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Cantor's diagonalization theorem, which proves that the reals are uncountable, is a study in contrasts. On the one hand, there is no question that it is correct. On the other hand, not only is it controversial, it attracts an inordinate number of cranks. Asaf Karagila compiles an excellent list of various "cranky" questions here.

I have come to the conclusion that math populizers have incorrectly concluded that the proof of Cantor diagonalization is accessible. For example, see this in the popular business press. My view is that all such "populist" proofs assume way too many sophisticated mathematical proofs to be viable.

Specifically, the standard Cantor proof assumes, without statement, that:

  • countable sets are enumerable; i.e. can be put into a sequence
  • sequences are valid ways to define new numbers (limits, sums), despite the fact that they contain an infinite number of terms
  • the constructed real not in the list of "countable" reals is the limit of a Cauchy sequence, and is therefore a real number and not rational (which explicitly references the completeness of the reals)

So my question is on either side of my examples:

  • Given my protestation, is the standard proof by contradiction salvageable for non-mathematicians; and
  • What amount of rigorous mathematics is needed to make this proof "crank proof"? I would say what I have provided is sufficient, realizing that serious cranks are impervious to anything not in their thinking!

EDIT: Thanks for the answers. I think I have a good idea where the line of (mis)understanding might be. Multiple people have expressed the notion that non-mathematicians understand the real numbers as infinite decimals; i.e. through their standard notation. This is where I think we mathematicians are wrong, at least for some people. I think lots of people doubt the existence of any such infinite object which seem to require an infinite construction, like Cantor. Specific reals, like $\pi$, have other definitions that seem "non-infinite". I accept that those who cannot grasp such infinite constructions may be a lost cause for popularization, but I do think that acknowledging this gap is important.

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closed as primarily opinion-based by Phira, Fixed Point, Thomas Andrews, Brian Rushton, J. W. Perry Jan 17 '14 at 4:24

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise.If this question can be reworded to fit the rules in the help center, please edit the question.

I think the proof is very accessible. The problem is more that people who aren't into math probably will find neither the theorem nor the proof very interesting, I think. In other words, you don't need much mathematical knowledge to understand the proof, but you need to be the kind of person who's delighted by posing and answering such problems. 'cause if you aren't, why would you want to spent time thinking about whether you can assign natural numbers to all the reals? – fgp Jan 10 '14 at 3:33
@fgp: I totally agree with your comment, but it backs up my thesis. We need points which are generally interesting. – trb456 Jan 10 '14 at 3:41
I don't think the problem is the proof, I think it is the theorem. There are certain theorems that attract cranks and for the most part the cranks don't really care about the proof, they are convinced the theorem is wrong. $0.999\dots=1$ is another and the proof is quite accessible, but it sure draws the cranks. – Ross Millikan Jan 10 '14 at 4:01
(I'm also not entirely convinced that this is a question as opposed to a rant/opinion-based discussion, although I won't nominate to close. Perhaps flagging for CW would be a good start?) – Stahl Jan 10 '14 at 4:15
@Stahl: I do hope this is not a rant! But I think your example of Banach-Tarski as accessible is what I'm getting at. Banach-Tarski is not accessible! Non-mathematicians might find it intriguing, but I really doubt they can "grok" non-measurability. There is a reason measure is taught in grad school and usually not before. It is precisely the line where intuition fails that I'm trying to explore. – trb456 Jan 10 '14 at 12:21
up vote 2 down vote accepted

What you have to realize is that non-mathematicians understand the real numbers AS infinite decimals. They don't feel the need for Cauchy sequences and what not. Also, these non-mathematicians are completely comfortable with things going on forever. A lot of the more recent work was just in trying to convince mathematicians that what they had been doing for centuries made sense. Non-mathematicians come from the point of view of the old mathematicians. Cantor's work is the reason I knew I was going to major in math. It should be popularized at least to H.S Calculus students IMO.

