More real numbers, all of which are distinct; again, it obviously does not yield an unbroken continuum. Put another way, the set of real numbers has a multitude or cardinality, which is exceeded by its power set; but a true continuum exceeds all multitude or cardinality. It is not composed of parts, it can only be divided into parts, all of which can likewise be divided into more and smaller parts of the same kind. — aletheist

Which real number is bigger?

1 or 0.999...

Right - the real numbers constitute an analytic continuum, but not a synthetic continuum; i.e., a true continuum in the Peircean sense, which cannot be represented by numbers at all. As he put it, "Breaking grains of sand more and more [even infinitely] will only make the sand more broken. It will not weld the grains into unbroken continuity." — aletheist

Breaking sand "infinitely" yields tiny bits of sand? Please!

Breaking the real numbers "infinitely" yields what?

The real continuum of the real numbers. You know parameters like "t" and "x" are real numbers, unless they are complex numbers.

I'm not sure what you're saying here. If you're only saying that consciousness depends on brain activity, then Chalmers would agree. He's a property dualist, after all, not a substance dualist. All he's arguing is that consciousness is not identical to brain activity (or any other physical thing).

And he's not saying that p-zombies are physically possible. He's saying that p-zombies are logically possible. — Michael

I don't understand how a property dualist could argue that p-zombies are possible, even logically. If they are, then p-minds are also possible. What does that even mean? I think our familiarity with dull people and zombie films lures us into the misconception that it is possible to remove properties from objects arbitrarily.

These questions don't make any sense. It is simply the case that the movement of the p-zombie's body (including the movement of the lungs and vocal chords) is causally explained by the laws of physics and prior physical states of matter. This must be true for the physicalist, as the physicalist doesn't allow for non-physical causes. The issue, then, is whether or not we can conceive of this situation without conceiving of this person having first-person experiences. Chalmers claims that we can; that we don't need to imagine that there's anything that it's like to be this person to imagine the purely mechanical series of causal relations that the physicalist must say actually explains the behaviour (e.g. electrical activity in the central nervous system). — Michael

Then I guess Chalmers would claim that a computer would continue to play chess after the deletion of the chess-playing program?

If one computer is physically identical in every way to a chess-playing computer, then it would also, necessarily, be playing chess.

P-zombies aren't physically possible.

However, certain traumatic brain injuries can result in a loss of subjectivity.

Is there an a posteriori way of determining whether there are any real continua vs. everything (including space and time) being discrete? — aletheist

Our fundamental theories are based on the continuum of space-time. What more do you want?

Quantum mechanics may be a theory that yields discrete observables, but the theory itself, and the dynamicas happen on the continuum.

As a structural engineer, I analyze continuous things using discrete models (i.e., finite elements) all the time, and it works just fine for that purpose. Of course, I also apply safety factors to the results, since I am not interested in having the underlying theory falsified. — aletheist

I hope you noticed that computers can't even instantiate the reals.

As I asked in the OP, is it possible to determine whether there are any real continua vs. everything (including space and time) being discrete? If so, how should we go about it? — aletheist

There is no a priori way of determining anything about reality: hence Zeno's paradox is solved.

There are some conjectures in physics about a granularity of space and time, but there is absolutely no evidence that such a state of affairs exists. According to our deepest, most fundamental and rigorously tested theories, we inhabit a space-time continuum.

Your insinuation that the real numbers cannot somehow model the physical continuum is rather odd.

You did not describe the infinity of the natural numbers, which is that they continue forever, endlessly. And no, the infinity between two real numbers, (no matter how large or small those numbers might be), is no bigger than this infinity. They are both infinite. One is not a bigger infinite than the other, that it nonsense. — Metaphysician Undercover

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.

But what I was objecting to was waving a physics textbook over the termite mound and saying "This explains everything." It doesn't, it can't. Not because life is beyond science -- but rather, our science isn't quite that capable yet. — Bitter Crank

Science is perfectly capable of explaining Life. It does not do so in terms of the Standard Model or General Relativity because that is not possible.

No, you can't count the natural numbers either, because they're infinite. That's the point I'm arguing with aletheist, they are by definition uncountable, because by definition they are infinite, and infinite is by definition endless, which is by definition uncountable. — Metaphysician Undercover

So, you agree you can't count any interval of the real numbers.

So, you agree you can count any (not too big or you'll get bored) interval of the natural numbers.

Given any interval of the natural numbers, you can calculate how many natural numbers are in that interval, even if the interval is too big to actually count in a lifetime.

One of these infinities is bigger than the other, much bigger. In fact the measure of the natural numbers on the continuum is zero.

I don't think so. It's just a simple question: does the golfball have to arrive at the center point before it can make it to its destination? Common sense says yes. Infinite regress appears.

