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.

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

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. He would prefer a different term altogether for the standard set-theoretical concept, like "foozlable" as suggested by @fishfry.

Maybe we should just go with "denumerable" or "enumerable." My dictionary says that these two words both always and only mean "capable of being put into one-to-one correspondence with the positive integers." Now I wish that I had looked them up several thread pages ago ... :(

There is no highest number, that's what makes the set of natural numbers uncountable — Metaphysician Undercover

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.

So no complete infinite set (if such a notion even makes sense) can be actually counted, obviously, any more than any complete infinite set can be actually placed in one to one correspondence with the complete set of natural numbers. There seems to have been much very silly confusion and argument over this very point.

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

I do not understand the concept of being reborn at each moment in time (exactly what is this moment and what is happening in between?), 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. Such a point of view helps things along mightily in understanding the nature of existence while admittedly it may be too obvious and easy to consume for those who are looking for academic debates about stuff.

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. In Daoism it all begins as one, the Dao (consciousness) which becomes Yin and Yang (the wave) which then moves as Qi (energy). The model is straightforward. No need for discontinuities or paradoxes.

Everything falls into place very quickly. Traits, inherited characteristics, inborn abilities are all nothing more than persistent memory.. Nature this becomes very concrete and real and rather than discuss what is occurring between discontinuities and paradoxes, we can get down to the business of understanding life more fully in a manner that encourages full exploration of life.

By this I mean, a self-fulfilling journey into the arts, for without such a journey, all I am suggesting will seem like gibberish. One can surely understand nature without mathematics but I believe such understanding without experiencing art is not possible. I will reiterate, science confuses and muddles by utilizing symbols to replace life.

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".

Do you think it is possible to count the elements of an infinite set — Metaphysician Undercover

You answered the question yourself in the previous paragraph. I think you are being unnecessarily pedantic about what is a self-evident and trivial point.

Of course it is not possible, either logically or actually, to finish counting an unending series. That is a matter of mere definition.

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.

Didn't I say in my previous post that it is not possible, actually or logically (in principle in other words) to finish counting an infinite series. It seems obvious that this is the same as to say that it is not possible in principle to count all the elements of an infinite series or set.

But it is not impossible, as you say, to count the elements of an infinite set, any finite number of them can in principle be counted.

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

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.

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

Is this the case, when physics uses discrete measurements to describe everything whether it be time, particles, etc. and then uses these discrete measurements to describe things? Physics has always taken the position of separate and measurable for practical reasons but such descriptions are just practical tools. Philosophers who take opposing views are quickly marginalized for not adhering to these mathematical descriptions. 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 because t appears in equations. Further on you discuss parts of things, further concessions to discrete mathematical equations.

That there is no beginning, no end, no parts, no moments, cannot be represented in mathematical equations. Infinities and division by zero create havoc in mathematical equations, yet they constantly occur. Such situations should make it quickly apparent that equations are not useful for understanding nature. One must learn to use consciousness to penetrate consciousness, but when was Mozart or Van Gogh, or Eastern meditation (Tai Chi) ever part of a philosophy curriculum. All of this would provide real experience as opposed to awful, inappropriate symbols. Philosophers have been relying on tools of science which are simply incorporate for penetrating nature which is why Bergson and Bohm should be read by philosophers who are interested in understanding the nature of nature.

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?

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

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.

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?

That the set of natural numbers has the same cardinality as the set of natural numbers is a tautology. As for other sets, I believe bijective functions are used to determine if two sets have the same cardinality.

I never switched definitions. I maintained my non-mathematical definition, which was "capable of being counted" (#3)...

I don't think "capable of being counted" is the same as "possible to enumerate the entire set". 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.

Of course, if by "countable" you mean "possible to enumerate the entire set" then obviously the set of natural numbers is not countable. But I don't think aletheist (or anyone else here) is saying that it is. As I said before, there's just a lot of talking past each other.

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

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.

But 'all of them' (in itself a meaningless term) could never in practice be so placed, any more than they could in practice all be counted. There just doesn't seem to be any reason why we could not keep placing them in one to one correspondence forever just as we could keep counting them forever. So, it is a meaningless exercise to compare the "size" of infinite sets; no infinite set has a size that is what is meant by 'infinite'.

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?

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".

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

This is actually a great mathematical question. If you assume the Axiom of Choice (AC), then all sets have a well-defined cardinality that is the smallest ordinal that bijects to the set. If you take AC as false, then there exist sets that can't be well ordered, hence sets that don't have well-defined cardinalities. Or if they do, they're not comparable to the standard Alephs. I'm a little fuzzy on that point.

