Cantor is wrong, yes, but you won't prove that to contemporary mathematicians who believe his ideas because this is an epistemological issue and can't be resolve with mathematical arguments.<p>This is really an age-old philosophic dispute that the mystics won decades ago. As David Hubert described it "No one shall expel us from the Paradise that Cantor has created." It is the "paradise" of deuces wild for the mathematicians and it rests on an equivocation of the meaning of "infinity". Cantor's idea of a "completed infinity" is a self contradiction if you grasp what the concept of infinity actually means and keep it tied to reality. It is the error of treating infinity as a real thing and not an abstraction. A similar error, for similar reasons, is made in the history of the philosophy by the mystics of nihil who wanted to treat nothingness as on par with existence via the Reification of Zero.
Your argument is indeed completely wrong.<p><i>Let me construct a new positive even number as follows. Take the first positive even number, then add the second positive even number, then add the third positive even number, etc.</i><p>The set of even number is infinite. And this is a divergent sum, it's infinite, not an even number.
The new string has to be placed in the assumed list of Reals. If you place it anywhere, you have to recast the Reals.<p>It's not just that it's new by algorithm, but it does not have a deterministic place within the assumed list of Reals.<p>It's a new "Cantor's Paradise" every time you place that new number. And it does not come into existence by the same procedure used to START generating the list. It's a metanumber -- exception to rule. This is why one has to have a metaphysics of number before one accepts the proof. Why Hilbert et al accept the proof on aesthetic grounds, and a whole class of mathematics on aesthetic grounds. The appropriate response is: That's just another real number.<p>Naturals, etc START somewhere but you are not forced to reindex with each new number.
In your set of positive even numbers example, you are only showing that a specific enumeration scheme does not generate the set of even numbers. A diagonalization argument requires that you show that no such enumeration scheme can exist (not just a particular chosen one). In the typical proof of diagonalization, it does not presuppose any particular scheme of listing the numbers. It just assumes that some arbitrary one exists, then derives a contradiction.<p>For your binary tree construction, a trancedental real number can be represented by a path of infinite length down the tree. Your argument is that since a breadth first traversal will eventualy exhaust that whole path, that the real number described by it will have been encountered. There are two ways to interpret what you are doing wrong:<p>- If we are indexing/pairing these nodes by time steps (an index), your construction is using a countably infinite time step to express the numbers described by an entire path (which defeats the point of being countable).<p>- For countable sets, you have to give me an index of finite size. If I give you a real number, you need to return a natural number (or equivalent) that indicates where it is. To test this, give me the index of pi in your claimed "countable" enumeration. The reason you wont be able to somewhat follows from Cantor's diagonalization scheme.<p>You might want to do some Googling before spending the time to write up an entire blog post about it (god forbid posting it to HN). The mistake you made is very common and has been discussed to death. You would have caught it.
Why do programmers assume they are mathematicians? There is a huge difference between those disciplines.<p>Trying to uphold the HN principles here, with a civil but pointed question. Computer science is deep, so is physics, and mathematics as well but they are distinct and quite different. Steven Hawking fumbled the proof of the infinitude of primes in his excellent survey of mathematical history "God Created the Integers," as usual for physicists he saw his way to the proof without traversing each step.<p>Infinity is a difficult but knowable concept. Calculus is a good teacher of this, and Euclid is a good teacher of proof. That's what's missing I say: the standard of proof. For physicists and computer scientists that standard depends on the outcome of physical events, but mathematics requires a purity that doesn't exist in the physical world.