I've never seen a definition of continuous, which refers to parts, but I'm sure you can find one, even if you just make it up yourself. My OED defines continuous as unbroken, uninterrupted, connected throughout space and time. Any philosophical definition I've seen describes continuous as unbroken, uninterrupted. And that's exactly what division does, it breaks, or interrupts. It is necessary therefore that the continuous is indivisible, because if it is divided it is not continuous. To divide the continuous would make it no loner continuous, so to describe it as consisting of parts is to describe it as non-continuous. It is a well known metaphysical principle, that the continuous is indivisible.

You want to describe the continuous as consisting of contiguous parts. But then you are describing a contiguity rather than a continuity. Do you recognize the difference between contiguous and continuous?

There are, sadly, quite a few people in this bad state. "Self-help" doesn't always deliver salvation. Major life changes which I didn't engineer gave me new life circumstances which solved my problem. — Bitter Crank

I tend to believe that self-help has a limited success. What is really successful is what you describe, major life changes. And, most often these cannot be engineered because it is far too complex to determine what is needed, and even if that were properly assessed, to bring it about would require a large amount of manipulating others. Even if one could determine the required major life changes to rescue oneself from the "bad state", most I think, would not choose these changes.

"Part" implies some but not all of. How do you propose that we can have some but not all of, any particular whole, or supposed continuum, without a separation? "Some but not all of" implies necessarily, separation, that's what "some but not all of" means, separation.

One more time: By definition, a continuum has parts, all of which have parts of the same kind. — aletheist

Clearly I do not accept your contradictory definitions. "Part" implies of necessity, a separation, and this negates any claim of continuity, which is a lack of such separation. You may proceed with your deluded metaphysics if you so desire.

Wrong. There is no separation (assumed or otherwise) between infinitesimals. Neighboring infinitesimals are indistinct; the principle of excluded middle does not apply to them. — aletheist

Then they are not infinitesimals, are they? They are united as one large continuum and it is false to refer to them as separate infinitesimals.

Wrong. A continuum - by definition - is that which has parts, all of which have parts of the same kind. What a continuum cannot have are indivisible parts, like points. — aletheist

A continuum is a continuity. It is the desire to model the continuum as a serious of parts, like we do in a mathematical model, which negates the essence of the continuity, rendering it as a series of discrete units. To say that a continuum has parts is contradictory. By saying it consists of parts, you no longer describe it as continuous.

Wrong. Discreteness requires separation and distinction; infinitesimals, as defined by synthetic differential geometry (a.k.a. smooth infinitesimal analysis), are neither separate nor distinct. — aletheist

I agree that discreteness requires separation, but what you seem to be failing to recognize is that "part" also requires separation. That is why it is contradictory to say that a continuity consists of parts. The true continuum must be indivisible, that's why it cannot be modeled mathematically.

So recognize what causes the ire, and prevent it before it builds, because then it's too late. Some people are sensitive, and sometimes it may be just a matter of reading one sentence and it's already too late. How would you know which sentences to avoid? I guess it's a matter of knowing which individuals to avoid. But if certain individuals just make a habit of provoking the ire of others, why not ban them? It's the easiest method of avoidance.

No one is talking about "infinitesimal points" except you. Infinitesimals are not separate dimensionless points, they are lines of extremely small but non-zero length that smoothly blend together so as to be indistinct. A continuum is that which has parts, all of which have parts of the same kind . A one-dimensional line cannot be divided into zero-dimensional points, only shorter and shorter one-dimensional lines. — aletheist

It doesn't matter how you lay the infinitesimal out, as a point, or as a line, there is still the assumed separation between it and other infinitesimals, and therefore it is necessarily a discreteness. A continuum cannot have parts, or else it is by virtue of those parts, not continuous, it is discrete.

That would be news to him. I guess you missed the part about the timelets "smoothly overlapping" such that "time is, so to speak, still passing" within each of them, rather than being frozen in a discrete instant. — aletheist

By saying "smoothly overlapping" you are speaking in terms of discreteness. You have identified separate parts which overlap. This is not continuity.

if you press the buttons in the right order, you'll invoke their ire...

...My preference is to ignore it until it goes away. — Wosret

Is it really possible to ignore that thing which invokes your ire? If it were, then how would it invoke your ire?

You should by now have noticed that the holier than thou attitude doesn't get you very far around here. So if it's an act, you might just drop it, because it only makes you appear like a bully.

First, that isn't so - hence why I said we're all equal around here. — Agustino

What you say, and what you display are distinct.

