I don't see how any form of 'jection' is possible, if you cannot lay out all the members of the set, which is the case with an infinite set.

You have good insight into this. For a set to have a sensible cardinality, it needs to be able to be enumerated. — fishfry

Thanks fishfry, it's rare to see a complimentary comment here. It's only taken me days to get to this point. Notice that the recognition that something is not quite right (insight) played a very small part in getting to this point, the big part was persistence in analysis, to determine exactly where the problem is.

I don't really understand this objection. I can say that no matter what apple I eat, there will always be apples that I haven't eaten. Therefore apples aren't edible? — Michael

Natural numbers are countable, just like apples are edible, but that's a generalization. The entire set of natural numbers, being a particular defined thing, is defined as infinite, and is therefore not countable. Apples are edible. The entire set of all apples, if it is infinite, cannot be eaten.

I am making a statement about the nature of being infinite, what it means if a particular thing is defined as infinite, not an inductive generalization about a thing being counted or eaten, such as numbers or apples. However, if we assert that the thing being counted, or eaten, is infinite, then we must constrain ourselves with respect to what it means to be infinite, when we go to make other assertions about those things, in order to avoid contradiction.

When you assert that all natural numbers are countable, this is an inductive conclusion. This conclusion contradicts the defined essence of the set of natural numbers, as infinite and therefore uncountable. When you proceed in your mathematical operations, from the premise that all natural numbers are countable, you proceed from an inductive conclusion rather than from the true defined essence of the set of natural numbers. And these two premises are contradictory. The defined essence takes into account what it means to be infinite, the inductive conclusion does not. Therefore when you proceed from the inductive premise you will inevitably produce false conclusions concerning infinities. The one I've already seen on this thread is that some infinities are "bigger" than others.

So what exactly do you mean by saying that there are numbers which are incapable of being counted? — Michael

I mean exactly what I said, no matter how high you count, or even how high of a number you can name, there will always be higher, unnamed or uncounted numbers. That's the nature of being infinite. It means that the set of natural numbers is uncountable.

Because cardinality is defined in such a way that to have one does not depend on it being possibile to enumerate the entire set. — Michael

Cardinality is defined on Wikipedia as " a measure of the 'number of elements of the set'. It seems quite obvious that it is impossible to have a measurement of the number of elements in an infinite set. "Infinite" is not a number, nor is it a measure.

That the set of natural numbers has the same cardinality as the set of natural numbers just is that the natural numbers can be placed in a one-to-one correspondence with the natural numbers. We don't just take it for granted that it can; we can mathematically show that it can. — Michael

That is false. Being infinite, you cannot establish a bijection, just like you cannot count them. You might assume that if you could count them, you could place them in a one to one correspondence, but you cannot count them, so such an assumption is irrelevant. It's like saying if the infinite were finite, then we could do this. It's just a contradictory assumption. I am quite convinced that such a bijection cannot be done, it is a falsity. You say it can be mathematically shown, and it is not just assumed. Let's see the demonstration then.

Even your description of time explicitly capitulates to discrete measurements by insisting on "moments of time". This is precisely the problem. The are no moments of time but philosophers adopt this point if view becauset appears in equations. Further on you discuss parts of things, further concessions to discrete mathematical equations. — Rich

I've studied the nature of time for very many years now, and I've read a lot of related material. Here is some speculation. For the longest time, I believed as you do, that time is continuous. It appears quite obvious that all divisions in time are artificial. We derive "the point" in time from our experience of the present. We know that the present is real because of the radical difference between future and past, so we assume a point in time which separates future and past. This is an abstraction, the point is abstract. We utilize this point, by moving it around, projecting it forward and back, to mark off particular durations, periods of time. The periods of time are completely arbitrary within what appears to be one continuous time without any such points in reality.

