... it is impossible that an infinite set has a cardinality. — Metaphysician Undercover

Not when "cardinality" is defined as a specific property of infinite sets.

I am ready to accept it, as soon as all inconsistencies and contradictions are removed. — Metaphysician Undercover

There are no inconsistencies or contradictions within the hypothetical realm of mathematical set theory.

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

Pot, kettle, black.

A one-to-one relationship between sets e.g. from the natural numbers onto itself (i.e. a permutation) is not a one-to-one function, but a one-to-one and onto function. An easy slip-up to make when moving from the language of relations to the language of functions, wile being brow-beaten by unreason.

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

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 above, cardinality is defined in such a way that infinite sets have one.

And following from this, a countable set is defined as a set that has the same cardinality as the set a subset of natural numbers. Which, if the notion of cardinality is so problematic for you, is just to say that a countable set is defined as a set that has an injective to the set a subset of natural numbers. Which, if the notion of bijection is so problematic for you, is just to say that a countable set is defined as a set upon which we can apply a particular formula to each member and uniquely pair it to some member of the set a subset of natural numbers.

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.

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" — Metaphysician Undercover

No, it's the same definition and both relate to bijection. It's just that the cardinal numbers used are different. In the case of finite sets we use the natural number that is equal to the number of elements in the set and in the case of infinite sets we use stipulated aleph numbers.

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. — Metaphysician Undercover

Sorry, I worded it wrongly above (although worded it correctly earlier). A countable set is defined as a set with the same cardinality as some subset of the set of natural numbers (which, again, includes the natural numbers themselves).

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

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

It's the same definition of cardinality: "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."

It's just that the value of the cardinality is different. We can't use natural numbers to denote the cardinality of infinite sets, just as we can't use natural numbers to denote the numbers greater than 0 but less than 1. So we use a different notation: in the latter case, fractions or decimals; in the former case, aleph numbers.

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?

No, it's the same definition and both relate to bijection. It's just that the cardinal numbers used are different. In the case of finite sets we use natural numbers and in the case of infinite sets we use aleph numbers. — Michael

It might be worth noting that Cantor proved that any interval of the Reals [a,b] cannot be placed in one-to-one correspondence with the Naturals, before he developed the idea of cardinality.

So, instead of getting bogged down with definitions, the remarkable discovery that there are more Real numbers in any finite interval than all the Naturals, indicates a fundamental difference between the continuous and the discrete.

Of course, this difference was analysed further and more remarkable discoveries were made, but it is the distinction between the continuous and the discrete that is of fundamental importance.

Another remarkable feature of the continuum is that in any interval [a,b] there are an uncountably infinite number of transcendental numbers.

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? — Metaphysician Undercover

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.

And I'm not sure what you mean by limited infinity.

A floozable set is a set with the same cardinality as some subset of the set of natural numbers. — Michael

The set {1,2,3,4,5} has the same cardinality as {6,7,8,9,10}, so we should call it foozable? How do you calculate the cardinality? Do you fooze the set, or do you perhaps count it's members?

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

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? — Metaphysician Undercover

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. 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!

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.

Without moving to exclude those ideals which are impossible, we have no way to increase accuracy and success. — Metaphysician Undercover

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."

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. — Metaphysician Undercover

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.

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 ... the way to succeed in inquiry is to narrow possibilities, by eliminating unjustified possibilities. — Metaphysician Undercover

It is a mistake to block the way of inquiry by ruling out possibilities too hastily. I would prefer not to exclude things which at first glance may appear to be impossible, but which are later shown to be possible. The key is to formulate retroductive conjectures that are amenable to deductive explication and inductive evaluation through experimental testing - often in the actual world, but sometimes in the imagination, as for example within mathematics.

It is a mistake to confuse mathematics with metaphysics. — aletheist

I think it is a bigger mistake to write off the great works of Georg Cantor and Nicolas Bourbaki (that makes at least 8 geniuses) on the basis of your personal inability to comprehend the first thing about it.

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. — aletheist

That does not follow. It must be proved. That's Cantor's theorem. Well worth looking at since it's a beautiful little proof that gives us an endless hierarchy of transfinite cardinalities.

I think it is a bigger mistake to write off the great works of Georg Cantor and Nicolas Bourbaki (that makes at least 8 geniuses) on the basis of your personal inability to comprehend the first thing about it. — tom

What exactly is it that you think I am not comprehending? Sincere question, I am eager to learn.

That does not follow. It must be proved. That's Cantor's theorem. — fishfry

If it is a theorem that has been proved, then it follows, does it not? What am I missing?

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.

If it is a theorem that has been proved, then it follows, does it not? What am I missing? — aletheist

Your full quote earlier was:

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. — aletheist

There is only one way to read this.

* 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.

That is flat out false. You said "therefore ..." and that's wrong. Surely you see that. The correct statement is: "And in fact we can prove that this holds for infinite sets as well." In fact the very definition of 2^N needs to be changed to make this work. Rather than talking about natural number exponentiation, we now change the meaning to redefine 2^N as the collection of subsets from a set of cardinality N to the set 2.

There's no "therefore" in this. You need to make a new definition of the notation and then prove a theorem. You started with the expression 2^N meaning natural number exponentiation, then changed definitions in midstream to redefine 2^N as the powerset of N.

