There's no need to list all of the elements. All this talk about constructivism, intuitionism and finitism misses the point ( I do not subscribe to any of these -isms nor do I have to in order to be internally consistent. )

PROOF

1) To say that S is larger than S' means that S' is a proper subset of S.
( A definition that applies to all sets, regardless of their size. )

2) N is a proper subset of N0.

3) Therefore, N0 is bigger than N.

This is an indisputable proof. As indisputable as 2 + 2 = 4.

However, if you're convinced by a fallacious proof, you will normally deny the validity of this one, like a cancer attacking healthy cells.

FALLACIOUS PROOF #1

The first fallacious proof they use to show that N and N0 are of the same size is the observation that, if you add 1 to infinity, you still get infinity. This is true but only in the sense that the result is also an infinite number ( i.e. larger than every integer. ) They make a mistake when they conclude that, just because "infinity" and "infinity + 1" are infinite numbers, it follows that they are equal. It's like saying that 4 equals 5 merely because 4 and 5 are integers.

FALLACIOUS PROOF #2

The second fallacious proof they use is grounded in the premise that, if you can come up with a symbol that is defined as bijection between N and N0, it follows that a bijection between N and N0 exists ( i.e. it's not a contradiction in terms. ) It's like saying that square circles exist merely because there exists a term called "square circle" that is defined as a square circle.

Here's the definition again — Banno

I can't quote the entire part, the LaTeX code gets messed up for some reason.

You are right that f( n ) = n − 1 by itself is not a complete definition of a function. A function definition requires a domain and a codomain. I conveniently left those out, assuming that they could be inferred from the context.

I was referring to the N -> N0 variant.

bijectivity would again depend on proof, not stipulation — Banno

When I say the function is bijective by definition, I do not mean that bijectivity is explicitly stated, but that it is an unavoidable consequence of the definition. The "proof" consists solely in unpacking what is already implied by the definition, not in adding any further stipulation.

f(n)=n−1 might be bijective, non-surjective, or non-injective depending on the domain and codomain. — Banno

That's correct. However, I was talking about f: N -> N0, f( n ) = n - 1. That specific variant does imply bijection.

The real problem is being ignored and that is that the fact that a definition implies bijection between N and N0 does not mean that such a bijection exists in the sense of being a logical possibility.

N0? — Banno

Not N0 but f(n) = n - 1. That function is a bijection by definition.

It is defined as f(n)=n−1 and then shown to be a bijection. That definition does not mention bijectivity at all. At this stage, the function could turn out to be injective, surjective, neither, or both. Nothing is being smuggled in. — Banno

Yes. It is not explicitly stated in the definition. However, the definition implies it. And because it implies it, it is a bijection by definition.

It's like defining the symbol "S" as "a closed figure with three straight sides". It does not explicitly state that it has 3 angles but it does imply it. So it is correct to say that S has 3 angles by definition.

"By definition" does not mean "explicitly stated by the definition". It means "fixed by the definition ( either explicitly or implicitly )".

In mathematics, functions are defined as sets of input-output pairs. f( n ) = n - 1, in this view, is a set of input-output pairs where every element from N is paired with exactly one element from N0 and vice versa. That makes it a bijection by definition. But that does not mean that the bijection between N and N0 exists, i.e. that it is a logical possibility. It merely means that's what the symbol can represent. It's similar to how simply saying that the term "square-circle" means "a shape that is both a square and a circle" does not mean that square-circles exist.

The seemingly devastating consequences of accepting that no bijection exists between N and N0 is that f( n ) = n - 1 does not exist either, i.e. that it is an oxymoron. The good news is that that's merely a consequence of an incorrect definition of functions. Functions aren't sets of input-output pairs. They are sets of rules.

It is defined as a bijection. The same way square-circles are defined as shapes that are both circles and squares. That does not mean they are logical possibilities, i.e. free from internal contradictions.

And there's a subtle difference between "You can pick any number from N and map it onto a unique number from N0" and "You can pick every number from N and map it onto a unique number from N0".

With finite set there's a contradiction.

With infinite set there isn't. — ssu

You're not responding to what's in the quote. You did not prove the following: that just because you can think of a function that is defined as bijection between N and N0, e.g. f(n) = n - 1, that it follows that a bijection exists between N and N0. You conveniently ignore that. And so does Banno. It's very convenient.