However, there is some "part" of time which is not arbitrary, and this is the present, as the division between future and past. We must consider this reality concerning time. This difference between future and past, we respect as real within all of our activities. So the continuity of time can be described as the continuous difference between future and past. And what we can see, if we look directly at time itself, is that future time is continuously becoming past time. The time designated as tomorrow (future time, 23/02/2017) will become yesterday (past time), as the future is continuously becoming the past. What about this division between future and past, which we've represented as a dividing point, and from which we've derived the highly useful "point" in time?

Now we can ask, what is actually occurring when the future is becoming the past. With reference to my prior post, and how we experience the physical world, and free will, we can conclude that the physical world is coming into existence, "becoming", as the future becomes the past. Consider that with respect to the physical world, you are looking backward in time only. If you turn around, and look ahead, toward the future, there is no physical world there, nothing, just predictions. There is what the Neo-Platonists would call Forms there, in the future, and these Forms are what is ensuring that the physical world which you see behind you, in the past, is consistent, and lawful.

But there proves to be an issue with the assumed continuity of time. We derive the continuity by looking at how time has passed. We produce an order which would start from the furthest back in time, continuing to now, and assign continuity to this. But in reality, the order of the real physical world is such that it begins now, at the present. However, time is passing, so it must begin again, and again, and again, at each moment. This necessitates that with respect to the physical world, there are real points in time, the points at which the physical world repeatedly comes into existence. It may be the case that there is a further continuity which underlies this, but with respect to the physical world, there must be real points in time. The continuity which we look at, is created by us. We look back toward the beginning of the physical universe, and produce a continuity from there until now. But this continuity has the real existence of the physical world backwards. The real existence of the physical world is such that it begins at the present, not at the so-called "beginning of time".

Set A is a subset of set B if every element of set A is an element of set B. Nothing about the definition requires that A containers fewer elements than B. Such a set is instead called a proper (or strict) subset. — Michael

OK, thanks for that good clear definition Michael. I was wrong to think of sets and subsets as parts and wholes, they are actually completely separate entities with a describable relationship.

That the set of natural numbers has the same cardinality as the set of natural numbers is a tautology. — Michael

The referred to tautology takes for granted that the set of natural numbers has a cardinality. If it does not have a cardinality there is no such tautology. Being infinite, I do not think that it is possible that the set of natural numbers has a cardinality. To be a countable set, according to the definition you provided, a set must have a cardinality. This could be the root of my disagreement with aletheist.

When I say that apples are edible – and by this I mean that they are capable of being eaten – I'm not saying that it's possible to eat every apple. — Michael

To say that all apples are edible is to say that each apple may be eaten. There is no apple anywhere which cannot be eaten. In the case of the natural numbers, by definition, there are numbers which are incapable of being counted. This is because no matter how high a number you take, there are always higher numbers. There will always be, necessarily, uncounted numbers. So your analogy is not good, because saying that all apples are edible does not allow that there are always some apples which necessarily cannot be eaten.

Of course, if by "countable" you mean "possible to enumerate the entire set" then obviously the set of natural numbers is not countable. — Michael

How could a set have a cardinality if it's not possible to enumerate that entire set?

The set of natural numbers is a subset of the set of natural numbers. A countable set is a set with the same cardinality as some subset of the set of natural numbers. Therefore, the set of natural numbers is a countable set.

You seem to keep switching in some non-mathematical definition of "countable". But the term "countable" that is being used here is the mathematical term. Nothing about the term "countable" in mathematics entails the possibility that we could enumerate the complete set. — Michael

I never switched definitions. I maintained my non-mathematical definition, which was "capable of being counted" (#3), and this was contradictory to the mathematical definition you've provided (#.1), according to the fact that an infinite number is not capable of being counted. So long as we keep the two completely separated, and there is no ambiguity as to which one we are using, and therefore no equivocation, then there is no problem. But aletheist wanted to bridge the gap between #1 and #3 with a #2. The proposal was that we could qualify #3 in a particular way, to produce #1, such that #1 would be a particular type of #3; we could say that #1 is "#3 in principle", "#3 potentially", or " # 3 logically".

