That said, the question of whether the natural numbers can be "counted" in any meaningful sense of the word -- stipulating that technical conditions in formal set theory are not necessarily meaningful -- is a good one. — fishfry

My point is that "the natural numbers" is defined in such a way that is impossible to count them all. No matter how many you count, there will always be more. The set of natural numbers is infinite and this means that it is impossible to count, uncountable, by definition.

Aletheist claims that it is logically possible to count them, therefore they are countable in principle. I don't agree because I see that they are uncountable, by definition, and therefore the idea that they are countable is contradictory.

Apparently you are in such a big hurry to reply that you are not even bothering to pay attention to what I actually post. In this case, you are mixing up the first two definitions that I so carefully spelled out. The first one, which directly quoted fishfry, is the one from set theory - not my set theory, but standard set theory - and if it helps, we can substitute "foozlable" as he just suggested (again). The second one - the one that I assume you are still criticizing - has nothing to do with set theory at all, as @fishfry helpfully pointed out a while ago. We simply disagree on whether "countable" always and only entails the ability to finish counting; you say yes, I say no. — aletheist

As I said, I accept the first definition, so long as we adhrere to the principles offered by fishfry. This definition has nothing to do with counting whatsoever, it is completely unrelated, that's why fishfry offered a completely different word. I have no problem, as long as we don't ambiguate between the first and third. You, for some reason want to create a bridge of relationship between the first and third, so you've offered the second. I see this as an attempt to ambiguate, to create the means for equivocation. Also, you've already argued that foozlable means "potentially countable", and "countable in principle". And you've continued to argue this long after fishfry offered the means for complete separation. So you continue to make efforts for equivocation.

Now, if we maintain this separation, and put foozlable aside for now, It is apparent to me, that you do not actually believe that the set of natural numbers, as infinite, is uncountable in an absolute sense. You keep wanting to say that in some sense it is countable. But it is defined in such a way as to be uncountable, absolutely. You keep implying that you believe that the infinity of natural numbers is logically countable, or countable in principle.

We simply disagree on whether "countable" always and only entails the ability to finish counting; you say yes, I say no. — aletheist

I really have no idea of what this means, what you are proposing here. As far as I understand what counting is, something must be counted in order to be counted, If you do not finish counting something then it is not counted. If you cannot finish counting it then it cannot be counted. If it cannot be counted, then it is uncountable. Are you proposing some type of partial counting? If so, then even if I accept this principle that something might be "partially countable", and that this is a meaningful principle, then you still need to produce an argument to support your claim that "partially countable" means the same as "countable".

If you do not finish counting something then it is not counted. If you cannot finish counting it then it cannot be counted. — Metaphysician Undercover

This right here is where we disagree. To count something is not the same as to finish counting it. Being able to count something is not the same as being able to finish counting something. In other words ...

Everyone who is not being deliberately obtuse understands what countable means - it means you can count elements of the set. No one, unless they are being deliberately obtuse, thinks that this fact has any bearing on whether anyone would be willing to embark on counting all the members of a very large or even infinite set. — tom

We simply have different non-technical definitions of "countable." It is not the case that yours is true and mine is false, or vice-versa; they are just different.

We simply have different non-technical definitions of "countable." It is not the case that yours is true and mine is false, or vice-versa; they are just different. — aletheist

Given the Naturals, I can count some of them, in order e.g. 999 1000 1001

Given the reals, please count the three members that come after 0.999... and tell me what they are, or even what the next number is so we know we have only two of them.

I think the point of counting is to quantify. Every step of the way in counting one has quantified the stuff up to that point (so it's weird to think of counting numbers. What?)

Anyway, there need be no counting in Zeno's Paradox. If you say the golf ball must arrive at the center point in order to make it to the hole, you just bought the whole enchilada.

This right here is where we disagree. To count something is not the same as to finish counting it. Being able to count something is not the same as being able to finish counting something. — aletheist

As I said, your definition appears like nonsense to me. To be able to do something, is to be able to complete that task. Being incapable of completing that task, is failure. When failure is guaranteed, then the claim of being able to do that task is completely unjustified.

You claim to be able to count something, when failure is guaranteed, and this is an unjustified claim of being able. The fact that you can attempt a task does not justify the claim that you are able to do the task.

But we've already solved the paradox: it is merely a confusion between an abstract attribute and a physical attribute of the same name.

Since it is possible to prove theorems about abstract mathematical attributes, which have the status of necessary truths, we are misled into assuming we have a priori knowledge of the real physical attribute of the same name. We don't.

