom above that necessarily implies that different infinite sets can have different cardinal numbers, we can have a definition of cardinal numbers that comprises finite cardinal numbers and a single infinitely large cardinal number, without resultant contradictions.
Absence of any proof
Besides the fact that the fundamental Axiom of cardinality does not prove that there can be different infinite sets that have different sizes - since, by definition, an axiom must be an unproven assumption - 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 did not have different quantities of elements, 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.
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 PDF On Considerations of Language in the Diagonal Proof and PDF 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.
She continues:
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.
and continues:
…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.
then
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 to sweep it under the carpet by claiming that it can be solved by applying an arbitrary axiom. 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 axiom. It isn’t inherently ‘true’, and it isn’t immune to change. It can be changed simply by changing the axiom. Perhaps one day mathematicians will acknowledge the folly of building an edifice on sand.
Footnotes:
Rationale: Every logical argument must be defined in some language, and every language has limitations. Attempting to construct a logical argument while ignoring how the limitations of language might affect that argument is a bizarre approach. The correct acknowledgment of the interactions of logic and language explains almost all of the paradoxes, and resolves almost all of the contradictions, conundrums, and contentious issues in modern philosophy and mathematics.
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