I have maintained that the two are inherently incompatible, contradictory on this point of infinity. However there is another possible route of reconciliation which we haven't explored yet. It is possible that #1 is the more general, and that #3 is a particular type of #1. This would require a description of what "countable" actually means in #1.

Here are the two dubious principles which I see are involved with your mathematical definition, which need to be justified. First, what does it mean to take a set, the set of natural numbers for example, and produce a subset which is the same as the set, and say therefore, that the set of natural numbers is a subset of the set of natural numbers. One is the "set", the other is a "subset", yet they are the same with two names. The names refer to something different. What is the reason for giving the same thing two distinct names? What I see is that the set is the whole, and a subset is a part. To represent the set as a subset is to class it as a part. But it is false to represent the whole as a part of itself, because it is not a part of itself, it is the whole of itself. So this is a category error, to make the set a subset of itself, without having some means to distinguish between the whole as whole, and the whole as part. To make them equivalent is category error.

The other dubious principle is the cardinality of the infinite set. If the set of natural numbers is a subset of the set of natural numbers, then this subset has an "infinite cardinality". Judgement of cardinality is required in order to designate a set as countable. Therefore to judge the set of natural numbers as countable, requires a judgement of its cardinality, to ensure that it is the same as itself (the set must have the same cardinality as some subset of the natural numbers). How would you judge this cardinality?

I'm asking you what you think. I believe that it is impossible to count the elements of an infinite set. I've only said that about twenty times. You seemed to believe, like aletheist, that it is possible, in principle, to count them.

Indeed, but Metaphysician Undercover only accepts this definition if we divorce it from the colloquial meaning of "countable," which according to him applies only to finite sets small enough that someone can actually finish counting all of their members. — aletheist

No, I think any finite set of natural numbers is in principle countable, it's the fact of being infinite which makes the whole set of natural numbers uncountable.

As someone already pointed out, countability is defined as the ability to place a set, whether finite or infinite, in one to one correspondence with the set of natural numbers or some portion of it. This is logically just the very same as to be able to count the elements of the set. — John

Do you think it is possible to count the elements of an infinite set?

But do I understand why one should even entertain such a concept when all is well and good by just acknowledging what one is experiencing, i.e. a continuous experience of consciousness which we experience within a duration. — Rich

Ever since Newton's laws, the discipline of physics has taken the continuity of physical existence for granted, it is a given. As such, continuity is apprehended as a necessity. It underlies the laws of physics. But in Aristotelian physics, continuity is assigned to matter, and matter is understood to have the nature of potential. As such, continuity is understood as possible.

When, as philosopher's, we come to understand the nature of intention and free will, we realize that the continuity of existence of any object can be interfered with, interrupted, even ended, at any random moment of the present, by means of a free will act. If any object can be annihilated at any moment of the present, then the continuity of existence of physical objects, at the present, cannot be taken to be necessary. This is why continuity must be classed in the category of potential.

Following this classification, that the continuity of existence at the present, is possible rather than necessary, we need to seek a cause of such continuity. Any potential which is actualized must have been caused to be actualized. This implies that at every moment in time, as time passes, there is a cause of existence, a becoming, or coming into being of each physical object. That is necessary to account for the assumption that the free will act can randomly annihilate the physical object at any moment of the present. The continuity of the physical object is not necessary.

