There are several approaches to the modern theory of “real numbers”. Unfortunately, none of them makes complete sense. One hundred years ago, there was vigorous discussion about the ambiguities with them and Cantor’s theory of “infinite sets”. As time went by, the debate subsided but the difficulties didn’t really go away.
A largely unquestioning uniformity has settled on the discipline, with most students now only dimly aware of the logical problems with “uncomputable numbers”, “non-measurable functions”, the “Axiom of choice”, “hierarchies of cardinals and ordinals”, and various anomalies and paradoxes that supposedly arise in topology, set theory and measure theory.
Some of the stumbling blocks have been described at length in my Math Foundations series of YouTube videos. In this blog we concentrate on the problems with real numbers and arithmetic with them.
The basic division in mathematics is between the discrete and the continuous. Discrete mathematics studies locally finite collections and patterns, and relies on counting, beginning with the natural numbers 1,2,3,⋯ and then extending to the integers, including 0, as well as -1,-2,-3,⋯, and to rational numbers, or fractions, of the form a/b with a and b integers, subject to the condition that b non-zero and with a particular notion of equality.
Continuous mathematics studies the “continuum” and functions on it, and relies on measurement, which these days involves also “irrational numbers” like √2,√5 and π that the ancients wrestled with, as well as more modern “irrational numbers” such as e and γ arising from integrals and infinite series. But what do these words and objects actually precisely mean and refer to? We should not presume that just because we use a common term or notion familiar from everyday life, that its mathematical meaning has been properly established.
Up to a hundred years ago, the notion of the “continuum” seemed intuitively straightforward, but difficult to pin down precisely. It appeared that we could rely on our intuition of space, following the philosopher I. Kant’s view that somehow humans had an innate understanding of three-dimensional geometry. However with the advent of modern physics, and in particular relativity and quantum mechanics, the true nature of the “continuum” grew increasingly murky: if time is relative and perhaps finite in extent, and space has an inherent graininess which renders it certainly not infinitely divisible, then what exactly are we modelling with our notion of the “infinite number line”?
While engineers and scientists work primarily with finite decimal numbers in an approximate sense, “real numbers” as infinite decimals are idealized objects which attempt to extend the explicit finite but approximate numbers of engineers into a domain where infinite processes can be ostensibly be exactly evaluated. To make this magic work, mathematicians invoke a notion of “equivalence classes of Cauchy sequences of rational numbers”, or as “Dedekind cuts”.
Each view has different difficulties, but always there is the crucial problem of discussing infinite objects without sufficient regard to how to specify them. I have discussed the serious logical difficulties at length around video 80-105 in the Math Foundations series.
For example the video Inconvenient truths with sqrt(2) has generated a lot of discussion. However not everyone approves of casting doubt on the orthodoxy: the video has more than 1000 likes, but also 316 dislikes. I doubt if I am saying anything in this video which is actually incorrect though — you can judge for yourselves.
Let’s return to the safe side of things. A finite sequence such as s = 1,5,9 may be described in quite different ways, for example as the “increasing sequence of possible last digits in an odd integer square”, or as the “sequence of numbers less than 10 which are congruent to 1 modulo 4”, or as the “sequence of digits occurring in the 246-th prime after removing repetitions”. But ultimately there is only one way to specify such a sequence s completely and unambiguously: by explicitly listing all its elements.
When we make the jump to “infinite sequences”, such as a sequence somehow implied by the expression “m = 3,5,7,⋯” the situation changes dramatically. It is never possible to explicitly list “all the elements” of an infinite sequence. Instead we are forced to rely on a rule generating the sequence to specify it. In this case perhaps: m is the list of all odd numbers starting with 3, or perhaps: m is the list of all odd primes. Without such a rule, a definition like “m = 3,5,7,⋯” is really rather meaningless.
We can say the words “infinite sequence”, but what are we actually explicitly talking about??