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I think this notion is exactly what I am doubting. I think some people think of real numbers this way, but I think others doubt the existence of anything that requires an infinite expression. And the Cantor proof seems to require an infinite construction, which is perfectly valid, but not to the second group. – trb456 Jan 10 '14 at 12:25
Yours is the answer I'm selecting. I think you have captured the standard view of mathematicians, and I think this view is unfortunately misguided. I think there is a notable subset of non-mathematicians who don't accept this definition of the reals, which is where the cranks come from. I don't know if there is anything that can be done about this, but I think this is the issue. – trb456 Jan 10 '14 at 12:42
@trb456: I think being a mathematical crank means you are already heavily interested in mathematics. In particular, I think doubting infinite expressions is only likely if you have asked questions like "What is mathematics?". Although I don't think being a finitist makes you a crank (although many cranks are finitists!). [You can tell me to stop discussing whenever you want]. – Jacob Wakem Jan 10 '14 at 20:33
I agree with all this, and thanks for the discussion! – trb456 Jan 10 '14 at 21:13
From an empirical point of view, when I was in high school all the juniors in my Calculus class were presented Cantor's work and it was a very positive experience. But we were all "potential mathematicians". So maybe you are right about the "general population". This view amongst mathematicians might be too self-reflective (I'll try and get some data soon!). – Jacob Wakem Jan 10 '14 at 22:11

I do agree that Cantor's theorem can sometimes cause confusion, although so can many other things. I don't agree with most of the reasons discussed in the OP.

That a countable set can be listed as a sequence is essentially the definition of countable: a sequence is a function with domain $\mathbb N$, and a set is countable if it admits a bijection with $\mathbb N$, i.e. can be enumerated as a sequence. I don't think this is so hard.

I don't think completeness of real numbers enters in a difficult way; the argument constructs a real number as an infinite decimal, and anyone who has learnt decimals accepts an infinite decimal as a real number. (Indeed, this may well be their working def'n of real number.)

In my experience, the most common confusion that one sees with the argument is the following: given a map $f: \mathbb N \to \mathbb R$, i.e. a sequence of real numbers, one constructs a new real number not in the image of $f$. This can elicit the confused reaction of "can't we just add that new real number to the sequence"? So the confusion seems to be with the set-up ($f$ was arbitrary, and however we chose it, it doesn't cover all of $\mathbb R$) and modes of argument that are standard in mathematics but perhaps unfamiliar to people ouside mathematics.

That's not surprising to me: quantifiers, and related issues like making general choices, or proving something for an object $x$ which was chosen generally (and thus concluding that the statement is true for all possible choices of $x$) are always a source of confusion to some people (and are not confusing at all to others). The role of $x$ as a dummy variable in integrals is confusing to some. The role of $x$ in the equation $x+3 = 5$ is confusing to some (at some point $x$ was just a symbol and $x+3$ was a linear polynomial, but suddently $x$ becomes the specific number $2$).

Working correctly with ideas of generality and specialization is always going to be a hard thing to teach to some people. (In geometry, which sometimes has suble notions of general position, or, in contrast, special position, even good mathematicians can get confused about these sorts of issues!) I don't think that Cantor's argument is particularly to blame for this.

A separate question is how important this result is, and whether there are other pieces of mathematics that it would be better to popularize. If I was to argue this point, I would argue on choice of subject matter (what is central to mathematics and what is more peripheral). But these choices are always somewhat subjective, and in any case don't seem to be the thrust of the OP (which is more about pedagogy, I think, then about issues of subject matter and taste; or have I misunderstood?).

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And even the point you mention as the most common source of confusion has more to it than just being hard to grasp for those not trained: The possibility of an infinite set not containing all its elements at a given time (potential infinity as opposed to actual infinity). This makes it more apparent that it is not just a delicate point that makes people confused. The question "can't we just add that new real number to the sequence?" has some substance. ( – user119256 Jan 10 '14 at 5:13

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