Note that the regress is headed back to the starting point, not the destination. — Mongrel

Yet nothing physically infinite happens, and what motion is possible is determined by the laws of physics alone, and not by the necessary truths about an abstraction that bears the same name.

Common sense dictates that Zeno's mistake was to PRESUME that a certain mathematical notion called "infinity" is physically relevant.

Perhaps I commit the fallacy of accident. — TheMadFool

The scientific explanation of life is fundamentally a theory of replicators undergoing variation and selection. Life could exist under any physics that supports that level of emergence.

How many are there? Can you count them? Or is it impossible to count the real numbers, making them uncountable?

Contrast that with the Naturals, which, by definition you can count. Just try it 1, 2, 3, 4, 5. How many was that?

As I said, your definition appears like nonsense to me. To be able to do something, is to be able to complete that task. Being incapable of completing that task, is failure. When failure is guaranteed, then the claim of being able to do that task is completely unjustified. — Metaphysician Undercover

Can you count the number of real numbers between 0 and 0.1? If so, how many are there?

Can you count the number of naturals between 1 and 50,000,000, if not, how many are there?

But we've already solved the paradox: it is merely a confusion between an abstract attribute and a physical attribute of the same name.

Since it is possible to prove theorems about abstract mathematical attributes, which have the status of necessary truths, we are misled into assuming we have a priori knowledge of the real physical attribute of the same name. We don't.

We simply have different non-technical definitions of "countable." It is not the case that yours is true and mine is false, or vice-versa; they are just different. — aletheist

Given the Naturals, I can count some of them, in order e.g. 999 1000 1001

Given the reals, please count the three members that come after 0.999... and tell me what they are, or even what the next number is so we know we have only two of them.

I can't explain how my own mind composes these sentences and sends instructions to my fingers to type these letters. — Bitter Crank

When the science of the mind reaches the level of sophistication where it can explain what you do and think, what do you think the fundamental object of study will be? Atoms? Quarks? Or maybe, just maybe, it will take "mind" as it's fundamental object.

The science of computation studies universal computers, not electrons. Biology studies Life, not quarks, and Information theory studies, well, information.

Reductionism has been an extremely effective methodology in science, but it is a mistake to think that when we deal with fundamental objects of a theory, they must be capable of being explained at a lower level. You simply cannot explain computation or animal behaviour or thought in terms of the Standard Model.

Look, we have been using at least three different definitions of "countable" in this thread:

The accepted mathematical one from set theory, "able to be put into bijection [one-to-one correspondence] with the natural numbers."
My notion of "potentially countable" or "countable in principle," which is that there is no particular largest value beyond which it is logically impossible to count.
The notion of "actually countable," which requires it to be possible to finish counting. — aletheist

That's not right.

Everyone who is not being deliberately obtuse understands what countable means - it means you can count elements of the set. No one, unless they are being deliberately obtuse, thinks that this fact has any bearing on whether anyone would be willing to embark on counting all the members of a very large or even infinite set.

Now, will someone pleas count the number of reals that exist in the range (0, 0.0000000000000001)?

Please locate that quote of mine, I can't find it and don't remember saying it. — fishfry

The reals in their usual order are a continuum. They can be reordered to be discrete. — fishfry

If the reals can be disordered, just do it. Can't be more than a few lines?

Take any real number, leave a gap, state where the gap is, and place it between any two other real numbers.

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.

Just keep counting.

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

Pierce was wrong. Simple!

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

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.

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.

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?

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.

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.

Doesn't science explain all physical phenomena? Why does life get a special status. We've been using science (biology and medicine) to understand life and look how much progress we've made. — TheMadFool

Our current best theory of life is Neo-Dawinism. It is a theory of replicators subject to variation and selection. Where's the physics in that?

We also have other theories whose fundamental objects are independent of underlying physics - e.g. the theories of information and computation. Even thermodynamics is a theory of heat engines, which is somewhat independent of the particular design of the particular heat engine i.e we abstract away a great deal of the physics.

So, while science may indeed explain everything, this cant be done in terms of physics(+ chemistry).

Of course there's a difference between a robot and living things. Living things are far more complex than the robots of today. However, in the finaly analysis, life is naught but a complex chemical reaction. Am I wrong? — TheMadFool

I think you might be wrong.

I don't think it is possible to explain life in terms of the laws of physics(+chemistry). In order to explain life you need to invoke at least replicators subject to variation and selection, and I don't think these concepts can be reduced to physics. Also a rather detailed history has to be invoked in order to explain present day biodiversity, and a great deal of that history will involve behaviour, which again cannot be expressed in terms of physics.

