Really? Cantor proved the reals constitute a continuum. Whatever they are, they are certainly not discrete. — tom

The reals in their usual order are a continuum. They can be reordered to be discrete. Counterintuitive but set-theoretically true. Order is important in this discussion. The rationals in their usual order aren't discrete, but we can line them up in bijection with the natural numbers and thereby identify the first, second, third, fourth, etc.

Also (not to hijack this thread further, but this relates to Zeno) what do we mean when we say the real numbers are a continuum? If by continuum we mean a particular philosophical idea of a continuous space, then the mathematical real numbers may or may not satisfy a philosopher. If by continuum we mean the standard mathematical real numbers, then we are being circular. Certainly the standard real numbers are not a proper model of the intuitionistic continuum. These are murky philosophical waters.

given enough time, someone or something could count up to and beyond any arbitrarily specified value. — aletheist

Which has absolutely nothing to do with the question of whether a given set is foozlable -- able to be bijected to the natural numbers. I don't want to pile on so I'll just refer to what I've already said. The reason it's important is because this thread was percolating along on the usual lines -- nature of time and space, whether the mathematical solution of Zeno in terms of infinite series is also the physical solution, etc. Once you conflated the technical meaning of countable with its every day meaning -- a logical fallacy -- the thread lurched off on a very unproductive tangent IMO. It is far from clear that "given enough time" you could count to any specified value. If time itself is part of the universe, then you will run out of time between the Big Bang and the heat death of the universe.

Besides: What you just said is that for any given number N, you can count up to N. That is manifestly false for the reason I gave. But even if I grant you that point you still can't count ALL of the natural numbers. You have just conflated counting up to some big finite number with counting ALL the natural numbers. A big logical fallacy. Consider that the statement "X is a finite set" is true for each natural number, but not for the collection of ALL natural numbers.

So even if you can count to a zillion, and a gazilloin, and a googolty-googol-gazillion, you can't count ALL the natural numbers using that same meaning of counting. Right?

I guess you must deny, then, that the integers are countable, since nothing and no one can actually count them all. And yet it is a proven mathematical theorem that not only the integers, but also the rational numbers are countable - i.e., it is possible in principle to count them - despite the fact that they are infinitely numerous. — aletheist

I went back through this thread from the beginning. Finally on page 11, this quote is the first mention of mathematical countability. The above quote is simply flat out wrong. It commits the fallacy (does it have a formal name?) of confusing a term of art with its everyday meaning. Countability as defined in mathematics simply has nothing at all to do with the everyday meaning of the ability to be counted. I already made this point but now I found the source of the recent confusion in this thread.

A child learning to count, "one, two three, four, ..." has absolutely nothing to do with mathematical countability. Saying that a set is countable does NOT mean "it is possible in principle to count them." It means exactly that there exists a bijection from the natural numbers to the set. Nothing more and nothing less.

You know the old joke. "Why can't you cross a mountain climber with a mosquito? Because you can't cross a scaler with a vector." That joke depends on conflating the engineering definitions of scalar, vector and cross (as in cross product) with the common English meaning of a climber -- a "scaler" -- and the medical meaning of vector -- a means of disease transmission, and the biological meaning of cross, as to cross-breed living things based on their genetic makeup.

But this is a JOKE, not something you can take seriously in a philosophical discussion. You can not, unless you being disingenuous, say that "The rational numbers are countable" and then say this shows that a child could count them in the every day sense of the word.

If you counted, in the sense of saying out loud "one, two, three ..." the natural numbers, starting at the moment of the Big Bang, at the rate of a number per second; or ten numbers, or a trillion -- you would not finish before the heat death of the universe.

You are simply conflating a term of art -- a technical term used with a specific meaning in a specific context by specialists -- with the everyday meaning of the term.

Sorry to be ranting now but really, the quote above is terribly wrong. You can't count the natural numbers in the every day meaning of the word. There are infinitely many of them. The natural numbers are countable, in the technical sense that there exists a bijection between the natural numbers and themselves. If you think to yourself, "The natural numbers, the integers, and the rational numbers are examples of foozlable sets," you will not confuse yourself or others by shifting the meaning of a technical term to its everyday meaning.

You can't count the members of an uncountable infinity. There is no such thing as a next member. — tom

You can certainly well-order an uncountable set. You need the Axiom of Choice to well-order the real numbers, but you do not need Choice to show the existence of the first uncountable ordinal. That Wiki page is light on detail but the idea is that the set of all countable ordinals is an ordinal (needs proof of course), and it can't be a countable one (because a set can't be a member of itself), hence it must be an uncountable ordinal. Such a thing is impossible to visualize but it exists.

An ordinal is an order type of a well-ordered set. A set is well-ordered if every nonempty subset has a smallest member. There's a first, then a second, then a third, etc. Clearly the natural numbers are well-ordered. Now to get to larger ordinals you have to allow limit ordinals, which are unions of upward chains of ordinals. I don't want to get technical, which is why in my earlier post I just wrote

And Cantor's transfinite ordinals let you count way past the natural numbers. — fishfry

That's why the usage of the everyday meaning of counting is totally out of place here. It's vague, and mathematicians can indeed well-order uncountable sets.

Right, this is all that I meant when I said that the natural numbers are countable by definition. — aletheist

No I'm afraid you are still missing my point. I defined a set as countable if it can be put into bijection with the naturals. You claim this "defines" the naturals as countable but I say, "I don't believe you. Prove it." And you say: "Aha, the identity function on the naturals is a bijection." You have PROVED directly from the definition that the naturals are countable. It's a theorem (admittedly so easy it's never stated explicitly) and not a definition.

That is obviously not what infinite means within mathematics, since the natural numbers and integers are very explicitly defined as countably infinite. — aletheist

This is not true. A set is defined as countable if it can be put into bijection with the natural numbers.

By this definition we can then show that the natural numbers, the integers, and the rationals are countable; and that the reals aren't. We define "countable" as a technical term, having no meaning other than that which we've given it. We then prove that the naturals and integers are countable. Formally, having defined the technical term "countable," we then note that the identity map on the naturals, which is a bijection, proves that the natural numbers are countable. Then we prove that the integers are countable by lining them up as 0, 1, -1, 2, -2, ...

As Tom mentioned earlier, much confusion would be avoided if Cantor had picked another name. If we say a set is foozlable if it can be bijected to the natural numbers, then we can prove that the natural numbers, the integers, and the rationals are foozlable; and that the reals aren't. But nobody would have to spend any time arguing about whether you can count the elements of an infinite set.

Surely we all agree that technical terms have specific meanings in context that do not necessarily correspond to their meaning in everyday language. An engineer and a doctor give very different meanings to the word vector. Nobody gets confused, because within their respective technical disciplines the word vector has a formal definition. In the legal profession such words are called "terms of art." A term of art is a word or phrase that has a specific technical meaning within a given discipline that is unrelated to any common meaning.

It's a mistake to think that countability has anything to do with the ability to be counted. That's way too vague. For one thing it's arguably false for the everyday meaning of the word "count." And Cantor's transfinite ordinals let you count way past the natural numbers. Better to simply realize that in set theory, "countable" means exactly one thing and one thing only: that a given set may be bijected to the natural numbers. What Cantor really meant to say is foozlable. Or in the original German, füzlich [That's a joke]. Now any semantic confusion goes away.