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
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
We simply disagree on whether "countable" always and only entails the ability to finish counting; you say yes, I say no. — aletheist
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
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. — aletheist
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. — Metaphysician Undercover
Can you count the number of real numbers between 0 and 0.1? If so, how many are there? — tom
How many are there? Can you count them? Or is it impossible to count the real numbers, making them uncountable? — tom
Contrast that with the Naturals, which, by definition you can count. Just try it 1, 2, 3, 4, 5. How many was that? — tom
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
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
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
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
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
To be able to do something, is to be able to complete that task. — Metaphysician Undercover
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
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
My dreams are very real to me. — Rich
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 — aletheist
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
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
One of these infinities is bigger than the other, much bigger. — tom
You appear to be making a category error. "Counting" is an activity of the subject, "countable" is a property of the object. — Metaphysician Undercover
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
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
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
But if you want to prove any of these assertions, you need to justify them. — Metaphysician Undercover
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
You haven't stipulated any reasonable definition of countable. — Metaphysician Undercover
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. — tom
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
the set of natural numbers is uncountable by definition — Metaphysician Undercover
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