To a computer scientist, an “infinite sequence” is modelled by a computer program, churning out number after number perhaps onto a hard drive, or in former years onto a long tape. At any given point in time, there are only finite many outputs. As long as you keep supplying more memory, or tape, and electricity the process in principle never stops, but in practice will run short of resources and either grind forwards ever more and more slowly (the next output will take two years, hang on just a while!) or just come to a grinding halt when power or memory is inevitably exhausted.
So in this case the sequence is not to be identified by the “completed output tape”, which is a figment of our imagination, but rather by the computer program that generates it, which is concrete and completely specifiable. However here we come to an essential difficulty with “infinite processes”: the program that generates a given “infinite sequence”: is always far from unique. There is no escape from this inescapable fact, and it colours all meaningful aspects of dealing with “infinity”.
A finite set such as {2 4 6 8} can also be described in many ways, but ultimately it too is only properly specified by showing all its elements. In this case order is not important, so that for example the elements might be scattered over a page. Finite sets whose elements cannot be explicitly shown have not been specified, though we might agree that they have been described.
An example might be: let S be the set of all odd perfect numbers less than 10^{100000}. [A perfect number, like 6 and 28, is the sum of those of its divisors less than itself, i.e. 6=1+2+3 and 28=1+2+4+7+14.] Such a description of S does not deserve to be called a specification of the set, at least not with our current understanding of perfect numbers, which doesn’t even allow us to determine if S is empty or not.
With sets the dichotomy between finite and infinite is much more severe than for sequences, because we do not allow a steady exhibition of the elements through time. It is impossible to exhibit all of the elements of an “infinite set” at once, so the notion is an ideal one that more properly belongs to philosophy—it can only be approximated within mathematics.
The notion of a “completed infinite set” is strongly contrary to classical thinking; since we can’t actually collect together more than a finite number of elements as a completed totality, why pretend that we can? It is the same reason that “Leprechaun heaven” or “hierarchies of angels” are not generally recognized as proper scientific entities. Infinite sets, angels and Leprechauns may very well exist in our universe, but this is a philosophical or religious inquiry, not a mathematical or scientific one.
The idea of “infinity” as an unattainable ideal that can only be approached by an endless sequence of better and better finite approximations is both humble and ancient, and one I would strongly advocate to those wishing to understand mathematics more deeply. This is the position that Archimedes, Newton, Euler and Gauss would have taken, and it is a view that ought to be seriously reconsidered. I believe it is also closer to the view of modern giants such as H. Poincare and H. Weyl, both of whom were skeptical about our uses of “infinity”.
Why is any of this important? The “real numbers” are where Cantor’s “hierarchies of infinities” begins, and much of modern set theory rests, so this is an issue with widespread consequences, even within algebra and combinatorics. Secondly the “real numbers” are the arena where calculus and analysis is developed, so difficulties with their essential arithmetic foundation lead to weakness in the calculus curriculum, confusion with aspects of measure theory, functional analysis and other advanced subjects, and are obstacles in our attempts to understand physics.
In my opinion, it is possible, perhaps even likely, that we need to understand mathematics in the right way before we will be able to unlock the deeper secrets of the universe.
By reorganizing our subject to be more careful and logical, and by removing dubious axiomatic assumptions and unnecessary philosophizing about “real numbers” and “infinite sets” we make it easier for young people to learn, appreciate and contribute.
This also strengthens the relationship between mathematics and computing.
It is time to acknowledge the orthodoxy that silently frames our discipline. We need to learn from our colleagues in physics and computer science, and begin the slow, challenging but important and ultimately rewarding task of restructuring mathematics properly.
njwildberger: tangential thoughts 
I somehow think linguistics may help us find equivalent definitions of a sequence and translate across programming languages snd dialects.
Dear Dr. Wildberger.
For 1000s of years, no mathematician had even considered the idea of uncomputable axioms. Then 200 years ago, Cantor was born. His ideas took some tweaking, but over time, 99% of mathematicians accepted that Set Theory was so interesting and powerful, that it was worth dispensing computability for it.
You are one of the 1% who doesn’t see it that way, and that’s fine. So do I. So do most of your viewers. It’s completely insane how the majority blindly worship the axiom of infinity and choice.