As I said, your definition appears like nonsense to me. To be able to do something, is to be able to complete that task. Being incapable of completing that task, is failure. When failure is guaranteed, then the claim of being able to do that task is completely unjustified. — Metaphysician Undercover

Can you count the number of real numbers between 0 and 0.1? If so, how many are there?

Can you count the number of naturals between 1 and 50,000,000, if not, how many are there?

Can you count the number of real numbers between 0 and 0.1? If so, how many are there? — tom

I already answered this days ago, I don't see the relevance. Why do you keep asking?

How many are there? Can you count them? Or is it impossible to count the real numbers, making them uncountable?

Contrast that with the Naturals, which, by definition you can count. Just try it 1, 2, 3, 4, 5. How many was that?

How many are there? Can you count them? Or is it impossible to count the real numbers, making them uncountable? — tom

As I said to you days ago, it's impossible.

Contrast that with the Naturals, which, by definition you can count. Just try it 1, 2, 3, 4, 5. How many was that? — tom

No, you can't count the natural numbers either, because they're infinite. That's the point I'm arguing with aletheist, they are by definition uncountable, because by definition they are infinite, and infinite is by definition endless, which is by definition uncountable.

The fact that you can be counting the natural numbers does not prove that they are countable. Does the fact that a person is walking on the earth, and claims to be walking around the earth, prove that the earth is walkable? We can only get to the conclusion which you and aletheist desire, through the fallacy of composition.

But we've already solved the paradox: it is merely a confusion between an abstract attribute and a physical attribute of the same name. — tom

I don't think so. It's just a simple question: does the golfball have to arrive at the center point before it can make it to its destination? Common sense says yes. Infinite regress appears.

Note that the regress is headed back to the starting point, not the destination.

This is questionable though. We can understand time as discrete units, or we can understand time as a continuity. We can also understand it as some kind of composition of both. What if real time, which we are experiencing, consists of discrete units, and it is just the brain and living systems which are creating the illusion of continuity? I tend to think that the only real continuity is the existence of the soul itself, and the soul, during the act of experiencing, renders the appearance of time as continuous, to make it compatible with its own existence, and therefore intelligible to the lower level living systems. Now, as highly developed life forms, we have developed mathematics, which will allow us to understand the true nature of time, as discrete, but we must get beyond the way that time is presented to us by our lower level living systems, (i.e, that intuitive impression of time) to be able to understand time mathematically. — Metaphysician Undercover

If one takes the position that duration (real time) is consciousness that endures - which is precisely what we experience - then it is difficult to explain the notion of discrete. Are we constantly dying and being reborn in some discrete firm of unknowable duration? It would seem that continuity more accurately reflects our actual experience.

I am inclined to subscribe to how Peirce addressed this.

'Real' is a word invented in the 13th century to signify having Properties, i.e. characters sufficing to identify their subject, and possessing these whether they be anywise attributed to it by any single man or group of men, or not. Thus, the substance of a dream is not Real, since it was such as it was, merely in that a dreamer so dreamed it; but the fact of the dream is Real, if it was dreamed; since if so, its date, the name of the dreamer, etc. make up a set of circumstances sufficient to distinguish it from all other events; and these belong to it, i.e. would be true if predicated of it, whether A, B, or C Actually ascertains them or not. The 'Actual' is that which is met with in the past, present, or future. — aletheist

There are experiences (memories) that are shared and those that are not. It is quite a task to separate those that are Real from those that are not. Best to just accept them as all being Real, with different attributes of firm. My dreams are very real to me. If I relate them, then they also become real to others though in a different qualitative sense. What all experiences share is the essential quality that they are memory of some sort.

I don't think so. It's just a simple question: does the golfball have to arrive at the center point before it can make it to its destination? Common sense says yes. Infinite regress appears.

Note that the regress is headed back to the starting point, not the destination. — Mongrel

Yet nothing physically infinite happens, and what motion is possible is determined by the laws of physics alone, and not by the necessary truths about an abstraction that bears the same name.

Common sense dictates that Zeno's mistake was to PRESUME that a certain mathematical notion called "infinity" is physically relevant.

Given the reals, please count the three members that come after 0.999... and tell me what they are, or even what the next number is so we know we have only two of them. — tom

Why are you asking me? You and I agree that the real numbers are not countable.

