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
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
There is no highest number, that's what makes the set of natural numbers uncountable — Metaphysician Undercover
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. — Metaphysician Undercover
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
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
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
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
Do you think it is possible to count the elements of an infinite set — Metaphysician Undercover
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. — Metaphysician Undercover
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. — Metaphysician Undercover
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
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. — Metaphysician Undercover
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 never switched definitions. I maintained my non-mathematical definition, which was "capable of being counted" (#3)...
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. — Michael
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
That the set of natural numbers has the same cardinality as the set of natural numbers is a tautology. — Michael
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
Of course, if by "countable" you mean "possible to enumerate the entire set" then obviously the set of natural numbers is not countable. — Michael
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
How could a set have a cardinality if it's not possible to enumerate that entire set? — Metaphysician Undercover
This is so only because in principle the set of, say, even numbers, can be placed in one to one correspondence with the set of both even and odd numbers. — John
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. — Metaphysician Undercover
How could a set have a cardinality if it's not possible to enumerate that entire set?
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.
You have good insight into this. For a set to have a sensible cardinality, it needs to be able to be enumerated. — fishfry
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
So what exactly do you mean by saying that there are numbers which are incapable of being counted? — Michael
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
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. — Metaphysician Undercover
When you assert that all natural numbers are countable, this is an inductive conclusion. — Metaphysician Undercover
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. — Metaphysician Undercover
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. — Metaphysician Undercover
It seems to me that you're trying to argue against mathematical terminology without actually understanding the mathematics involved. — Michael
Sorry, I got my terms mixed up. It's surjection, not bijection when it comes to infinite sets. — Michael
Not that it matters one jot due to the level of willful ignorance on display, but I think you were quite correct to use the term "bijection" = one-to-one and onto.
I'm no longer finding it amusing to follow the obtuse denial of well established mathematical truths, so I can't say for sure because I can't be bothered to tease out the remnants of sanity in this thread, but I get the feeling that a surjection will not suffice for your purposes. — tom
No, it is a deductive conclusion that is necessarily true, given the standard mathematical/set-theoretic definition of countable/denumerable/enumerable/foozlable. — aletheist
Not if cardinality/multitude is defined in a particular way that specifically pertains to infinite sets. — aletheist
For any set with N members, there is a "power set" that consists of all of its subsets, and that power set has 2N members. — aletheist
He might very well understand it, he just refuses to accept it. — aletheist
I'm no longer finding it amusing to follow the obtuse denial of well established mathematical truths, so I can't say for sure because I can't be bothered to tease out the remnants of sanity in this thread, but I get the feeling that a surjection will not suffice for your purposes. — tom
Yes, I think you're right. aletheist clarified earlier that I was wrong to admit to being wrong. — Michael
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.