But who defines what mathematics is, isn’t up to you, or me. It’s not up to us…
It’s up to everyone.
And the consensus is clear, even if you or I don’t like it.
So why is it so important for us to win back the title of “mathematics”?
They have a label for people like us – “finitists”. Finitists do not accept axiomatic foundations that rely on infinite processes. That’s exactly what we are.
We have our own label, we have our own ideas. Isn’t that good enough?
Thanks for the comment Zac. I think rather the last few generations have accepted this philosophizing somewhat unthinkingly. I believe that the decision to accept set theory has not been made logically, but rather as a fait accompli sidestepping the necessary process of careful consideration and debate. While it is a bit late for that admittedly, better late than never! There is too much at stake, and the gap between the dreamings of modern pure mathematics and computational reality is now getting so big that once younger people wake up to what is really going on, they will quickly come around to the computational, concrete point of view. Let’s see!
Ok so imagine if we did succeed. Imagine if we got everyone to finally admit that set theory is illogical and mathematics is restricted by computability.
Then modern mathematicians would just make up a new word, like “meta-logic” (meaning the same thing as normal logic except it allows for what humans can’t realistically compute).
And they would call themselves “meta-mathematicians”. “Meta-mathematicians” who construct “meta-theorems” from this “meta-logic”. And then they’d go on about their business, studying set theory because it is part of “meta-mathematics”, and nothing would actually change. Nothing except the labels. Us “true mathematicians” would still be the 1% minority, and the “meta-mathematicians” would be the 99% majority. Same structure, different labels. It’s an isomorphism!
But you don’t believe this would happen. You believe they would also quickly come around to our side. Why?
Great words! I agree with you man! So do I.
Zac, you are arguing like a man who says, “There’s a hidden pitfall in the popular road, causing generation after generation of travelers to suffer lifelong injuries, but our alternative road is safe even though only a few take it. Isn’t that good enough?”
Generation after generation of mathematicians, physicists, and really the public at large are being misled into deep intellectual injury from which most never recover, induced to spin their wheels in a lobotomized fantasyland for their entire productive life.
This would be an atrocity if it were deliberate. It is a great public harm likely costing millions of lives as the larger society is deprived of new technologies and a general rational approach to intellectual discourse due to the neutered and spayed king and queen of the sciences.
The public must be alerted to the fact that they have been led into a trap, a fatal conceit that had done untold harm to the world. The mythmaking of modern mathematics and physics must be driven into the barbaric past to which it belongs. It is religion dressed in a lab coat, like most of the rest of institutional science.
Steve Patterson is completely correct when he says we are in a true intellectual dark age, and never has this been more apparent. People taught to believe contradictions will swallow anything, as recent events show. Everyone who understands this has a moral obligation to rescue those ensared, even if like the good Samaritan they will come under fire for it.
Talking about numbers is not possible without at the same time also calculating arithmetically [includes the comparison]; therefore the latter has no meaning for a single number; so the talk must be about two (or more) numbers. Given two numbers a,b and the comparison between them (digit by digit from left to right, according to Western convention), that must yield a first pair of digits (a.digit_x b.digit_y) where the digits x and y are different (otherwise a,b are not 2 numbers, not a valid premise); now cut off the sequence of digits already compared (up to and including x,y) and take half the sum of these two partial sequences: that is a new rational number between two [arbitrary] reals. The count of real numbers cannot be greater [*non-trivial*] than the count of rational numbers.
With kind regards,
Klaus D. Witzel
Lower Saxony
You proved was that given 2 different “real numbers”, there exists a rational number between them. That’s a Calc I theorem lol.
But then you somehow jumped from that to “the count of real numbers cannot be greater than the count of rational numbers”. Without any definitions, your conclusion is meaningless. So let’s start with – can you define what the “count of rational numbers” is?
you first: define what your somehow jumped is.
Chaeremon, for some reason there is no reply button to you.
I can’t define “somehow jumped” mathematically.
And that’s fine. Because I’m not claiming “you jumped to conclusions” is a mathematical theorem.
My claim that “you jumped to conclusions” is mathematically meaningless. It’s true.