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

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

I don't know what "in principle" means. If we are in standard set theory, we can biject the naturals to the evens by mapping each natural n to the natural 2n. If we are not in standard set theory, then you'd have to say what the rules are for that system.

It doesn't make any sense to say that "in principle" you can do something that's legal in set theory. If set theory says you can do it, you can do it. If you are using some other framework for talking about numbers and sets, you have to say what that is.

To make this clear, I interpret the phrase "in principle" as indicating a lack of clarity in specifying what domain we are in. If we're in set theory we can biject, and if we're not, what are the rules? It's like sitting down to play chess and they won't tell you how the pieces move. Tell me what your rules are for defining functions between sets, and I'll tell you whether there's a bijection between the naturals and the evens.

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

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?

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

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

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

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.

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.

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

Sorry, I got my terms mixed up. It's surjection, not bijection when it comes to infinite sets.

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.

It's a function, so in this case f(x) = x. And with even natural numbers, f(x) = 2x.

I don't see your point. What's a function?

It's a thing in maths.

It seems to me that you're trying to argue against mathematical terminology without actually understanding the mathematics involved. Seems like a layman trying to argue against the physicist's definition of "charm" or "strange" when it comes to quarks.

You said "it's a function". I asked what does "it" refer to.

If you have something constructive to say, then address the issues. I didn't come this far in this thread just to have you piss me off with ad hom. The fact is, as I explained, that you are dealing with inductive conclusions concerning "the natural numbers", but these inductive conclusions are inconsistent with the defined essence of "the natural numbers", as infinite. Therefore whatever you say about infinity is completely unreliable. Referring to functions only brings you deeper into inductive territory, without first recognizing that any conclusions you make concerning infinity cannot be respected

When you assert that all natural numbers are countable, this is an inductive conclusion. — Metaphysician Undercover

No, it is a deductive conclusion that is necessarily true, given the standard mathematical/set-theoretic definition of countable/denumerable/enumerable/foozlable.

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

Not if cardinality/multitude is defined in a particular way that specifically pertains to infinite sets. For any set with N members, there is a "power set" that consists of all of its subsets, and that power set has 2^N members. For any value of N whatsoever, 2^N > N. Therefore, a power set always has more members than the set from which it is derived. This is true even if N is infinite; the power set of the natural numbers must have more members than the (infinite) set of the natural numbers. That is all it means to say that the set of real numbers is "bigger" than the set of natural numbers. I will grant that it is counterintuitive, but it is defined this way in set theory, where it is not problematic at all since mathematics is the science of drawing necessary conclusions about ideal states of affairs; it has nothing to do with actual states of affairs.

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

Being able to finish counting them has absolutely nothing to do with establishing a bijection, or one-to-one correspondence. @Michael's subsequent reply is incorrect - bijection applies to infinite sets, as well as finite sets; it is a specific type of surjection.

Michael's subsequent reply is incorrect — aletheist

I knew I should have stuck to my guns. MU's comment had me doubt my interpretation. Oh well. Thanks for your clarification.

It seems to me that you're trying to argue against mathematical terminology without actually understanding the mathematics involved. — Michael

He might very well understand it, he just refuses to accept it. He is committed to the presupposition that only the actual is real, so if something is actually/nomologically impossible, it just is impossible, full stop.

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.

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

Yes, I think you're right. aletheist clarified earlier that I was wrong to admit to being wrong.

No, it is a deductive conclusion that is necessarily true, given the standard mathematical/set-theoretic definition of countable/denumerable/enumerable/foozlable. — aletheist

Well, perhaps we'll find that definition. So far, "cardinality" is incapable of producing your desired conclusion, because it is impossible that an infinite set has a cardinality.

Not if cardinality/multitude is defined in a particular way that specifically pertains to infinite sets. — aletheist

So your claim is that there is a different definition of cardinality for infinite sets then there is for finite sets? Then what is true of finite sets in relation to cardinality is not necessarily true of infinite sets, because cardinality would mean a different thing.

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

This doesn't solve the problem. It assumes a set with N members. An infinite set has indefinite members.

He might very well understand it, he just refuses to accept it. — aletheist

I am ready to accept it, as soon as all inconsistencies and contradictions are removed. As of now, there is a necessity to resolve the incompatibility between the definition of cardinality, and the definition of infinite.

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

Those who are "in denial" will always refuse to face the fact that their "well established truths" are actually falsities.

Yes, I think you're right. aletheist clarified earlier that I was wrong to admit to being wrong. — Michael

Are you ready to address my post now, and show me how an infinite set has a cardinality?