If indeed we are all just accumulated memory within a universal field, with the brain acting as a reference wave that perceives the holographic-like images within this field (as opposed to someone storing images within it), then the soul is nothing more than the persistent wave pattern which we call memory coupled with the same consciousness that consumes it. Conscious, memory and the field are aspects of one. — Rich

I feel there is something inverted about this perspective, which I cannot quite put my finger on. Memory is a function of the continuity of the physical world. When things go to memory, they are held there by the continuity of this part of the physical world remaining the same through a duration of time. But this continuity is of the essence of potential, and must be caused to actually occur in the way that it does. So whatever type of thing, which you might infer the existence of, which must actually cause the physical continuity, it must be proper to the part. The problem is that continuity is proper to each object individually, each part, and not proper to the whole. A part may stay the same, in continuity, but the whole always changes. So continuity, and therefore its cause, must be sought by understanding the part rather than the whole. Rather than modeling the part as being derived from the continuity(as a field or such), the continuity must be derived from the part. So from this perspective, a similar thing to which causes memory, must also be the cause of continuous physical existence, but this should be found within the parts themselves, not within the "field".

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. — tom

OK, if that's what you think, then maybe you could explain how one boundless or endless (infinite) thing is bigger than another. I'd be very interested to see this explanation.

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. — Michael

Actually we've been through all this. It's taken a few days, and numerous pages, but we haven't agreed on any conclusion. Here, you are defining "countable" in relation to any "subset of the set of natural numbers". Aletheist kept wanting to commit the fallacy of composition, assuming that what is true of the part is true of the whole. So we cannot assume that because a subset of the natural numbers is countable, then the complete set is countable. And, since the set of natural numbers is defined as infinite, boundless, endless, then by that definition, it is impossible to count, and therefore uncountable.

It is possible that "continuity" is just an imaginary notion, a fiction conjured up by the human mind. If this is the case, then there is really no need to model continuity. So if it is true that mathematics can only model the discrete, this is not a problem if continuity is just a fiction anyway.

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 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.

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.

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.

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?

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 then, also arguably, one of the reasons the 'new age' exists is because of the shortcomings of the Christian mainstream - it's authoritarianism, inflexibility, dogmatism, and the rest. — Wayfarer

This is likely the most difficult issue of religion, to establish compatibility between the idea that the human being has real freedom of choice, yet there is also real objective authority. The answer is not to oppress freedom of choice with authoritarianism (as Agustino implies), because we must respect the fact that the human race progresses through advancements in knowledge, and evolution, such that what was once believed as true, in the past, may not be believed as true anymore. Nor is the answer to proceed forward with completely unprincipled decision making.

So there is a very awkward need to allow the free thinking human mind to reach out into the fringes, groping in the dark, as it may be, grasping at straws in the realm of the unknown, in order to find principles to cling to, as leverage, to pull the unknown into the realm of becoming known. This is the activity through which knowledge progresses. But this activity, whereby the unknown becomes known, which can only be carried out by the freest minds, must itself be principled in some way.

In addition to say life(I'm assuming you have that in mind) is not explicable in terms of science is special pleading. — TheMadFool

It's not special pleading, it's just reality, a statement of fact.

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".

And yet set theory explicitly says otherwise. — aletheist

Some theories are false. It's very hard to convince the people who believe in false theories, that they are false. That's life. It appears like your set theory, if it really is as you describe, relies on the fallacy of composition. You should investigate this, and if the theory is as you describe, quit believing in it so strongly, because it's false. Or else it isn't as you describe, then you should develop a better understanding of what the theory really says.

And which number would that be? I asked you to identify it, not describe it. — aletheist

There is no highest number, that's what makes the set of natural numbers uncountable. If I could identify the highest number, we could count to it, and count all the numbers. I cannot, and nor can you, or anyone else, and so the natural integers remain uncountable, as they always will be.

The accepted mathematical one from set theory, "able to be put into bijection [one-to-one correspondence] with the natural numbers."
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.
The notion of "actually countable," which requires it to be possible to finish counting.


You have made it quite clear by now that you reject the first two, but that does not render them false or contradictory - just different from yours. — aletheist

I accept the first one, but that defines "countable" relative to the natural numbers, so it is insufficient to tell us whether or not the natural numbers, are countable. It is a definition used to judge things in relation to the natural numbers so it cannot be used to judge the set of natural numbers itself. We need a definition which we can apply to see whether or not the set of natural numbers itself is countable.