And in the exact same way, your claim that “The count of real numbers cannot be greater than the count of rational numbers” is mathematically meaningless, because you haven’t defined it mathematically yet.
Whether you believe in real numbers or not, if you can’t provide a mathematical definition for your terms, it’s just not mathematics. It can be philosophically true, or scientifically true, or whatever. But not mathematically.
What separates Wildberger from other math cranks, is he understands the importance of mathematical definitions. And he sticks to them. He doesn’t waffle about his terms, like you’re doing.
I have read and watched a number of writings and videos of Dr. Wildberger opposing the use of infinities in mathematics. I have tried my best to understand his objections.
I have not been able to understand any basis for his objections. It is clear that he believes most or all uses of infinities in mathematics are not valid. But except for his repeatedly expressing that view, I have never been able to glean his *reasons* for those objections, just the fact that he doesn’t like the use of infinities one bit.
I can only conclude that he doesn’t *have* a basis for his objections beyond his intuitive feeling that he doesn’t like infinities.
Hi Dan, Sorry that you have not been able to understand my objections. I do hope you won’t give up on watching my videos: who knows, perhaps in the fullness of time things will slowly become clearer. It is not easy to have cherished notions seriously challenged. You might like for example my recent debate with Daniel Rubin, perhaps that will clarify some things.
This has nothing to do with “cherished notions seriously challenged”. It has to do with my not having read or heard any *reasons* for your objections. Just a repetition of your belief that certain things are invalid or do not exist.
Question: Have you written your reasons for your objections down in one place that I can consult? Thanks.
Hi Dan, You can for example read my paper “Set Theory: Should you believe?” at
Click to access SetTheory.pdf
Dear Mr. Wildberger,
I found some of your videos on youtube and I felt relieved. I am not a mathematician but I started to feel very uncomfortable with the idea I was thought in the university that we could “build” a line out of points. It just did not look right. The problem ended in the Real numbers and I talked to some mathematicians I know to help me. Although they could not explain anything to me, they said there was no discussion on this topic anymore. They said I should study Cauchy sequences and “Dedekind cuts… . When I found your videos I was relieved! Thank you!
Recently, I read I book about a French philosopher called Louis Lavelle. Lavele, says that there are two kinds of existence: (i) being as a data or fact – something that has been created and (ii) being as an act – something that can create or that is under creation. In this sense, we would be “acts” in this world as we create ourselves during our lives and become a fact or data after we die. I believe there is a similarity between this way of seeing the world and the numbers: rational numbers are facts, irrational numbers are acts. They have a different nature. The same difference is found between finite and infinite sets. Looking to irrational numbers and infinite sets as if they were a data or a fact is the problem… .
I do not know if you agree. If this is the case, you should read Louis Lavelle.
Best,
Petrônio.
Hi Petronio, Thanks for your nice comment, I will have to look up the work of L. Lavelle. Thanks, N
I agree, Petronio, in epistemology we must cleanly delineate objects from events, things from actions/occurrences.
An “infinite set” tries to be both a “collection” and a “collecting” at the same time, yet static and dynamic notions are incompatible with each other. A contradiction results and we have nothing but nonsense, given the appearance of solidity by math-cultural rules about what counts as an acceptable proof and what doesn’t. Karma Peny’s channel on youtube has a satisfying account of one such math-cultural rule, which he calls a “cat-and-mouse argument,” an argument that is logically reversible but math-culturally taboo to reverse.
In physics we have the same problem when “a wave” is invoked, because wave is what something does, not what something is. Water waves. A wave is movement, not an object. Likewise, space is nothingness, thus it cannot bend or warp as Einstein would have it. This miscarriage of semantic justice is held together, as in modern math, by a culturally enforced system of equivocations. One must see past the word matrix to notice anything amiss, but once seen it can’t be unseen – indeed, the entire edifice falls to ruin.
All that is needed for clarity is to keep careful track of how words are used, but human social dynamics make the necessary semantic rigor feel socially awkward and even gadflyish, thus eloquent glib arguments tend to prevail over persnickety rigorous ones and the problem perpetuates through the ages. At least until we come to use something other than words for our scientific communications.