The victim has to perceive an imbalance of power between themselves and the bully, not the bully. In actual fact the bully quite often feels, inside, inferior compared to his victim. — Agustino

Reread your own quote: "One essential prerequisite is the perception, by the bully or by others..."

No, you perceive yourself as the one with power, as is evident from your holier than thou attitude.

Agamben puts it thus: "the example is characterized by the fact that it holds for all cases of the same type, and, at the same time, it is included among these. It is one singularity among others, which, however, stands for each of them and serves for all. On one hand, every example is treated in effect as a real particular case; but on the other, it remains understood that it cannot serve in its particularity ... Neither particular nor universal, the example is a singular object that presents itself as such, that shows its singularity." (The Coming Community). In other words, examples have a self-referential function. — StreetlightX

The problem though, with "the example", is that it is never a perfect representation of the category, it too has accidentals which make it nothing more than a particular instance, in essence. So we get 'the metre" paradox which Wittgenstein refers to. The metre rule cannot be said to be a metre, in the sense of an actual instance of the general, i.e. real existence of the rule. But without that actual instance of the general, as the example, nothing is a metre, so at the same time, it "must" be said to be a metre.

The subject is governance. The feedback I am giving is that the governance of the forum, like the governance of many places in the world is failing. I notice that people are confused about the subject, are tending on one side to hasty reactions, and on the other to hasty dismissals. — unenlightened

I agree, I find the amount of "loose talk" is increasing. Being a serious philosopher, I think loose talk is garbage and detrimental to the site.

My wish for the forum is that it should be more interesting and readable, and more significant than the comments section of a youtube video. This requires governance, it doesn't happen on its own. Such governance needs to be in the interests of, and acceptable to, the averagely interesting and readable contributor. (excuse me for stating the blindingly obvious, but in the circumstances it seems I need to start from first principles). — unenlightened

Again, I agree, as much as I have always been one to buck the authorities, I've come to realize that governance is necessary.

The problem, then, is how to raise the tone of debate, how to prevent good posters from being discouraged and silenced, how to maximise freedom, given that laisse faire does not lead to the desired result, but to the degeneration of the site. So this means we have to reach some sort of consensus about what makes a good post and what makes a bad post, and this is not all that easy, because, as I pointed out above, someone can honestly think that this discussion is fruitless, and they might be right, so I ought to at least consider it. — unenlightened

I don't believe that this discussion must necessarily be fruitless. I believe that it is one which must be had, but since it goes to the deepest philosophical levels, it may end up being fruitless, as many such discussion are. Ironically this is the topic of some of those deleted posts. Agustino claimed that the way to bring out the goodness in people is to religiously enact laws of governance and oppress them with authoritarian forces, making them into moral individuals. I claimed that this only drives them away from your reliion, and the only true way to bring out the good in people is to culture the desire to be good.

Now one of those issues, which has come up here as well, is 'just ignore it'. Well, no. We have been doing that and so have the moderators, and it does not go away, but proliferates. Engage, ignore, report, moderate, withdraw. there is I think no clear rule to be made as to what is best to do in every circumstance, But I am quite sure that ignore as a general policy does not work. — unenlightened

Obviously then, what is needed is a balance. This is because each individual is different. Some individuals just need to be encouraged to do what is good, and having that desire, they will do so, so long as they are not interfered with. Others have no such desire. They cannot be "cultured" to bring about such a desire because this is a long term principle which must be applied at childhood. These people must be dealt with in an authoritarian way. At the ripe old age of most posters, it is impossible to change their personality, so moderators are really forced with the issue of judging character.

But the point is, one cannot simply accept some formula, one has to keep questioning oneself, Am I making a contribution, or am I just being irritating. One has to examine this carefully, because, and I hope this is a case, sometimes being irritating is a valuable contribution. — unenlightened

Finally, it is quite evident that most likely, some participants are incapable of making such judgements, as am I being irritating, of themselves. How can I ask myself am I being irritating when I have no regard for what irritates another. Furthermore, and this is the real problem, there are some members of society who have an innate desire to be irritating. It's that childish emotion, which was never overcome in maturation, of wanting to be the centre of attention. And this may incline one to stir the shit pot, create the shit storm, all for the sake of naught.

That's the issue which I believe must be dealt with, intentional disrespect. It's very difficult too, because you mentioned humour, and it usually first appears as humour. There it lies substantially undetected disguised, where it festers, then it explodes when poked. I think that the use of humour in a site like this should be thoroughly examined, and perhaps limited to particular threads where it might be encouraged. Sure, proper use may lighten things up, and provide brief amusement, but one can never be sure of what lies underneath, and the negatives may outweigh the positives. In general, I think that in philosophical discussion there is no real need for humour, and it might be deletable as a matter of principle.

