You know the old joke. "Why can't you cross a mountain climber with a mosquito? Because you can't cross a scaler with a vector." That joke depends on conflating the engineering definitions of scalar, vector and cross (as in cross product) with the common English meaning of a climber -- a "scaler" -- and the medical meaning of vector -- a means of disease transmission, and the biological meaning of cross, as to cross-breed living things based on their genetic makeup.

But this is a JOKE, not something you can take seriously in a philosophical discussion. — fishfry

I'll bet that it's a big hit with the ladies on the bar scene, though. :D

Countability as defined in mathematics simply has nothing at all to do with the everyday meaning of the ability to be counted ... If you counted, in the sense of saying out loud "one, two, three ..." the natural numbers, starting at the moment of the Big Bang, at the rate of a number per second; or ten numbers, or a trillion -- you would not finish before the heat death of the universe ... You can't count the natural numbers in the every day meaning of the word. — fishfry

I have acknowledged this repeatedly - the natural numbers (and integers) are not actually countable, in the sense that someone or something could ever finish counting them. However, they are all countable in principle, in the sense that there are no natural numbers (or integers) that are uncountable; given enough time, someone or something could count up to and beyond any arbitrarily specified value. As I said before (with sincere gratitude), you have stated more accurately what I meant all along.

given enough time, someone or something could count up to and beyond any arbitrarily specified value. — aletheist

Which has absolutely nothing to do with the question of whether a given set is foozlable -- able to be bijected to the natural numbers. I don't want to pile on so I'll just refer to what I've already said. The reason it's important is because this thread was percolating along on the usual lines -- nature of time and space, whether the mathematical solution of Zeno in terms of infinite series is also the physical solution, etc. Once you conflated the technical meaning of countable with its every day meaning -- a logical fallacy -- the thread lurched off on a very unproductive tangent IMO. It is far from clear that "given enough time" you could count to any specified value. If time itself is part of the universe, then you will run out of time between the Big Bang and the heat death of the universe.

Besides: What you just said is that for any given number N, you can count up to N. That is manifestly false for the reason I gave. But even if I grant you that point you still can't count ALL of the natural numbers. You have just conflated counting up to some big finite number with counting ALL the natural numbers. A big logical fallacy. Consider that the statement "X is a finite set" is true for each natural number, but not for the collection of ALL natural numbers.

So even if you can count to a zillion, and a gazilloin, and a googolty-googol-gazillion, you can't count ALL the natural numbers using that same meaning of counting. Right?

Not quite, since even the real numbers are still discrete despite being uncountable; they thus form a pseudo-continuum. — aletheist

Really? Cantor proved the reals constitute a continuum. Whatever they are, they are certainly not discrete.

Really? Cantor proved the reals constitute a continuum. Whatever they are, they are certainly not discrete. — tom

The reals in their usual order are a continuum. They can be reordered to be discrete. Counterintuitive but set-theoretically true. Order is important in this discussion. The rationals in their usual order aren't discrete, but we can line them up in bijection with the natural numbers and thereby identify the first, second, third, fourth, etc.

Also (not to hijack this thread further, but this relates to Zeno) what do we mean when we say the real numbers are a continuum? If by continuum we mean a particular philosophical idea of a continuous space, then the mathematical real numbers may or may not satisfy a philosopher. If by continuum we mean the standard mathematical real numbers, then we are being circular. Certainly the standard real numbers are not a proper model of the intuitionistic continuum. These are murky philosophical waters.

f you think to yourself, "The natural numbers, the integers, and the rational numbers are examples of foozlable sets," you will not confuse yourself or others by shifting the meaning of a technical term to its everyday meaning. — fishfry

I think the issue here has been metaphysical - so neither everyday, nor mathematical. Although the mathematics of course has to have some grounds for finding its own axiomatic base "reasonable".

So the Zeno paradox is about a particular difficulty between a mathematical operation and the world we might want it to describe. The math seems to say one thing, our experience of the world another.