Thanks for yet another helpful clarification.

What exactly is it that you think I am not comprehending? Sincere question, I am eager to learn. — aletheist

I may have been responding to you, but I thought it obvious I was not referring to you!

I may have been responding to you, but I thought it obvious I was not referring to you! — tom

Ah, okay; it was (obviously) not obvious to me, since I generally assume - unless there is a clear indication otherwise - that a reply to one of my comments is directed at me. Thanks for clarifying.

We know that the present is real because of the radical difference between future and past — Metaphysician Undercover

I would agree that the future is much different in that it is a concept rather than a concrete event in memory. In memory, we have a possiblity of an action which defines the future. In regards to past and present things get much more problematic. Both are simply memory being apprehended. There is only a qualitative immediacy difference between that which feels like now and that which feels like before. I am watching a TV show but that apprehension is really what is known in my memory add qualitatively more recent than the show I watched yesterday. There is no hard line between any aspect of memory including an action with intent into the future. There is no reason to make one, it just muddies the water and creates problems in explaining such a line.

Adopting this approach creates a single aspect of life called Memory that endures in duration (time). An utterly continuous flow pushed on by consciousness (or Bergson's Elan Vital).

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

I always get a little uppity when people try to dismiss Zeno's paradoxes with the fact that an infinite series can have a sum. It misses the point entirely. And the paradox can even be worded to INCLUDE the summation idea within itself. If I tell you I want to walk 10 feet and that I'm going to walk 5 feet first, and then 2.5 feet, and so on... Haven't I just implicitly stated that I believe an infinite series has a finite sum? Namely the 10 feet I talked about at the beginning? It's just lazy to think that the idea of an infinite series is actually a good response to Zeno's charge of the impossibility of motion.

Just stating that an infinite series can have a sum, as some have done in this thread is not enough to resolve the paradoxes of Zeno or the other examples on that wiki page, in fact Thomson's Lamp is a brilliant example of just how ineffective that argument is, at least in this case.

I think the question that ought to be asked, in light of Zeno's criticisms of the pervasive idea of a divisible world, is whether motion is a supertask, or not.

Is motion a supertask?

Then, if we can answer that question, we might move on to the possibility of supertasks themselves, about which there has been much disagreement.

Further, I'd like to point out that Zeno's Paradoxes can be tweaked to not only attack the continuity and coherence of space, but also time. As has been alluded to already in this thread, time and space are central to Zeno's line of dialectic, and dealing with time takes us away from the pesky and ultimately fruitless Planck length explanations that crop up just as regularly as the sum-of-series ones do.

An example: Suppose you wanted to microwave your frozen tv dinner. You set the timer for one minute (this is one of those fast cooking dinners that you love so much), and then you wait. But before you wait a full minute, you have to wait a half minute, and before a half a quarter, and so on all the way down... And in the end, of course, you end up not starting at all, because there is no smallest amount of time that you can actually wait. And then you go hungry.

I've seen it argued that Zeno's Paradoxes are an indication that space is not continuous, but I haven't seen the same said of time. Maybe I just missed it?

And finally, I think it might be worth examining two of the modern "successors" (I use the term loosely) to Zeno's paradoxes, in the Thomson's Lamp Paradox and the Ross-Littlewood Paradox, both of which can be found in the wiki link I provided.

The is no paradox if one a treats time and space as indivisible - which is clearly the case. Only those trapped in the works of numbers would agree otherwise. Of course, the is motion and duration always flows, but for some their experiences are not as real as numbers.

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

Why is motion a supertask rather than a hypertask?

The is no paradox if one a treats time and space as indivisible - which is clearly the case. — Rich

It might be indivisible at a certain scale, but it's not indivisible at every scale. There is a half way point between the start of a 100m line and the end, and this is true even if we don't plot it, which is why I don't understand aletheist's and apokrisis' objection at the start.

But as for space (or at least motion) being indivisible at some fundamental scale, I'd agree. The paradox is avoided if space (or at least motion) is discrete rather than continuous. And I believe atomic electron transition is a known example of discrete motion in nature.

I always get a little uppity when people try to dismiss Zeno's paradoxes with the fact that an infinite series can have a sum. It misses the point entirely. — Voyeur

While I welcome your approach and think that it is among the more promising ways of looking at the problem, I must object to the remark about "missing the point." The problem is that most statements of Zeno's so-called paradoxes not only fail to make your points, but fail to make any cogent points, as has been largely the case in this thread.

The point about the convergence of infinite series, to which you take an exception, is an effective response to those statements that boil down to the thesis that an infinite sequence of events necessarily, as a matter of logical reasoning, takes an infinite amount of time. But I agree that the response is often given reflexively to statements that either rely on different assumptions or are so vague that one cannot confidently make out their core assumptions.

So I would set aside the two questions that you formulated - is motion a supertask? and are supertasks (metaphysically?) possible? - as open questions that, prima facie at least, are not incoherent or trivial. Other things that you mention, such as Thompson's lamp, might actually be less problematic than you think, being ultimately language problems rather than problems of metaphysics.

But anyway, if you want to talk about the point, a good way to start would be to give a crisp statement of the alleged paradox.

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.