The second definition of countable, your definition, makes no sense. "There is no particular largest value beyond which it is logically impossible to count." If this were the case, then no subsets of integers would be countable, because each of these has a particular largest value.

The third is the obvious choice as a definition to apply in order to determine whether the natural numbers are countable. It would be false to say that something which is not capable of being counted is countable. Therefore we can conclude that the set of natural numbers is not countable.

Yes, there is such thing as continuity and we experience it quite concretely as duration (real time). — Rich

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.

Here's a question then. Do you think that there is such a real thing as a continuity, and if so what would be its nature? Remember the premise, it cannot be understood with mathematics.

As far as I can tell, mathematics is totally reliant on the discrete and because of this limitation constantly makes philosophical ontological errors. — Rich

This is exactly the problem with the Zeno paradox of the op. Zeno's premise is that space is continuous. Then he introduces mathematics to deal with this assumed continuity. It fails. The conclusion which should be drawn, is that mathematics is incapable of dealing with the continuous.

If we move to the ontological implications, then if there is no such thing as a real continuity there is no problem. But if there is a real continuity then we have a problem, if we are trying to understand that continuity mathematically. So either space and motion are discrete, and there will be no problem to understand them with mathematics, or they are continuous, in which case they cannot be understood with mathematics.

And if we assume a real continuity of any sort, then we should not expect to be able to understand it mathematically.

Please identify a natural number or integer that is not capable of being counted. — aletheist

That's a very simple question to answer. The highest number is the one that's not capable of being counted.

Instead, what I am arguing is that it is possible in principle to count all of the natural numbers (and integers) because it is possible in principle to count up to and beyond any particular natural number (or integer). — aletheist

Again, you are saying that because it is possible to count any particular number, it is therefore possible to count all the particular numbers, and this is known as the fallacy of composition.

No one claimed that. — tom

Aletheist claimed that.

If it is possible in principle to count up to any particular natural number (or integer), then it is possible in principle to count all of the natural numbers (and integers). — aletheist

Fallacy of composition.

If it is possible in principle to count up to any particular natural number (or integer), then it is possible in principle to count all of the natural numbers (and integers). — aletheist

This is a textbook case of the fallacy of composition. And, you've also forgotten one premise here, that any particular natural number has numbers higher than it. And, this is the premise which makes it impossible, in principle to count all of the natural numbers.

I don't know if it's a "substantial" difference. It's certainly a difference. The rationals are foozlable and the reals aren't. Even in countable models of the real numbers, and yes such things exist, the reals are not foozlable. So yes it's a pretty important difference in math. — fishfry

I agree that there is an important difference between natural integers and real numbers, and even an important difference between rational numbers and real numbers, what I disagree with is that there is a difference in the infinities which arise in all these different situations. I think that the infinite itself is the same in each situation, but it is applied differently.

So here's an example. We could take a point like zero, or any other integer, and count the integers toward the positive and toward the negative, from that point, and assume two distinct infinite quantities, one negative and one positive from that point. Or we could take two points, like 1 and 2, 3 and 5, or 6 and 10, and assume an infinite quantity of real numbers in between. In these two different ways of using "infinite quantity", there is no difference in the meaning of "infinite". One is not is not a larger quantity than the other, just like the infinite quantity of real numbers between 6 and 10 is no bigger than the infinite quantity between 1 and 2. We could then take an irrational ratio, like pi, and say that it extends to an infinite quantity of digits, and this use of "infinite" is still the same.

You do agree they're foozlable, right? I just want to make sure I'm understanding you. — fishfry

Yeah sure, that's the name you gave instead of the name "countable". But I'm not sure that I would agree with the assumption that there is a substantial difference between a foozlable infinity, and an unfoozlable infinity. We can call them countable and uncountable infinities if that's easier.