Bijection is great. It replaces the need for a global act of quantification (demonstrating an example of infinity by showing a sequence is measurably unbounded) with a local demonstration of a quality (if bijection works for this little bit of a sequence, then that property ensures the infinite nature of the whole). So bijection doesn't do away with the notion of counting or a syntactic sequence. But it does extract a local property that rationally speaks for the whole.

No problems there.

And then we get back to the metaphysics on which even the mathematical intuitions are founded. Which was the issue the OP broaches and which you are side-tracking.

Again, incorrect. You evidently have a rather idiosyncratic personal definition of "infinite." My dictionary provides several widely accepted definitions, and none of them state or imply that it means "not countable." Besides, as I keep noting, the concept of being "countably infinite" is well-established and well-understood within mathematics. — aletheist

Infinite: endless.

Now you seem to be confusing "countable" with the idea of being finished counting. — aletheist

Countable means capable of being counted. If it cannot be counted, as is the case with something infinite, or endless, it is not capable of being counted. Therefore the infinite is not countable.

I have acknowledged this repeatedly - the natural numbers (and integers) are not actually countable, in the sense that someone or something could ever finish counting them. However, they are all countable in principle, in the sense that there are no natural numbers (or integers) that are uncountable; given enough time, someone or something could count up to and beyond any arbitrarily specified value. As I said before (with sincere gratitude), you have stated more accurately what I meant all along. — aletheist

Back to your contradictory notions "the natural numbers(and integers) are not actually countable ... However, they are countable in principle... "

You refuse to face the facts of the situation, the entire set of natural numbers is, in principle, not countable. That's what Infinite means, endless, so no matter how hard you try the infinite set is not countable. What is countable in principle, is any finite set of natural numbers. But it is false to claim that the entire infinite set is countable in principle, what is countable is finite subsets.

Countable means capable of being counted. If it cannot be counted, as is the case with something infinite, or endless, it is not capable of being counted. Therefore the infinite is not countable. — Metaphysician Undercover

You do agree they're foozlable, right? I just want to make sure I'm understanding you.

I think the issue here has been metaphysical — apokrisis

I really miss your bug-eyed avatar from the old forum :-)

You do agree they're foozlable, right? I just want to make sure I'm understanding you. — fishfry

Yeah sure, that's the name you gave instead of the name "countable". But I'm not sure that I would agree with the assumption that there is a substantial difference between a foozlable infinity, and an unfoozlable infinity. We can call them countable and uncountable infinities if that's easier.

Yeah sure, that's the name you gave instead of the name "countable". But I'm not sure that I would agree with the assumption that there is a substantial difference between a foozlable infinity, and an unfoozlable infinity. We can call them countable and uncountable infinities if that's easier. — Metaphysician Undercover

I don't know if it's a "substantial" difference. It's certainly a difference. The rationals are foozlable and the reals aren't. Even in countable models of the real numbers, and yes such things exist, the reals are not foozlable. So yes it's a pretty important difference in math.

But if you insist that "countable" is to be used with its everyday meaning, then we should be careful not to confuse this with foozlability, which is a technical condition used by specialists in set theory.

The reals in their usual order are a continuum. They can be reordered to be discrete. Counterintuitive but set-theoretically true. — fishfry

If you think you can disorder the reals, then pleas indicate the number following this one, and suggest between which two numbers you might place it:

0.999... What comes next?

The rationals in their usual order aren't discrete, — fishfry

Yes they are. They are countable, therefore discrete (and of measure zero on the reals).

These are murky philosophical waters. — fishfry

Only if you make them so.

But if you insist that "countable" is to be used with its everyday meaning, then we should be careful not to confuse this with foozlability, which is a technical condition used by specialists in set theory. — fishfry

If I give you a natural number, you can count it, and the next ...

Try 999 for size.

If I give you a real number, 0.999... where do you go next?

