It appears very obvious to me that if it is impossible to count them, then it is false to say that they are countable. — Metaphysician Undercover
No, to say that one is infinitely bigger than the other is nonsense, unless you are assigning spatial magnitude to what is being counted. We are referring to quantities, and each quantity is infinite, how could an infinite quantity be greater than another infinite quantity? — Metaphysician Undercover
Why don't you just look it up, or Google it? Plenty of stuff on cardinalities, countable and uncountable infinities, the diagonalization argument, Cantor ... — tom
It appears very obvious to me that you do not understand the accepted meaning of the word "countable" and, more fundamentally, the distinction between logical possibility and nomological possibility — aletheist
Your worldview is too small because it limits the real to the actual and the finite. — aletheist
But just because it's called "countable" doesn't means it's actually countable. You seem to believe that it actually does mean that it's countable. — Metaphysician Undercover
I'd rather a smaller world view which distinguishes fact from fiction, than a larger world view which doesn't distinguish fact from fiction. — Metaphysician Undercover
Exactly - it actually does mean that it is countable, but it does not mean that it is actually countable. See the difference? — aletheist
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. — Metaphysician Undercover
To say that there is a difference between actually countable and potentially countable is nonsense. — Metaphysician Undercover
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
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
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
See, you say that no one can actually count them, yet it has been proven that it is possible in principle to count them. — Metaphysician Undercover
Counting all of the integers is logically possible, but actually impossible. Infinitely dividing space is logically possible, but actually impossible. — aletheist
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. — Metaphysician Undercover
No. counting all the integers is not logically possible, it is impossible. — Metaphysician Undercover
That's what infinite means, that it is impossible to count them all, you never reach the end. It is such by definition. — Metaphysician Undercover
That is obviously not what infinite means within mathematics, since the natural numbers and integers are very explicitly defined as countably infinite. — aletheist
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. — fishfry
Fair enough, but the fact is that you can count members of a countably infinite set. — tom
You can't count the members of an uncountable infinity. There is no such thing as a next member. — tom
A set is defined as countable if it can be put into bijection with the natural numbers. — fishfry
It's a mistake to think that countability has anything to do with the ability to be counted. — fishfry
No, the fact is that you cannot count an infinite set, that's what "infinite" means ... 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. — Metaphysician Undercover
"Countable" is just a name, as @fishfry explained, it has no other meaning. — Metaphysician Undercover
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. — Metaphysician Undercover
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
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
They are not synonymous, but infinite is by definition not countable. — Metaphysician Undercover
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. — Metaphysician Undercover
How does this imply that all the natural numbers are countable? — Metaphysician Undercover
Every number you count has a larger number, therefore it is impossible that all of the natural numbers are countable. — Metaphysician Undercover
I think you really believe that it is possible to count infinite numbers, because this statement seems to be an attempt to justify this. — Metaphysician Undercover
The assumption that space is divisible is a matter of convenience, it does not reflect experience. — Rich
I should add that with this simple observation, that space is indivisible, all mathematical theories about nature that rely on mathematical divisibility of space, automatically lose all ontological meaning. — Rich
I do not know that anybody struck the true note before Benjamin Peirce, who, in 1870, declared mathematics to be 'the science which draws necessary conclusions' ... the essence of mathematics lies in its making pure hypotheses, and in the character of the pure hypotheses which it makes. What the mathematicians mean by a 'hypothesis' is a proposition imagined to be strictly true of an ideal state of things. In this sense, it is only about hypotheses that necessary reasoning has any application; for, in regard to the real world, we have no right to presume that any given intelligible proposition is true in absolute strictness.
You can't count the members of an uncountable infinity. There is no such thing as a next member. — tom
And Cantor's transfinite ordinals let you count way past the natural numbers. — fishfry
Right, this is all that I meant when I said that the natural numbers are countable by definition. — aletheist
Utmost is the issue that science way over steps it's bounds when it begins to replace everyday experiences with symbolic equations and declaring the equations to be more real. — Rich
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
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