I've never liked these thought experiments with infinity in them. They always have "and so on" in them. But if you're dealing with an infinite set, "and so on" can't complete during the life of the universe (or well, ever). This in itself seems like it solves the problem - you can never get pass the first step. It's an event horizon, literally.
if you're a programmer, disbelieving in infinity is a crippling flaw, because big-O analysis of algorithms faces the exact same kind of "event horizon" that you speak of.
I've never liked people who don't understand math and dress it up with philosophical gook. Math is a practical matter, not high philosophy. If you don't know math, your bridges will fall down and your airplanes won't fly.
The application of mathematics is a practical matter; I am not entirely sure what "high philosophy" is, but mathematics is also a conceptual, abstract discipline which can be investigated with no concern whatsoever for practical applications.
The reliance of analysis on Cauchy sequences is not "philosophical gook". zemaj may be naïve in her conclusions (or at least unaware of just how much mathematics one must give up when one takes a constructivist view), but she is hardly alone in her hesitancy about infinity. I wouldn't be sure what to make of the claim that Brouwer, Markov or Martin-Löf didn't know mathematics.
I wouldn't call not understanding the theoretical underpinnings of big-O analysis a "crippling flaw" for a programmer. As long as you understand what it means that an algorithm is O(2^n) and what limitations that entails, then you're grand. If you can also look at an algorithm you've written and work out what its big O complexity is then you know more than most programmers. Both of those can trivially be done without ever touching infinity.
Equally if all you care about is working out if your bridge is going to fall down, you don't really have any need for infinity. In fact you can spend an entire career using math to do all kinds of really awesome things without ever actually understanding math. I'd say that a good 95% of engineers fall into that last category, and I'm certainly not worried about driving over bridges because of it.
Right. I was about to comment that that's because you're assuming an infinite domain, but the above definition becomes useless unless you do.
The point, though, was that the behavior characterized by the notation is perfectly reasonable even with finite domains (even if the specific notation is not, which was a fatal flaw).
Not only that but the obviousness of "and so on" is often elided. "And so on" assumes unbounded time and/or resources and also assumes you know intimately the (undoubtedly recursive) algorithm and can codify it. These proofs often blatantly ignore that _in practice_ someone or something must perform this algorithm. (Er, not sure if I used elided correctly back there.) Anyhow, if you have an innate distaste for these types of proof then rest assured that you are not alone.
Also check out intuitionism by keerazy Dutch mathematician L.E.J Brouwer which asserts that we must essentially modify some preconceived logical tenets/laws in the face of infinity.
Apologies if you were aware of all this already. I have found it very helpful to respect my suspicions. Regardless of what people may tell you, this stuff is neither simple nor straight-forward once you start thinking deeply enough about the minutiae.