Once you conflated the technical meaning of countable with its every day meaning -- a logical fallacy -- the thread lurched off on a very unproductive tangent IMO. — fishfry

I am not the one who took us down that road by repeatedly insisting that "countable" must always and only mean the same thing as "actually countable." On the contrary, I carefully maintained the distinction between these two concepts throughout.

It is far from clear that "given enough time" you could count to any specified value. If time itself is part of the universe, then you will run out of time between the Big Bang and the heat death of the universe. — fishfry

Now you are the one conflating the mathematical with the actual. Any natural limitations on my ability as a human being to count up to very large numbers, whether there was a Big Bang, whether there will be a heat death of the universe, and how much time will have passed in between these two posited events are all completely irrelevant to the discussion. We are drawing necessary conclusions about an ideal state of affairs, so actuality (including time) has nothing to do with it.

You have just conflated counting up to some big finite number with counting ALL the natural numbers. — fishfry

I have done no such thing. I have noted, rather, that no matter how big a finite number you specify, it is possible in principle to count up to and beyond that number. In other words, you cannot identify a largest natural number (or integer) beyond which it is impossible in principle to count. If it is possible in principle to count up to any particular natural number (or integer), then it is possible in principle to count all of the natural numbers (and integers).

Your point is that this is not part of the relevant technical definition of "countable." I have accepted that clarification, and even thanked you for it, so I do not understand why you keep harping on it. Even so, there is presumably a reason why the standard term is "countable" and not "foozlable" or anything else.

I don't know if it's a "substantial" difference. It's certainly a difference. The rationals are foozlable and the reals aren't. Even in countable models of the real numbers, and yes such things exist, the reals are not foozlable. So yes it's a pretty important difference in math. — fishfry

I agree that there is an important difference between natural integers and real numbers, and even an important difference between rational numbers and real numbers, what I disagree with is that there is a difference in the infinities which arise in all these different situations. I think that the infinite itself is the same in each situation, but it is applied differently.

So here's an example. We could take a point like zero, or any other integer, and count the integers toward the positive and toward the negative, from that point, and assume two distinct infinite quantities, one negative and one positive from that point. Or we could take two points, like 1 and 2, 3 and 5, or 6 and 10, and assume an infinite quantity of real numbers in between. In these two different ways of using "infinite quantity", there is no difference in the meaning of "infinite". One is not is not a larger quantity than the other, just like the infinite quantity of real numbers between 6 and 10 is no bigger than the infinite quantity between 1 and 2. We could then take an irrational ratio, like pi, and say that it extends to an infinite quantity of digits, and this use of "infinite" is still the same.

If it is possible in principle to count up to any particular natural number (or integer), then it is possible in principle to count all of the natural numbers (and integers). — aletheist

This is a textbook case of the fallacy of composition. And, you've also forgotten one premise here, that any particular natural number has numbers higher than it. And, this is the premise which makes it impossible, in principle to count all of the natural numbers.

I have done no such thing. I have noted, rather, that no matter how big a finite number you specify, it is possible in principle to count up to and beyond that number. In other words, you cannot identify a largest natural number (or integer) beyond which it is impossible in principle to count. If it is possible in principle to count up to any particular natural number (or integer), then it is possible in principle to count all of the natural numbers (and integers). — aletheist

I made the mistake of stating, way back in the thread, that the DEFINITION of a countable set is that it can be put in one-to-one relation with the Integers, when I should have said the Natural Numbers, sorry!

This does not change the fact that, you can sit down and count members of a countable set, but you cannot do that with an uncountable set.

This is a textbook case of the fallacy of composition. And, you've also forgotten one premise here, that any particular natural number has numbers higher than it. And, this is the premise which makes it impossible, in principle to count all of the natural numbers. — Metaphysician Undercover

No one claimed that. You can count as many members as you like. Given an uncountable set, you can't count any members.

I recall doing this at school at the age ~16.

No one claimed that. — tom

Aletheist claimed that.

If it is possible in principle to count up to any particular natural number (or integer), then it is possible in principle to count all of the natural numbers (and integers). — aletheist

Fallacy of composition.

