I was very relifed to discover later in life that I wasn't alone in disagreeing with Cantor. That, along with a few other proofs and axioms, were a major source of angst for me as a kid in school.<p>I find Wittgenstein's take on it very interesting.<p><a href="http://plato.stanford.edu/entries/wittgenstein-mathematics/" rel="nofollow">http://plato.stanford.edu/entries/wittgenstein-mathematics/</a><p>I wish some teacher had told me that other maths are possible if we agree with different rules, even though schools are always going to teach and require the status quo.
A key part of Turing's computability paper hinges on the diagonal argument. It's really well covered in 'The Annotated Turing' by Charles Petzold.<p>In Cantor's version, we prove that numbers (real numbers?) are not enumerable by creating a new number.<p>In Turing's paper, he enumerates all valid Turing machines in a similar way, but crucially, we can't know if the new constructed machine is valid or not, so the outcome is different.<p>It's quite possible I'm botching elements of these, but it was a beautiful read.
Intro to Real Analysis (or whatever your school calls it) -- where Cantor's proof is normally covered -- is really a gem of a course and the first time since 8th/9th grade geometry that students are clearly shown why math is the queen of the sciences.<p>Unfortunately, in my experience, a lot of future high school math teachers skipped it and not because it's hard but because the class before it (intermediate calc) was a huge step up from high school calculus (yes they should take intro to calc at college level but kids who do well in high school calc always want to skip it). What ends up happening is you get a lot of high school calc teachers who don't really understand calculus all that well and can't answer questions and so they just end up teaching to the AP exam and the vicious cycle reinforces itself.
I read this for the first time in the excellent and surprisingly accessible The Annotated Turing[0], which I can highly recommend. If you're vaguely interested in things like proofs like these or about computability or just Turing's and others' contributions and approaches, the book approaches these things very well without presuming a deep pure mathematical background.<p>Seriously, read it.<p>[0] <a href="http://www.amazon.com/Annotated-Turing-Through-Historic-Computability/dp/0470229055/ref=sr_1_1?s=books&ie=UTF8&qid=1383018325&sr=1-1&keywords=the+annotated+turing" rel="nofollow">http://www.amazon.com/Annotated-Turing-Through-Historic-Comp...</a>
One of my favorite proofs. It blew my mind when I understood it; it's so simple in hindsight. I've tried to explain it to non-technical people and haven't has as much luck conveying why it's so cool.
"The digits of every rational number repeat after some finite number
of digits, so the "period" of every rational number is finite.
However, there is no upper bound on the period of rational numbers,
i.e., the periods are all finite, but there is no largest period.
Thus, in a manner of speaking, the least common multiple of this set
of strictly finite things is infinite."<p>Got lost here, what is the LCM of this set; which set?
Here's a proof of Cantor's Theorem in Metamath, which is one of the simplest formal systems in the world: <a href="http://us.metamath.org/mpegif/ruc.html" rel="nofollow">http://us.metamath.org/mpegif/ruc.html</a>
<a href="https://gomockingbird.com/mockingbird/#xl6a68x" rel="nofollow">https://gomockingbird.com/mockingbird/#xl6a68x</a><p>Here I use the Diagonal Proof to talk about Observation Statements (or Observa) / Mental States (of Observers) such that Diagonal Statements (DiaSt), once normalized for all Observers, can be measured along arbitrary distribution of duration ranges (T) of a non-local universe. Given this, we can predict the upper/lower bound of periodic collapse of quantum systems (our Model).<p>I've also seen Cantor's Proof applied as a metaphor for the analysis of Web Artefacts: Hypermedia Types (X axis) / Web Components (Y axis). (I see "soft objects" as a species of Web Artefact.)<p>There are many interesting applications. For me, personally, I think Cantor's Proof gives us a means of assuming the periodic collapse of quantum systems w/r/t quantum observers (it is a myth that "your" mind and "my" mind are discrete entities) such that quantum field theory can be salvaged.