Hey NJ,
Great post as usual, really highlights the problems with ZFC as an axiomatic system.
One question, I’d love to know what you think of the Peano Axioms as an axiomatic system.
In particular, the Peano Axioms with the predicate version of the induction axiom (not the set theory version, as that has all the obvious flaws of set theory).
Dear Joe,
I am not a big fan of any “axiomatic systems”. I reckon we should just make precise definitions and go from there. Why do we need to assume anything? That is almost always a cheat of some kind.
No particular reason to put the question here, but how do you think about Brouwer? I find it interesting Brouwer and Einstein corresponded intensively over many years. Brouwer persuaded Einstein to read a book which resulted in Einstein writing the God letter to the author, really hilarious stuff.
Hi, I think that Brouwer was a courageous and independent mathematician, and he made valuable contributions in outlining alternative approaches to the foundations of mathematics. However, he did still believe in “real numbers”. So my praise has to be qualified.
If we generously accept that the set of natural numbers has no upper bound, then we can also accept the arithmetical theory of real numbers.
If the absolute difference of two infinite sequences of rational numbers is less than 1/n, for every given natural number n, then the two infinite sequences represent the same real number.
The continuum is a self-evident model of a geometric line; every point-like hole would be a new real number and thus one should find another hole etc.
What does it mean to “have precise definitions” without having axioms? What can we use to construct these definitions? How can we make sure that we all will understand them the same way?
However, I have seen some critics of extreme finitism too. One of them is that “To define a number we need more characters than the number itself, so there doesn’t exist the final number”. Also In my opinion, the most difficult question for finitists to answer is that “What is the largest (finite) number and can we access it”.
If the largest number didn’t exist, then numbers would be potentially infinite.
If the largest number existed but we couldn’t get it, then would it be equivalent to say that the largest number is infinite like the actual infinity, which we can’t access forever?
If the largest number was finite and we could get access to it, then what kind of magical force would prevent us from adding one to that number, since that (extremely large!) number X and its successor X+1 are almost identical?
It’s possible that Kant has negated any possibility of a universally true theory of finiteness and infiniteness in his first two Antinomies.
What’s your opinion on the question asked above? Or maybe the only solution is that the concept “largest number” is fundamentally flawed since it refers to itself when we define it?
Also recently I’m thinking about the rationality of the concept “point at infinity” in projective geometry. Do you think it’s related to the existence of actual infinity, and is that concept reasonable?
What happens if we start counting to bigger and bigger numbers is that eventually we run out of puff: it gets too hard, too much memory is required, not enough time to make the computations etc. Its like the claim that you can’t walk forever? Then comes the rebuttal:
“Oh yah? Then what’s the furthest I can walk?”
Just because we can’t nail down when you will eventually quit doesn’t lessen our confidence that you WILL eventually quit.
I intended to comment this on your post “let alpha be a real number”, but couldn’t, perhaps because the post is too old. My comment is relevant to your whole problem with infinity and the real numbers, so I think leaving it here is fine.
You haven’t actually pointed out any problems with defining the set of real numbers to be the set R of certain sets of Cauchy sequences of rational numbers, where two Cauchy sequences are in the same set in R if and only if the sequence of their differences converges to 0. All you did was make up a conversation between you and a pretend mathematician who told you what the real numbers are and then you said “what if I don’t like?” and then said that Wittgenstein also doesn’t like the “infinite shenanigans” of maths, without stating any of his arguments or being anything but hopelessly vague about what his criticisms are. Perhaps he has valid criticisms, but there are certainly none to be found in this post.
In your post about the horned sphere you said that the only reason students of mathematics don’t have problems with the horned sphere is because they are “meek”, scared to “admit failure”, and that they are pretending to understand as a result. This is an extremely cynical viewpoint, and is not grounded in reality. Perhaps these students simply do understand. To assume that everyone other than you is just intellectually dishonest because they understand something you don’t is pure arrogance. It’s okay to be a finitist, but don’t pretend that everyone who isn’t a finitist is inferior to you.