Cantor proved the reals constitute a continuum. — tom

As he defined "continuum," yes. However, Peirce argued (and I agree) that the real numbers still do not conform to our common-sense notion of a true continuum as "that of which every part has parts of the same kind," such that it cannot be understood as a collection of individuals, no matter how dense. A truly continuous line can be infinitely divided into smaller and smaller lines, but never into points. As you have noted, the real numbers form a multitude larger than that of the rational numbers; but a true continuum exceeds all multitude.

Pierce was wrong. Simple!

The real numbers have been proved to for a continuum, even in the Peirceian sense.

Just keep counting.

Now list the natural numbers between 0 and 0.00000000000000000001 so we can count them.

I have noted, rather, that no matter how big a finite number you specify, it is possible in principle to count up to and beyond that number. In other words, you cannot identify a largest natural number (or integer) beyond which it is impossible in principle to count. — aletheist

Again this is an example of rationally seeking a way for the part to speak for the whole. What can't be achieved via actualisation can be supported by appeal to the existence of a local property - in this case, not bijection but a quick demonstration that any nameable number implies in its own syntactic construction a number immediately larger (or immediately smaller).

Tom is also employing this local syntactic property.

So yes, bijection seems more abstract a level of definition because it maps maths to maths rather than maths to physics (ie: syntactic spaces where time is still part of the deal - as in saying any time you name a number, the next higher number awaits). But still, the general mathematical tactic is the same - seek a local property that constructive principles will guarantee stands for the truth of the whole. And thus, the very nature of this tactic reveals the deeper questionable presumptions that metaphysics would be interested in.

It is the idea that reality is perfectly constructible that is questioned by a synechetic or holistic point of view.

But then even a simple holism falters - the idea of the continuum being instead " the foundational". The continuum is that to which an infinity of cuts can be made. If a division is possible, another one right next to it ... but spaced by the infinitesimal of some continua ... must be possible. So simple holism is simply the inverse problem. Although - like division as an arithmetic operation - there is an advantage that at least it is being flagged that there is a more primitive presumption about there being in fact a pre-existent whole (that gets cut or divided).

So simple holism brings out the fact that simple constructionism is presuming an infinite empty space that can be filled by an unbounded act of counting. The standard atomistic approach presumes its numerical void waiting to be filled. And even bijection just illustrates the presumed existence of this numerical void as a waste disposal system that can swallow all arithmetical sequences. You can toss anything into the black hole that is infinity and it will disappear without a splash.

So the simplest view treats infinity as the void required by atomic construction. The next simplest view treats infinity as a continuum - a whole that is in fact an everything, and so able to be infinitely divided.

Then obviously - as usual - there is the properly complex view where instead of an atomistic metaphysics of nothingness, or even the partial holism of a reciprocal everythingness, we arrive at the foundational thing of a vagueness as that deepest ground which can be divided towards this reciprocal deal of numerical construction vs numerical constraint, the filling of a numberless void vs the breaking of a numberful continuum.

Of course, none of this deep metaphysics need trouble those only concerned with ordinary maths. They can believe that Cantor fixed everything for atomistic construction and the story ends there.

But deep metaphysics makes the argument that the very act of trying to cut is what produces the divided that appears to either side. The continuum arises because it is cuttable. Which like the Chesire Cat's grin, sounds really weird to those only used to everyday notions of logic or causality where something - either everything or nothing - has to be the starting point or prime mover for any chain of events.

Countable means capable of being counted. If it cannot be counted, as is the case with something infinite, or endless, it is not capable of being counted. Therefore the infinite is not countable. — Metaphysician Undercover

Please identify a natural number or integer that is not capable of being counted. You are still conflating the notion of counting with the notion of being finished counting.

But it is false to claim that the entire infinite set is countable in principle, what is countable is finite subsets. — Metaphysician Undercover

Only if "countable" means "capable of being finished counting," which is not how the term is defined within mathematics, nor how I have ever used it in this thread.