There is no imbalance of social or physical power between us two - therefore at most there can only be conflict. — Agustino

I would draw your attention to the actual stipulation "perception" of power. There is much evidence that you preach the precepts of your church with a "holier than thou" attitude.

You still have it exactly backwards. Space, time, and motion are all continuous; we only model them as being discrete. — aletheist

We measure space, time and motion as discrete, because that's the only way we can apply the numbers. But we tend to believe that these are continuous. It is this false belief, that space and time are continuous, which give rise to Zeno's paradoxes. So long as you hold this belief, that space, time and motion are continuous you will have paradoxes.

The concept of "infinitesimal points" is incompatible with continuous motion, it is only compatible with discrete motion. An infinitesimal point must be separate from another infinitesimal point or else it is not a point, and this negates any possibility of continuity. A series of "timelets" is a description of something discrete. Your quote from John Bell has provided a description of discrete motion, not continuous motion. He has perhaps recognized that our belief in continuous motion must be adapted to be represented as discrete.

What he can't understand is logical definion in terms of itself-. — TheWillowOfDarkness

IN short there is no such thing as "self-definition"; what you actually mean is 'self-identity'. — John

"Self-definition", or "self-identity", whatever you want to call it, what do you mean by this?

But if you examine it closely, you are not cutting space. The mark simply dissolves into space as more precision is required. There is no materiality but there a continuum of substantial or density of the underlying field. — Rich

The problem though is that we actually are marking things, with a ruler and other forms of measurement. But when we think about space in our minds, we think about dividing it, making geometrical figures, and whatever, this is an imaginary space. The imaginary space, which we can divide infinitely is not consistent with the space full of substance which we work with, which we cannot divide infinitely. In other words our concept of space is inadequate, because we don't know how space is really divided. We only actually divide space by dividing up substance, and substance divides quite differently from the way that we divide space conceptually.

The is simply no way to create units within continuity and if one tries to, out pops Zeno. — Rich

But don't you have this reversed? What exists is units, objects, but we want to talk about space in terms of a continuity. So it's not like we're trying to create units within a continuity, what exists is units and we are trying to make these units into a continuity. That's what Zeno shows us. It's not that motion is continuous, and we are trying to understand it as units, it's that it is not continuous, but we are trying to model it as being continuous. And this creates the paradox.

When I examine time, all I sense is a feeling of flow memories. I don't feel and units of measurment. Time sometimes feel like it is passing slowly and sometimes quickly and sometimes it seems to disappear into something else when I am dreaming or call unconscious, this last experience being particularly interesting. — Rich

Do you think that you can sense a feeling of time when you are unconscious? I don't think so. Do you think you sense a feeling of time when you are dreaming?

2 is not included in what is less than two. — Banno

Nor is it include in what is greater than two, it is included in what is not greater than two. If it's not included in what's less than two, and not included in what's greater than two, then what is it? I know what you'll say, it's 2.

Well the subject of this thread appears to be what to do about off-topicness. There is first, an issue as to what constitutes off-topic. Shouldn't it be the originator of the thread, first and foremost, who decides what is off and what is on topic?

I don't see a problem. Nor does one appear when we make a second cut at >2. We now have three pieces: <2, 2, >2. — Banno

2 is the boundary between all that is less than two and all that is greater than two. But what is 2?

From a staff perspective, that's just hypothetical. It's down to the staff to make that judgement, notify members, and take any action deemed necessary. The judgement of the originator doesn't have the same standing, and they are unable to take action in the ways that staff can, although they can flag any posts they think ought to be flagged, and we encourage them to do so. — Sapientia

Are you saying that it is inappropriate for the originator of a thread to tell participants that they have stepped outside the bounds of the topic, as many presently do?

If your statement is true, then the next question is whether motion is a supertask. And if it is, doesn't that mean motion is logically impossible? — Voyeur

No, I don't think that motion is a supertask, I think a supertask is an impossibility. I do not believe that motion is impossible though. We observe motion. What I believe is that motion is not well understood. That's what Zeno's paradoxes indicate, that we do not have an adequate understanding of motion, it was this way back then, and it remains so, now. We like to think that human knowledge of motion has greatly advanced with the special and general theories of relativity, and mathematical equations but we really persist in an extremely inadequate understanding of the relationship between space and time. As is evident in modern metaphysics, we have a woefully deficient understanding of what it means to exist, and motion is what existing things do. Until we understand what existence is we will not understand what moving is.