To be able to do something, is to be able to complete that task. — Metaphysician Undercover

See, this appears like nonsense to me. One can be able to do something on an ongoing basis, such that whether one is able to complete that task is irrelevant. I am able to be thinking about elephants, so elephants are thinkable. I am able to be breathing earth's atmosphere, so earth's atmosphere is breathable. I am able to to be walking on the earth, so the earth is walkable. And I am able to be counting the natural numbers, so the natural numbers are countable.

Yet nothing physically infinite happens, and what motion is possible is determined by the laws of physics alone, and not by the necessary truths about an abstraction that bears the same name.

Common sense dictates that Zeno's mistake was to PRESUME that a certain mathematical notion called "infinity" is physically relevant. — tom

Zeno's intentions aside, you aren't solving the paradox by saying this. You're merely restating it. Infinite regresses appear from time to time. Philosophers usually take them as a sign that something's wrong. Consider Frege's and Quine's reactions to the regresses they discovered. No one says, "Oh that's just a fluke of the mind... I'll proceed on as if I never noticed that."

No. We pay attention to regresses because philosophy is the domain where we're free to take note of such impractical doo-dads.

If you're planning a trip to the Grand Canyon, feel free to ignore Zeno's Paradox. It has nothing to do with your trip. And by the way, why are you going this time of year? Don't you know the road to the North Rim is probably closed?

No, you can't count the natural numbers either, because they're infinite. That's the point I'm arguing with aletheist, they are by definition uncountable, because by definition they are infinite, and infinite is by definition endless, which is by definition uncountable. — Metaphysician Undercover

So, you agree you can't count any interval of the real numbers.

So, you agree you can count any (not too big or you'll get bored) interval of the natural numbers.

Given any interval of the natural numbers, you can calculate how many natural numbers are in that interval, even if the interval is too big to actually count in a lifetime.

One of these infinities is bigger than the other, much bigger. In fact the measure of the natural numbers on the continuum is zero.

My dreams are very real to me. — Rich

In my view (and Peirce's), they are either real (full stop), or they are not. Something is real if and only if has properties that do not depend on what any one person or finite group of people think about it. The properties of your dreams depend entirely on what you think about them, so they are not real by this definition.

I believe on analysis Pierce's definition is impossible to implement, e.g. defining properties independent of a person or a group of people. For this reason, I don't embrace it. I prefer viewing things exactly as experience warrants it, i.e. properties are revealed by individual observation followed by group consensus subject to contiguous change as observations and memories of those observations change.

I believe on analysis Pierce's definition is impossible to implement, e.g. defining properties independent of a person or a group of people. — Rich

The issue is not defining properties independent of a person or group of people, it is things having properties independent of what any person or group of people thinks about it.

The issue is not defining properties independent of a person or group of people, it is things having properties independent of what any person or group of people thinks about it — aletheist

This then goes into defining what is a property that is independent of that which is defining the property. From my position, everything is simply "fields" given definition by consciousness. In this regard, the internal dream field shares similarities with external fields but are different in how they are shared.

If one takes the position that duration (real time) is consciousness that endures - which is precisely what we experience - then it is difficult to explain the notion of discrete. Are we constantly dying and being reborn in some discrete firm of unknowable duration? It would seem that continuity more accurately reflects our actual experience. — Rich

What I think is that it is necessary to assume that the entire physical world is reborn, comes into existence anew, at each moment in time, and this is discrete existence. But as I said, the soul provides continuity, so it is not the case that we are constantly dying and being reborn, the soul is immaterial and not part of this discrete material existence. So as living souls, continuity is our actual experience. But when we deny dualism we suffer from the illusion that the physical world is continuous as well as our own existence as living beings.

See, this appears like nonsense to me. One can be able to do something on an ongoing basis, such that whether one is able to complete that task is irrelevant. I am able to be thinking about elephants, so elephants are thinkable. I am able to be breathing earth's atmosphere, so earth's atmosphere is breathable. I am able to to be walking on the earth, so the earth is walkable. And I am able to be counting the natural numbers, so the natural numbers are countable. — aletheist

You appear to be making a category error. "Counting" is an activity of the subject, "countable" is a property of the object. In order to deduce from the activity of counting, what it is that is countable, requires that you identify what it is that is being counted. If; it is just a part of the set of natural numbers which is being counted, then it is that part which is countable. If it is part of the earth's atmosphere that you are breathing, then it is that part which is breathable. If it is a part of the earth's surface that you are walking on, then it is that part of the earth's surface which is walkable. If you proceed from what is known about a part, to make a conclusion about the whole, then you commit the fallacy of composition.