This is a textbook case of the fallacy of composition. — Metaphysician Undercover

Only if I were arguing that it is possible in principle to count all of the natural numbers (and integers) merely because it is possible in principle to count some of the natural numbers (and integers). Instead, what I am arguing is that it is possible in principle to count all of the natural numbers (and integers) because it is possible in principle to count up to and beyond any particular natural number (or integer). Again, there is no largest value that is countable in this sense, so they must all be countable in this sense.

However, note that - per @fishfry's helpful clarification - this argument of mine has nothing to do with the technical definition of "countable" (or "foozlable"), which pertains only to sets and is easily proved to be a property of the natural numbers (and integers), but not the real numbers.

The real numbers have been proved to for a continuum, even in the Peirceian sense. — tom

He certainly did not think so. Could you please point me to the proof? Note, I acknowledge that the real numbers serve as a useful mathematical model of a continuum.

He certainly did not think so. Could you please point me to the proof? Note, I acknowledge that the real numbers serve as a useful mathematical model of a continuum. — aletheist

Does mathematics actual model a continuum? I don't think so. If it did, it wouldn't lead to do many paradoxes and incorrect descriptions of experiences. Mathematics, I believe, provides a rough models of discrete, measurable actions , which in themselves are practical for certain applications, but are also quite distant from experiences. For example, it is impossible to divide space or time (duration). This simple observation, which addresses the OP, makes all the difference in the world. Far from being parenthetical, it totally changes the way one views the nature of life and the universe.

Understanding this, addresses the question of why Einstein was incapable of understanding what Bergson was presenting and why Bergson understood Einstein but disagreed. And as the OP points out, with this understanding Zeno's paradox is dissolved.

If you think you can disorder the reals, then pleas indicate the number following this one, and suggest between which two numbers you might place it: — tom

We can well-order the reals. https://en.wikipedia.org/wiki/Well-order#Reals

I mention this because it's a counterexample to the intuition that a set can be "counted" if its members can be lined up so that there's one after another. The real numbers can be well-ordered. That means that they can be ordered such that every nonempty subset has a least element. So there's a first real, a second real, and so forth. Now when you run through the natural numbers, you take a limit ordinal and keep on going. Cantor worked all this out.

As I say, the only relevance of this example is to attack the argument that only the natural numbers may be well-ordered. In fact any set may be well-ordered. And even if you don't believe that because you reject Choice, we can still find at least one uncountable set that can be well-ordered without Choice. So the idea of "lining things up one after another" is far stranger than it first appears, and is not a reliable intuition for what can be "counted," whatever that might mean. I don't think it means anything at all, which is why there are now several pages of confusion derailing the original topic of this thread.

I mention this because it's a counterexample to the intuition that a set can be "counted" if its members can be lined up so that there's one after another. — fishfry

No, you claimed the reals can be disordered and made discrete.

You are wrong on both counts, but of course if you would like do demonstrate?

Take the number 0.999... and place it between two adjacent reals out of order.

No, you claimed the reals can be disordered — tom

Please locate that quote of mine, I can't find it and don't remember saying it. I probably said reordered and definitely well-ordered, but not disordered.

I may have misspoken myself to say that a well-order would be discrete. I don't believe this is true.

However you can certainly put a discrete topology on the reals, even in their usual order. Just define the discrete metric d(x,y) = 1 if x ≠ y and d(x,x) = 0. This metric induces the discrete topology on any set. In the discrete topology every point is an isolated point.

Does mathematics actual model a continuum? I don't think so. — Rich

Of course mathematics can and does model a continuum. However, the accuracy and usefulness of such a model depend entirely on its purpose, and that is what guides the modeler's judgments about which parts and relations within the actual situation are significant enough to include.

As far as I can tell, mathematics is totally reliant on the discrete and because of this limitation constantly makes philosophical ontological errors. Admission to this major limitation would allow philosophy to move ahead. As long as philosophers are pinned to mathematics then paradoxes will continue to confound. To put it concretely, discrete can never model continuous and any confusion will necessarily to paradoxes.