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The marks in space themselves are also symbolic since nor cannot truly divide space with a mark. — Rich

There is always substance though, a surface which we mark, or a ruler, or some such thing. So we measure space by referring to material substance, but we can only go so small with material substance, that is the point.

Time, or Duree as Bergson called it to avoid confusion, is not created by motion (this is the scientific time of a repeatable motion in space), but is a feeling that we capture via existing. It comes from consciousness not repeatable movements. I exist and feel my existence flowing as a duration whether or not can see the sun rise and set, or hear a clock. Real time is a psychological feeling of enduring in memory. — Rich

I do not believe that we get a sense of time simply from existing. I think we get the experience of time from sensing things. If you go to a quiet place and meditate, you can lose track of time. I do believe that you always know that time is passing, regardless of any sensing, but this is not "time" as we know it, i.e. time as a measure of duration, and maybe this is why you are distinguishing it from "scientific time". I do believe that we must recognize two distinct concepts, but time without measure needs to be better defined if we are going to call this "time".

This duty is only reasonably overturned if the thread-originator vanishes. — mcdoodle

What constitutes "vanishing"? If the originator must be present in order that the thread topic be followed, then why not leave the judgement of what is acceptable discussion, to the discretion of that originator? If that originator believes a particular post is not conducive to good discussion concerning the stipulated subject, then notify the poster to stay out, or start a new thread. That's common practise by some already.

Space and time must be thought of in a different way as not being divisible. An object doesn't travel half-way. It moves from here to there in one indivisible motion. There is no half in a continuously flowing and changing space. — Rich

I see how this makes sense with space, but I don't think it makes sense with time. With space it only makes sense to claim that there is a half distance if we can actually identify the real existence of that half distance, to say that the object travels that distance. So if we start with 100m we can mark this, and see that the object travels that spatial unity. We can mark a 50m unit, a 25m unit, and so on, and see the object travel these units. Inevitably there will be a point where we can no longer mark the distance, or observe the object travel it. So it doesn't make sense to speak of space in terms of divisibility like that.

Time however is different. Time is a concept derived from the motions of objects. It relates one motion to another. Because of this, it is not the property of any particular motion. This abstractness provides that it must be inherently divisible in order that we may apply it to ever faster and ever shorter duration of motion. So in the case of time I think we must always allow that even in the shortest identified time period, there is still a possible shorter time period, to provide us the capacity to identify even faster and shorter motions, in the future.

There is no such thing as zero duration. If there was, then the flow of duration (time) would have to stop and then what. Stop for how long? How does it restart? Duration (real time) is continuous and heterogeneous. It never stops and cannot be seen as stopping. Scientific time (clock time) is just a movement in space (not real time) that is symbolic and is used to approximately establish simultaneity. This is something different and shouldn't be given ontological significance. Doing so leads to all kinds of paradoxes such as those associated with Zeno's and Relativity's. — Rich

Actually, my suggestion was a change (operation) which requires zero amount of time. This implies that state A is simultaneous with state B, but are contradictory, such that state A changes to state B without any time passing. I agree that this is random nonsense, and incomprehensible, just like your description of "zero duration", but I was just trying to make sense of the proposed "supertask", which also appears to be nonsense.

That's interesting because then you are trying to convert the uncountable infinity to a countable infinity. I don't think that this is possible, and it may demonstrate that there is a real difference between the uncountable and countable, and I was earlier arguing against any such difference.

What this would require is a starting point, a definable, discrete, unit of time. Where would you get that from? An arbitrary designation would not do, because it would be just like saying we're making 1/64 our smallest divisible unit, arbitrarily. We would need an actual smallest unit, demonstrably indivisible, as the starting point. The Fourier transform indicates that the smaller the period of time, the more uncertainty there is in determining it, so you would have to get beyond that problem.

This is why I find the use of the aforementioned geometric series to address the paradox to be nothing more than trickery. It assumes from the start that it takes a finite amount of time to travel some finite distance (e.g. 10 seconds to reach the half way point), and then extrapolates from there. But obviously if your reasoning assumes that it takes 20 seconds to get from A to B (which you have done if you've also assumed a constant speed), then you're going to conclude that it takes a finite amount of time to get from A to B. — Michael

We could begin with the assumption that there is no such thing as a finite amount of time. I think this is a reasonable assumption, and those who argue that time is continuous would agree. Any application of a point, to divide time, is an arbitrary application, and is not actually being applied to any real point in time, so there is no real indication as to where that point actually is.