Again, incorrect. You evidently have a rather idiosyncratic personal definition of "infinite." My dictionary provides several widely accepted definitions, and none of them state or imply that it means "not countable." Besides, as I keep noting, the concept of being "countably infinite" is well-established and well-understood within mathematics. — aletheist

Infinite: endless.

Now you seem to be confusing "countable" with the idea of being finished counting. — aletheist

Countable means capable of being counted. If it cannot be counted, as is the case with something infinite, or endless, it is not capable of being counted. Therefore the infinite is not countable.

I have acknowledged this repeatedly - the natural numbers (and integers) are not actually countable, in the sense that someone or something could ever finish counting them. However, they are all countable in principle, in the sense that there are no natural numbers (or integers) that are uncountable; given enough time, someone or something could count up to and beyond any arbitrarily specified value. As I said before (with sincere gratitude), you have stated more accurately what I meant all along. — aletheist

Back to your contradictory notions "the natural numbers(and integers) are not actually countable ... However, they are countable in principle... "

You refuse to face the facts of the situation, the entire set of natural numbers is, in principle, not countable. That's what Infinite means, endless, so no matter how hard you try the infinite set is not countable. What is countable in principle, is any finite set of natural numbers. But it is false to claim that the entire infinite set is countable in principle, what is countable is finite subsets.

Oh I see, you're talking about the principles. They're made by human beings as well, creative expressions of human language.

But, mind you, not made by humans. — TheMadFool

What, robots and iPhones are not made by humans? What are they made by, robots?

However, I agree with tom that "you can count members of a countably infinite set"; again, there is no largest natural number or integer beyond which it is (logically or actually) impossible to count, so all of the natural numbers and integers must be countable. — aletheist

How does this imply that all the natural numbers are countable? It actually implies the very opposite. Every number you count has a larger number, therefore it is impossible that all of the natural numbers are countable. I think you really believe that it is possible to count infinite numbers, because this statement seems to be an attempt to justify this.

Incorrect; "uncountable" and "infinite" are not synonyms in mathematics, since there are countable infinities and uncountable infinities. This is a fact, not an opinion. — aletheist

They are not synonymous, but infinite is by definition not countable. There could be something else uncountable which is not infinite. As we've already discussed, when you refer to countable and uncountable infinities, you use "countable" in a different way, with a different meaning. This way of using "countable" does not imply that a countable infinity is actually countable (according to the other sense of countable), nor does it mean that it is potentially countable, according to the other way of using countable. It is a completely different way of using "countable".

I suggest that you continue to use "countable" in your way, and I'll use "countable" in my way, the two being very obviously incompatible with each other. But you should not claim that you can make the two compatible by saying that one refers to an actuality and the other to a potentiality, because this is not the case. Your sense of "countable infinity" does not equate with "potentially countable" according to my sense of countable, because infinite is neither potentially nor actually countable according to my sense of "countable", it is absolutely uncountable.

The principles we (robots, fish, iPhones, humans) work on e.g. the laws of physics and chemistry are same. The difference I believe is that of degree not of kind. — TheMadFool

Robots, iPhones, etc., work on principles known to human beings, and applied by human beings. Living beings work on (as of yet) unknown principles. If we ever want to learn these principles, we might have to change the other principles.

Fair enough, but the fact is that you can count members of a countably infinite set. — tom

No, the fact is that you cannot count an infinite set, that's what "infinite" means. You can count a finite subset, but you cannot count the infinite set. "Countable" is just a name, as fishfry explained, it has no other meaning.

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

Nor can you count the members of a countable infinity. "Countable" is just the name of the set.

The point I made earlier is that there is actually no difference between the countable infinity and the uncountable, as "infinite", they are the same. What is different is the thing which we are attempting to count, one is a continuity the other discreet units. The continuity cannot be counted, the discrete units can.