Also, physics does tell us that there is a difference in kind between a human and fish brain, not degree. In this respect, our brains are more like a laptop or an iphone than a fish.

As Tom mentioned earlier, much confusion would be avoided if Cantor had picked another name. If we say a set is foozlable if it can be bijected to the natural numbers, then we can prove that the natural numbers, the integers, and the rationals are foozlable; and that the reals aren't. But nobody would have to spend any time arguing about whether you can count the elements of an infinite set. — fishfry

Fair enough, but the fact is that you can count members of a countably infinite set. You can take a subset of a countably infinite set of any number you wish. You can order the set, and you can count from one member to the next.

You can't count the members of an uncountable infinity. There is no such thing as a next member.

Exactly - it actually does mean that it is countable, but it does not mean that it is actually countable. See the difference? — aletheist

A different word could have been chosen - how about "integer-like" or "zahlen", but that would change nothing. The non-zahlen infinities are vastly bigger, and that is an astronomical understatement.

(I rushed this post because my food was delivered half way through; I hope it makes any sense at all) — Efram

I love the Khan Academy with pizza

https://www.khanacademy.org/math/math-for-fun-and-glory/vi-hart/infinity/v/proof-infinities

No, to say that one is infinitely bigger than the other is nonsense, unless you are assigning spatial magnitude to what is being counted. We are referring to quantities, and each quantity is infinite, how could an infinite quantity be greater than another infinite quantity? — Metaphysician Undercover

Why don't you just look it up, or Google it? Plenty of stuff on cardinalities, countable and uncountable infinities, the diagonalization argument, Cantor ...

Yes of course, but a subset of integers is not infinite. The difference here is with respect to the thing being counted, what is within the set, real numbers versus integers, one is assumed to be divisible, the other is not. It is not a difference in the infinity itself. With respect to the infinity itself, one is no different from the other. — Metaphysician Undercover

OK, so you can count integers, but you cannot count the real numbers, even in a tiny subset. There is an uncountable infinity of reals within any subset - hence it is a continuum. The countable infinities do not have this property. They are different and one is at least infinitely bigger than the other.

Mass was said to be a fundamental property of matter, weight or some such thing, which is quantifiable. — Metaphysician Undercover

Mass was so fundamental that they needed two different types of it.

It can't be done, but that doesn't mean that the natural numbers are countable. Neither real nor natural numbers are actually countable, because of the nature of infinity. One has no beginning point, the other has no ending point, but neither, as an infinity, is actually countable. — Metaphysician Undercover

Countable and uncountable infinities are different.

You said you couldn't count a subset of the reals.

Do you think you might be able to count a subset of the integers?

That's the point, they are not countable, so to call them "countable" is just a name, a label, it doesn't mean that they are actually countable. You might differentiate natural numbers from real numbers by saying that one is countable and the other not, but that's just a name, in actuality neither are countable. — Metaphysician Undercover

Try counting the real numbers between 0 and 1.

That's right. It appears very obvious to me that if it is impossible to count them, then it is false to say that they are countable. Why would you accept the contradictory premise, that something which is impossible to count is countable? That makes no sense to me. This is the basic nature of infinity, that it is not countable. To believe otherwise is very clearly to believe a contradiction. The notion of infinity may be useful, but it's a fiction, a useful fiction. — Metaphysician Undercover

Countable infinities are precisely those which can be put into one-to-one correspondence with the integers. This is a definition, and no, no one expects you to count them all.

The real numbers, or any finite interval on the real line, cannot be put into one-to-one correspondence with the integers. This type of infinity is much larger than the countable infinity, and is the first in a sequence of uncountable infinities.

While there are many cardinalities (could be infinite number of them as far as I know) the distinction the countable and uncountable infinities is the most important.

Indeed. I do not see the relevance of being 5'11", unless as part of an argument to say that Max Planck was infinitely tall or something, or couldn't grow, and was born fully formed. — bert1

Appealing to a degree of granularity to space probably makes matters worse. How do you get from one step to the other if there is nothing in between?

But I'm not really interested in defending the notion of discrete motion. What I'm interesting in is showing that Zeno's paradox proves continuous motion to be illogical, and that any attempt to save continuous motion from Zeno's paradox by referring to being able to calculate the sum of a geometric series misses the point. — Michael

An alternative view is to accept Zeno's paradoxes as lessons of the impossibility to deduce how reality behaves a priori, from mathematics or any other way.

What can happen is determined by the laws of physics alone, and if they say that an uncountable infinity of points in space will be traversed, then that is what will happen. Something mathematically infinite in the process may have occurred, but that involves nothing physically infinite.

It seems that Zeno's mistake is to assume that a particular mathematical notion of infinity somehow determines what can and cannot happen in reality.