So instead of halving the unit of time for each successive half way point, why not double the unit of time for the previous half way point (e.g. by defining a new unit of time for each successive half-way point and considering that to be the unit that is used to measure the time spent)? The logic is the same, but the maths doesn't work out the way the "solution" wants it to.

E.g. when considering 0 - 0.5m, define the time as 1 unit. But then when considering 0.5m to 0.75m, define that time as 1 unit and so the time from 0 - 0.5m is 2 units, and so on. What's the sum then? — Michael

Now I don't understand what you are doing here. A unit of time is set out, determined, by some physical activity. You cannot just randomly change your unit of time, so that the same activity which takes 1 unit of time will later take 2 units of time, and then 4, etc.. What kind of measurement of time is that? We will just end up with an infinite amount of time in a finite motion, which resolves noting.

Off topic material stifles debate, by turning every discussion into the same discussion, of everything and nothing. The thread linked above illustrates this. But hopefully, this thread will not be diverted too much into a debate about that thread, nor about the state of modern politics. Rather, I am hoping to look in a more abstract way at how our conversations need to be ordered to maximise freedom, given that absolute freedom is both impossible and undesirable. In this sense, it might be better classified under politics, or metaphysics than feedback, but I feel that the latter classification best communicates the particular knottiness of a discussion about discussion. — unenlightened

I look at off-topicness as a necessary evil. It's bad because its digressive, wastes time, and distracts from the overall flow of the thread. It is necessary though, because the points of disagreement, and misunderstanding are usually not within the scope of the subject of the thread, though they manifest as disagreement in the subject of the thread. So if we adhere to the subject, we will just have people talking past each other, restating their own opinions, over and over, without getting to the root of the disagreement, what causes the difference of opinion, concerning the subject.

Of course it happens that one fundamental disagreement, one root cause, will manifest in a difference of opinion on many different subjects, so we might keep revisiting that fundamental issue. The goal might be to find one of these related subjects where there is agreement of opinion, and then bring that agreement down to bear on the fundamental disagreement, determining where the inconsistency lies.

Moving back toward the original question of this thread, I'm eager to introduce the notion of Supertasks to the conversation. A great summary with examples of Supertasks can be found here — Voyeur

Supertask is defined like this "a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time". Since you are asking whether a supertask is possible, I would say no. A countably infinite sequence is one which is endless, never completed, according to the definition of "infinite". Any operation requires a duration of time in order to occur. Therefore in a finite amount of time that sequence of operations would not be completed. A supertask is logically impossible.

Even if we are to assume an operation which requires a zero duration of time, then there could be an infinite sequence of these operations, but it would be in a zero amount of time. So to make an infinite sequence of operations in a finite period of time would require inconsistency within the sequence of operations, some operations taking time, to make a finite period, and some operations taking no time, to make an infinite sequence.

Sexual orientation determines whether one is sexually attracted to men or to women, homosexually or heterosexually. There's more to it, but we can go into all that another time. — Bitter Crank

If a man is attracted to white women and not black women, or blondes and not brunettes, or women of a certain colour eyes, is this a matter of sexual orientation as well?

The property that pervades through all god-beliefs is temporal in nature. All of them reference god in the past or the present. ''God exists'' or ''god created the universe''. — TheMadFool

I think that "exists" means to partake in all three aspects of time, past, present, and future. If you're just in the past, you existed, and just in the future you will exist. And the present separates the past from the future.

I don't think that a God which is just in the future is consistent with any concept of God that I know of.

It is a mistake to treat accuracy and success in the actual world as the only legitimate objectives of inquiry. For one thing, it is inconsistent with what most people mean when they talk about "ideals." — aletheist

OK, you can consider the individual who produces fictitious fantasies to be successful, I have no problem with that, it may be a pleasant and fulfilling activity. But to consider such fantasies as metaphysical successes, I will not agree with you there.

It is a mistake to confuse mathematics with metaphysics. Many things are possible within mathematics that are not actually possible. I also happen to believe that there are real possibilities that are not actually possible, but that is not at all the same thing as allowing that anything is possible. — aletheist

We are talking about the nature of the infinite here, and that is a metaphysical issue. If mathematics treats the infinite in a way that is metaphysically unacceptable, then we have an epistemological problem. Either the metaphysics is wrong, or the mathematics is wrong. But I'm not about to adapt my metaphysical principles just so that mathematicians can be understood as correct in the way that they deal with the infinite. If you are convinced that mathematicians are correct, then please justify their method of measuring the infinite.