In your article “Set Theory: Should You Believe?” you state what you think are the axioms of modern mathematics. One of the axioms you wrote was “there is an infinite set”. This is NOT an axiom! For one thing, the axioms are formal statements and the things you wrote down are informal statements. As somebody who spends a lot of time complaining about set theory and has written articles about set theory, you really ought to know the difference. More importantly, however, the formal statement which is the axiom of infinity does not correspond to the informal statement that there is an infinite set. Rather, the axiom postulates the existence of one particular set, and, once one develops a suitable notion of what it means for a set to be finite, one can prove that this set is not finite. You then complain that the axioms you wrote down are awash with difficulties because you used words like “property”, “function”, and “family of sets”, and you don’t know what those words mean. No wonder! You wrote the axioms down wrong! This is point of having the axioms be formal statements. You don’t need to know what a function is, or what a property is, or what a family of sets is, to set up the axioms of set theory. The only concept you need to take for granted is that of a variable, which you may take issue with, but one needs to start somewhere (you cannot offer me a development of mathematics that does not take any concepts for granted). You can then define rigorously what formal statements are, and then write down all the axioms you need as formal statements. So it seems that you just don’t understand set theory or how it is developed rigorously, and you’re blaming your lack of understanding on mathematics, and claiming that everyone who does understand is inferior to you because they really don’t understand and they’re just pretending to understand (because they’re “meek”, right?).
Hi Ben, Thanks for the comment. I understand you are an advocate of the usual position, which relies on philosophical statements like
“the set of real numbers to be the set R of certain sets of Cauchy sequences of rational numbers, where two Cauchy sequences are in the same set in R if and only if the sequence of their differences converges to 0”. To tighten the discussion, can you give us a simple non-trivial example of such a “real number”? I mean the whole thing, not just some convenient Greek letter or such that “labels” it.
And then you might like to meet the usual challenge that I give: what is the sum of “pi + e + sqrt(2)”?? What I want to do is to encourage believers such as yourself in moving away from abstract “set theoretical” pronouncements, and rather to explicit specifications of examples of the kinds of things you are ostensibly talking about.
In your video about rational numbers you define them to be equivalence classes of ordered pairs of integers, and these equivalence classes are infinitely large. So you have no problem with numbers being infinitely large equivalence classes. Yet you have a problem with real numbers being infinitely large equivalence classes of Cauchy sequences.
Give me a simple non-trivial example of a rational number. I mean the whole thing, not just some convenient letter that labels it. What is the rational number 1/2? According to your definition, it would have to be something like {(1,2),(2,4),(3,6),…}, but that’s not the whole thing. You can talk about decimal expansions and say it is 0.5, but really this is just another label. Moreover, one runs into serious difficulty with the number 1/3, whose decimal expansion you cannot write down in its entirety. And you need to be clear what an infinite decimal expansion even is, because in my mind it is a Cauchy sequence (uh oh).
I meet your usual challenge with a challenge of my own: what is “1+1”? The whole thing, not just a convenient label for it like the symbol 2. You see how we can play games like this with anything.
You need to point out exactly what the error with the construction of the real numbers as a set of equivalence classes of Cauchy sequences is. You keep claiming that it exists, and making audacious statements such as “the whole of maths built upon the real numbers is a logical sham” (your video about fundamental problem 19a), but you have never actually explained what that problem is. I find it hard to believe that there is such a problem, especially given that you are clearly okay with defining the set of rational numbers to be a set of infinitely large equivalence classes of ordered pairs of integers.
If what you’re looking for is a discussion of the arithmetic of the real numbers, you might like to read this: http://pi.math.cornell.edu/~kahn/reals07.pdf
Having watched your video on Cauchy sequences and real numbers (among various other videos you’ve made), you don’t point out a single issue with the definition. All you do is explain the definition while insulting it and saying “isn’t it ridiculous that this is how we define real numbers? Hardly looks like it’s going to work out, does it?” Yet it does work out.
Dear Norman
I have emailed you on the email address you gave me a few months ago. Thank you for your time and regards,
Peter