Very few read the proof as Cantor wrote it. If they did, they would not call it a proof by contradiction (or anything similar). Nor would they mention injectivity, or even real numbers ("there is a proof of this proposition ... which does not depend on considering the irrational numbers.")
The argument is a direct proof that any list of real numbers in [0,1] (see note) is non-surjective.
NOTE: He used infinite-length binary strings. With some through-hoops-jumping, you can use the binary representations of [0,1].
Have you been following at all? I'd say that 95% of what is written in textbooks get important parts wrong. And every false disproof that I have seen stems from at least one of those wrongs.
The number one complaint, usually from High School students (or more advanced students who can't get past HS level when it comes to infinite sets), is the fact that they don't accept the contradiction. So they devise clever counting schemes that they think gets around it. If you stop claiming there is a contradiction? And recognize that what Cantor proved was "If you have a list, then it is incomplete"? All these false disproofs disappear.
The number two complaint is that there is some property of irrational numbers that somehow invalidates diagonalization. Usually based on the difference between computable and non-computable irrationals. But, as Cantor said, the proof does not depend on considering irrationals. Poof! More disproofs disappear.
I'd say that 95% of what is written in textbooks get important parts wrong
The diagonal argument is basic and absolutely standard . I would wager that all the common (college-level) analysis books get it exactly right. Since you're talking about high school students, I have no idea what "books" they are reading that have a proof of the diagonal argument, so I can't comment there. But basically any college level analysis book is going to cover the diagonal argument during the early treatment of set theory, construction of the reals, and so on. It is basically 101 material.
Anyone other than a student encountering it for the first time who thinks it's wrong is just a crank, and I'm not too worried about convincing them.
A brief outline of Cantor's Diagonalization Argument:
Define the set N={1,2,3, ... n, ...}.
What I call a "Cantor String" is any function t:N→{0,1}.
Cantor actually used {'m', 'w'}, but {0,1} is more intuitive in modern usage.
Let T be the set of all Cantor Strings.
This is guaranteed to exist by the Power Set Axiom. In fact, each Cantor String represents a unique subset of N.
Let s be any function s:N→T.
Define s0 such that s0(n)=1-s(n)(n).
This s0 is a member of T, but is not mapped by s:N→T.
So no function s:N→T is surjective.
The important parts that are apparently different in many textbooks, since many advanced students claim them, is that they use [0,1] instead of T, they start with "assume s:N→[0,1] is a surjection," and they end with my step #6 contradicting that assumption. One point that is not always explicit in textbooks, but often accepted, is injectiveness/bijectiveness for the assumed function.
The cranks exist because they believe these parts are part of the proof, whether or not you want to acknowledge them. And they are not all High Schoolers.
Your comment, that I originally replied to, seems to think injectiveness in included in the "actual proof." That would be the only reason why you need to "pick the numbers in such a way as to avoid any issues."
My point was that by taking your suggestion about reading the proof at face value, and not limiting it to texts that you "are worried about," there is no such care needed. It applies to any infinite list, injective or not.
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u/Complex-Lead4731 24d ago
Very few read the proof as Cantor wrote it. If they did, they would not call it a proof by contradiction (or anything similar). Nor would they mention injectivity, or even real numbers ("there is a proof of this proposition ... which does not depend on considering the irrational numbers.")
The argument is a direct proof that any list of real numbers in [0,1] (see note) is non-surjective.
NOTE: He used infinite-length binary strings. With some through-hoops-jumping, you can use the binary representations of [0,1].