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Comment by tgb | Hacker News Reader

I'm a mathematician by training (though this is thoroughly outside the scope of a mathematician's job) and completely agree. That said, there's a real topic in the philosophy of mathematics called "finitism" that argues we should only consider finite mathematical objects and would exclude almost every real number. [1]

[1] https://en.wikipedia.org/wiki/Finitism

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Almost every is a very understated way to put that.

“Almost every” is a term of art in mathematics. It means basically “100% wherever you look, but not really 100%”

https://en.m.wikipedia.org/wiki/Almost_everywhere

Expressed as a percentage, it would be 100%, though ;)

Well, almost...

Nope, exactly. The measure of the interval [0, 1] excluding any countable subset (for example all rational numbers) is still exactly 1, that's usually what mathematicians mean by "almost every", here "almost every real number in [0, 1] is irrational" :)

I was joking, but to a point.

The problem is that "100%" =/= "All". But most of the time, it is considered to be equivalent. The "almost all" phrase is used instead of "100%" precisely because dividing by infinity doesn't yield a useful result in some contexts.

In more... "conventional" cases, if I tell you that 100% of the items are blue, you can conclude that none of them are red. With the intermingling of countable and uncountable sets, this statement fails. You have a situation where 100% of the items in the superset are blue, and there are also some red ones.

So it's "almost 100%" in the same way it's "almost all". I'm using the word "almost" to mean "infinitesimally close to"

Really, exactly 100%.

Yeah. But “100%”, and “all” or “every”, are considered synonyms most of the time. Not in this case though.

Isn't it the opposite, more like 100% everywhere you don't look, and nowhere you do? I think the really real numbers are the countable subset of them which are in some way expressible, and the uncountable subset which are in no way expressible except in vague generalities like "they're not shaved by the barber" are in a sense less real.

I like your definition of "look".

I was going with where a dart* would land (100% of the time it will "really" hit a point that can't be expressed using computable numbers), but you're using it to mean anything we can compute or express. That is, if we're looking somewhere, it has to be somewhere we can say that we're looking, and therefore expressible. If we take your meaning, then yeah, that's what I mean as well :)

* (Yes, the dart is an abstract one. No comments about the Planck length, please)

The example with the dart plays into the «random numbers» notion, i.e. you have to use some kind of random process to get hold (non-constructively) of such a number.

nice catch

How could you exclude "almost every real number", if such a thing should not even be considered?

You just leave the axiom of infinity out.

You simply choose not to invent any of them.

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