One of these infinities is bigger than the other, much bigger. — tom

You did not describe the infinity of the natural numbers, which is that they continue forever, endlessly. And no, the infinity between two real numbers, (no matter how large or small those numbers might be), is no bigger than this infinity. They are both infinite. One is not a bigger infinite than the other, that it nonsense.

You appear to be making a category error. "Counting" is an activity of the subject, "countable" is a property of the object. — Metaphysician Undercover

More nonsense. You are the one who wants to define "countable" entirely on the basis of whether it is actually possible for a subject to finish "counting" the object.

If you proceed from what is known about a part, to make a conclusion about the whole, then you commit the fallacy of composition. — Metaphysician Undercover

When I say that elephants are thinkable, or that earth's atmosphere is breathable, or that earth's surface is walkable, or that the natural numbers are countable, I am not reasoning from part to whole. I am not referring to any particular part of each thing, I am stating a general property of each thing. Elephants in general are thinkable, earth's atmosphere in general is breathable, earth's surface in general is walkable, and the natural numbers in general are countable. This is a perfectly legitimate and common use of language.

You are the one who wants to define "countable" entirely on the basis of whether it is actually possible for a subject to finish "counting" the object. — aletheist

That's not true, because I've claimed that the object, being the set of natural numbers is uncountable by definition, that means nothing, not even God could count it.

When I say that elephants are thinkable, or that earth's atmosphere is breathable, or that earth's surface is walkable, or that the natural numbers are countable, I am not reasoning from part to whole. I am not referring to any particular part of each thing, I am stating a general property of each thing. Elephants in general are thinkable, earth's atmosphere in general is breathable, earth's surface in general is walkable, and the natural numbers in general are countable. This is a perfectly legitimate and common use of language. — aletheist

You are making unjustified assertions, and this is perfectly legitimate, common use of language. But if you want to prove any of these assertions, you need to justify them. And you cannot prove a general conclusion about the whole, by demonstrating that it is true of a part. So if you prove that a part of the set of natural numbers is countable, this does not prove that the whole is.

And if you make the unjustified assertion that the natural numbers are countable, we have to juxtapose this to the contrary, and justified claim that the natural numbers are uncountable.

But if you want to prove any of these assertions, you need to justify them. — Metaphysician Undercover

Why would I want or need to "prove" or "justify" a generic definition of "x-able" that I have (repeatedly) stipulated?

You haven't stipulated any reasonable definition of countable. You made a broad description concerning the activity of counting, and the assertion that the natural numbers are countable. Earlier you made some form of definition of countable:

My notion of "potentially countable" or "countable in principle," which is that there is no particular largest value beyond which it is logically impossible to count. — aletheist

But this renders all sets with a largest value as uncountable.

You haven't stipulated any reasonable definition of countable. — Metaphysician Undercover

Your opinion is duly noted. :-}

You did not describe the infinity of the natural numbers, which is that they continue forever, endlessly. And no, the infinity between two real numbers, (no matter how large or small those numbers might be), is no bigger than this infinity. They are both infinite. One is not a bigger infinite than the other, that it nonsense. — Metaphysician Undercover

It's a very important result in mathematics. The continuum has the cardinality of the power set or the natural numbers. It's a much bigger infinity.

You are of course free to deny knowledge and maintain your willful ignorance.

It's a very important result in mathematics. The continuum has the cardinality of the power set or the natural numbers. It's a much bigger infinity. — tom

What do you mean by "bigger?" Bearing in mind that there are countable models of the reals? https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem

As is the unfortunate custom in this thread, a word with a highly technical meaning in mathematics is being conflated with the same word in its everyday usage. In math one set is "bigger" than another if the smaller set can be injected but not surjected to the larger. But that does not actually correspond to the everyday notion of "bigger," which the Lowenheim-Skolem result shows.

Along these lines ...

One of these infinities is bigger than the other, much bigger. In fact the measure of the natural numbers on the continuum is zero. — tom

What do you make of the Cantor set, an uncountable set of measure zero?

https://en.wikipedia.org/wiki/Cantor_set

Bijection does not preserve measure. You can see that by simply multiplying each element of the unit interval by 2. Now you have a bijection between sets of measure 1 and 2, respectively.

the set of natural numbers is uncountable by definition — Metaphysician Undercover

Actually, the set of natural numbers is countable by definition, as in mathematics a countable set is defined as a set with the same cardinality as some subset of the set of natural numbers.

Of course, if by "countable set" you mean something else, then the disagreements here are just a case of talking past each other.