You don't seem to understand what "in principle" means. It is impossible to count the infinite, and this is what infinite means, that no matter how you try, you will never ever count it, that's what infinite is. If you now introduce a principle, and say that this principle states that the infinite is countable, such that you can say "it is possible in principle to count them", all you have done is introduced a contradictory principle. It is a false principle

Counting all of the integers is logically possible, but actually impossible. Infinitely dividing space is logically possible, but actually impossible. — aletheist

No. counting all the integers is not logically possible, it is impossible. That's what infinite means, that it is impossible to count them all, you never reach the end. It is such by definition. To say that it is possible to count them all is contradictory. Therefore it is not logically possible.

I hate to be disagreeable, but I really think you're mistaken about this. It's a question for history and philosophy of science, of course, but how could there be (for instance) such a division in Cartesian substance dualism, where there are two kinds of substance? The whole point about Newtonian and Galilean substance was reduction to those attributes which could expressed in numerical terms. The distinction that emerged was not between 'matter and property' but between 'primary and secondary qualities', where the latter were associated with the observing mind (colour, etc) and the former (including mass) were primary attributes of the object of measurement. — Wayfarer

Yes, yes, that's exactly the point, the distinction between 'matter and property' was lost, because matter was taken for granted. If you read Newton, he had a lot of respect for the concept of matter, and discussed it a lot. But what happens with his laws, is a focus on this particular property of matter, mass. So in a sense, matter is equated with mass, all matter has mass, and all mass has matter. After that, the focus is just on that attribute, mass, because the matter is taken for granted. The equivalency of mass and matter was taken for granted, matter could be represented as mass, so there was no more need to question the existence of matter itself.

Show me one genuine contradiction in any of my previous posts, without conflating "countable" (as defined in mathematics) with "actually countable." They are two different concepts. — aletheist

Try this:

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

See, you say that no one can actually count them, yet it has been proven that it is possible in principle to count them. It's not possible in principle to count them, that's the point, that's what infinite means, that it is impossible to count them. You only contradict yourself.

To say that there is no difference between actually countable and potentially countable is simply incorrect. Do you really not understand the distinction between the actual and the potential? between the nomologically possible and the logically possible? — aletheist

I know very well the difference between potential and actual, as well as many different senses of "possible". It really appears like it's you who has no understanding of this. But if you really believe this is the case, then try to explain the difference between actually countable and potentially countable. Just don't give me contradictions or falsities. If it is impossible to count it, then it is impossible that it is "in principle" countable, because that principle would be a false principle.

I really don't think that's correct MU. The equations of matter work for matter in a generalised sense, it doesn't matter which type. The whole point about Aristotelean 'substance' is that it is a complex concept, and isn't really part of modern natural philosophy, except by analogy. — Wayfarer

As you said, the equations refer to mass, not matter itself. Mass is a measurable property of matter. The duality, or complexity, of substance is inherent within Newton's laws, because it is assumed that there is matter, and it is assumed that matter has mass. These are two distinct things, matter and its quantifiable property, mass. The working premise at that time, was that there is no matter without mass, and no mass without matter. Some may have thought that matter and mass are the same thing, but the two are not the same thing, as mass is clearly a property, and that is not consistent with the concept of matter.

Matter, in Aristotle's physics was that which persists, does not change, through a change. The law of conservation of mass removes this designation from the matter itself, and puts it on the mass, which is a quantifiable form. But then it was found that matter could exist in the form of energy as well, and this required a law of conservation of energy. Now we have two distinct fundamental forms of matter, two distinct types of substance, one is mass, the other is energy. That is only because the designation of "that which persists" has been removed from matter itself, and applied to the form.

Exactly - it actually does mean that it is countable, but it does not mean that it is actually countable. See the difference? — aletheist

No I don't see the difference, and you've already tried to explain, but all you do is contradict yourself. "Countable" means possible of being counted. To say that there is a difference between actually countable and potentially countable is nonsense. What would potentially countable mean to you, that it's not countable but could be made to be countable? That's nonsense.