This right here is precisely the reason why we have been at such loggerheads throughout this discussion (and others). As I keep saying over and over, mathematics is the science of drawing necessary conclusions about ideal states of affairs; it does not pertain to anything actual, except to the extent that we use it - with varying degrees of accuracy and success - to model the actual. — aletheist

The way to produce, and increase accuracy, in modeling what is real, reality, is to determine and exclude as possibilities, those "ideal states of affairs" which are actually impossible. Without moving to exclude those ideals which are impossible, we have no way to increase accuracy and success.

You have an idiosyncratic metaphysical prejudice that requires something to be actually possible in order to be considered possible in any sense. Again, your worldview is too small; there is much more to mathematics than merely counting and measuring things, and the value of pure mathematics - like that of pure science - is not limited to its practical usefulness. Do not block the way of inquiry! — aletheist

If you want a metaphysics which allows that anything is possible, you go right ahead and adopt that metaphysics, but it's not for me. I would prefer to exclude things which at first glance may appear to be possible, but which are later shown not to be possible, as impossible. If that makes my "worldview too small" for your preference, then so be it. And I don't want to spoil your party, but the way to succeed in inquiry is to narrow the possibilities, by eliminating unjustified possibilities.

You're getting too tied up with the term "countable". As suggested above, just consider the term "floozable". A floozable set is a set with the same cardinality as some subset of the set of natural numbers. — Michael

As I've said earlier, I am satisfied with two completely distinct definitions, but there are always those who what to bridge the gap. Furthermore, I believe that if we adhere to this separation, mathematics will be rendered useless, because we will not be able to use it to actually count or measure anything. If the principles of mathematics have no relation to what it means to actually count, or measure something, then what good are they?

So now you've introduced a further problem, despite your protestation, you've given "cardinality" a new definition, one quite distinct from that used when dealing with finite sets. As you recall, from Wikipedia, cardinality was defined as " a measure of the 'number of elements of the set"'. Now you "measure" the infinite set in relation to aleph numbers, instead of in relation to natural numbers. So I need a demonstration that this is a valid form of measurement. An arbitrary determination is not a valid form of measurement, so show me that you can actually measure something with aleph numbers rather than just making arbitrary judgements. We demonstrate the validity of the natural numbers by actually counting and measuring things

So the set of natural numbers is "countable" according to the value of the aleph numbers? I assume that the aleph numbers negate the infinity of the natural numbers, by stipulating a limit to that infinity. What kind of infinity are we left with then, if it is a limited infinity? Isn't this a contradictory infinity? Of what value are the aleph numbers, if they do nothing other than produce a contradictory infinity?

Any one can see that the "aleph numbers" are not the same as the "natural numbers", therefore it is not the same definition.

As explained here, "Cardinality is defined in terms of bijective functions. Two sets have the same cardinality if, and only if, there is a one-to-one correspondence (bijection) between the elements of the two sets."

So, if there is a bijection between elements of two sets then these two sets have the same (and so have a) cardinality. In the case of finite sets, the cardinality is equal to the finite number of elements in the set. In the case of infinite sets, the cardinality is a stipulated aleph number, which in the case of the natural numbers is ℵ0.

Which is why your question doesn't make sense. Infinite sets have a cardinality by stipulation. As aletheist explains about, cardinality is defined in such a way that infinite sets have one. — Michael

OK, so there is a different definition of cardinality for finite sets then there is for infinite sets, the former relates to bijection, the latter to stipulation. Which one is used in the determination of "countable"?

And following from this, a countable set is defined as a set that has the same cardinality as the set of natural numbers. — Michael

Yes, well I already went through this with aletheist. Aletheist produced a definition of "countable" according to which, no finite sets are countable. And that's what you have done here. The "cardinality" of the infinite set is not the same as the "cardinality" of the finite set. If "countable" is defined relative to the cardinality of the infinite set, then by definition, the finite sets are not countable.

It appears like you are unwilling to accept the fact that there is a substantial difference between a finite set and an infinite set. So you are unwilling to accept that what is true of the finite set is not necessarily true of the infinite set, and vise versa. You desire that the same principles are true for both finite and infinite, and so you are attempting to stipulate that this is the case. However it is not the case, and this appears to be a real problem for you. It is a problem because you will proceed to produce all kinds of false conclusion concerning the infinite, such as Tom's insistence that one infinity is "bigger" than another.

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?

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

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