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Comment by andrewla | Hacker News Reader

The Cantor conception of "infinity" is bad and should be retired. It's a pure naval-gazing exercise that has set back practical mathematics significantly. The notion of the "continuum" is fundamentally flawed. Intuitionalism and constructivism are a much better framework for reasoning about mathematics.

The response to the idea that the natural numbers and the even numbers and the rational numbers and the computable numbers have the same "cardinality" should be "who cares". All it means is that Cantorian cardinality is a bad standard against which to judge the "size" of an infinite set.

It ends in the same trap as the Axiom of Choice, where you can produce lots of obviously incorrect results and remark about how amazing they are; similarly you can make nonsensical statements about things being true "almost everywhere" that are clearly actually true nowhere.

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> [Intuitionism] and constructivism are a much better framework for reasoning about mathematics.

Just for the sake of non-mathematicians, allow me to note here that despite intuitionism/constructivism being around for many decades, 99%+ of mathematics research done today is not, in fact, performed in these frameworks. These are just a curiosity that few working mathematicians actually care about. These approaches do have some certain philosophical benefits, but they have some extreme practical disadvantages that result in overwhelming majority of mathematicians rejecting the notion that these are “better frameworks”.

Your comment to me reads like saying that autogyro is a better framework for powered aviation than fixed wing or helicopters: regardless of your actual arguments in its favor (which may in fact be good), the fact that 99%+ of the industry disagrees is rather telling.

In addition to this, I'd like to add that intuitionistic logic is consistent if and only if classical logic is. This follows from the Godel-Gentzen negative translation[0], which implies that for any contradiction in classical logic, you can get the same contradiction in intuitionistic logic more or less by adding "not not" before both sides of the contradiction. The same applies to the axiom of choice: set theory with choice is consistent if and only if set theory without choice is consistent[1].

This means that you don't get any safety by rejecting the law of excluded middle, nor by rejecting the axiom of choice. For this reason, I think intuitionistic logic is trading away a lot of power for basically no gain.

[0] https://en.wikipedia.org/wiki/G%C3%B6del%E2%80%93Gentzen_neg...

[1] https://en.wikipedia.org/wiki/Axiom_of_choice#Independence

> All it means is that Cantorian cardinality is a bad standard against which to judge the "size" of an infinite set.

But do you have something better? It’s not as if mathematicians immediately accepted this as the way forward. There was a long struggle accepting that lots of statements that are true about finite sets do not extend to infinite ones (examples: “adding an item to a set makes it larger”, “when summing a set of numbers, the result doesn’t depend on the order you do it”)

I think that you either have to accept this as the best way to treat infinite sets, or have to give up the notion of infinite sets, and that has its problems, too. For example, it would mean there’s a largest integer.

We don't have to give up on infinite sets, but instead think of them as limiting processes instead of a complete object. Basically sacrifice the notion of "set" as a realized object. Just like sqrt(2) is not a rational number, but it is a computable number, which means that there is a computation that can yield an approximation that matches reality to whatever precision is requested.

Similarly the natural numbers are a process that can produce more natural numbers, and can do so for however large a limit you choose.

The problem of rearranging the elements of a set to get a different sum is handled here because you are looking at the limit of the sum, not the sum of the abstract set. Just as in classical math you have to make sure that to compute { sum_i=0^inf (-1)^i / i } that you are actually computing { lim N->inf sum_i=0^N (-1)^i / i }, with intuitionist math you can't help but do the latter, because the former statement makes no sense except as the latter.

The "adding an item to a set makes it larger" is only problematic if you're trying to define "larger" for non-terminating sets, and in addition trying to treat them as "sets" divorced from their underlying structure. The rationals and the integers may look the same to a set theorist, but they are clearly very different objects and it's not at all clear what additional power you get from introducing morphisms that don't preserve any underlying structure.

Cantor's work on infinity and the diagonal argument was hugely important in mathematics, paving the way for important results like Godel's incompleteness theorems, the halting problem, the creation of modern set theory which allowed unifying effectively all of known mathematics into one theory, etc.

> It's a pure naval-gazing exercise

> The response to the idea [...] should be "who cares".

If anything sets back mathematics, it's when people have this kind of attitude towards the parts of math they find unintuitive.

Mathematics is fine. Uncountable infinity is really only a problem in physics.

David Hilbert was a fan. Cantor's paradise he called it.

https://en.m.wikipedia.org/wiki/Cantor's_paradise

I'm not really familiar with math beyond linear algebra and googling was not very helpful; could you please give an example of an obviously incorrect result enabled by the axiom of choice? No need for detail, just the name of an example is good enough for me.

As far as axiom of choice goes, they were probably referring to https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

Whether or not it is "wrong" is in the eye of the beholder. Why should infinite things be intuitive? After all even "real" things in the universe are highly unintuitive - e.g. quantum mechanics sounds obviously "wrong" at first glance.

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