The problem is not with the definition of the word "same size", the problem is with the wrong conclusion that N is the same size as E or that N is the same size as Z. They aren't equal. And once you plug that in, the contradictions are gone. But neither you nor Banno nor mathematicians want to admit that you made a mistake that you've been stubbornly preserving for decades, so instead, you blame it on the definition, as if definitions can be logically proven to be wrong ( they can't, they are prior. ) The redefinition of the term "same size" merely covers up the fact that you screwed up.

OK, you really don't understand the Hilbert Hotel. — ssu

And you really don't have the ability to determine whether or not other people understand something.

And then when one gest, let's say G1, leaves, it's still full (meaning there's a bijection) because: — ssu

It's not full. There's no longer one-to-one correspondence between the two sets.

R1 R2 R3 ...
---- G2 G3 ...

R1 is unpaired. And it cannot be paired with any other guest because they are all already paired.

And if another guest comes, that G0, then the hotel fills up:

R1 R2 R3 ...
G0 G1 G2 .... — ssu

Nah, that's not what you get when G0 comes.

What you get is this:

R1 R2 R3 ...
G0 G2 G3 ...

G1 left. He's not in the hotel.

Their argument is that you can create an empty room for G0 by moving all the guests one place to the right. But of course, that's impossible. They employ sleight of hand to MAKE IT LOOK like it's possible. The trick is to hide the unpaired guest in the ellipsis and to never reveal it.

And understand that I went over this many times in the past.

they are not appeals to authority — Banno

Bullshit. You're literally copy-pasting textbook arguments. Zero thinking on your part.

I think ↪Banno has done a fine job of showing the inconsistencies that arise if we don't. — Esse Quam Videri

Not really. Banno's argument is flawed because it is based on the erroneous premise that I already covered:

"If we can think of a function that is defined as a bijection between A and B, it follows that there exists a bijection between A and B."

Because he can think of a function that is defined as a bijection between N and E, namely, f(n) = 2n, he concludes that N and E are the same in size. He never actually proves that such a function is not a contradiction in terms. And I can easily show that it is.

The inconsistencies that he speaks of do not come from "my" use of the term "same size" ( as if anyone else is using it any differently ) but from his own logical mistakes ( which aren't really his own, he merely copied them from a textbook. )

You’re treating “number of elements” as a notion whose inferential rules must be fixed by finite counting, and on that assumption the infinite case does look contradictory. Mathematics takes a different route — Esse Quam Videri

They don't. It's called ad hoc rationalization.

↪Magnus Anderson I think part of what’s driving the disagreement here is that two different notions of “same size as” are in play, and they come apart precisely in the infinite case. — Esse Quam Videri

I appreciate your response but I disagree with your conclusion, namely, that we're using two different definitions of the term "same size". I am quite confident that we're using the same definition.

When speaking of sets, the word "size" simply means "the number of elements". The word is defined the same way for both finite and infinite sets. That's how the word has been used for ages and it's the way it is used today.

With that in mind, the term "same size" simply means "equal number of elements". That's it.

The terms "bijection" and "one-to-one correspondence" refer to a relation between two sets A and B where every element from A is paired with exactly one element from B, and vice versa.

The observation is that, for every two sets A and B, if they are equal in size, they can be put into one-to-one correspondence with each other. And if they aren't, then they can't.

This means that, if we know that there's a bijection between A and B, it follows that A nd B are equal in size.

The problem they faced is that, with finite sets, one can determine whether or not they are equal in size simply by counting the elements and then comparing the resulting numbers; but with infinite sets, this isn't the case.

So they came up with the idea that, if we can show that infinite sets can be put in one-to-one correspondence with each other, we can conclude that they are equal in size.

So far so good. That is all true.

The problem lies in HOW they go about establishing whether or not any two sets can be put in one-to-one correspondence.

Their method is based on a hidden, and an erroneous, premise that, if we can think of a function that is defined as a bijection between A and B, it follows that there exists a bijection between A and B.

That's akin to saying that, if there exists a symbol that is defined as a shape that is both a square and a circle, then we can safely conclude that such shapes exist.

You've probably heard of Hilbert's Paradox. Hilbert's Paradox exposes a very serious contradiction in the way infinites are normally dealt with. Unfortunately, most pretend it's not a real contradiction, justifying themselves will sorts of silly rationalizations.

Suppose we have a hotel with a number of rooms equal to the number of natural numbers.

Suppose each room is occupied by a single guest.

That gives us a nice bijection between the set of guests and the set of hotel rooms.

R1 R2 R3 ...
G1 G2 G3 ...

Guest #1 ( G1 ) is in room #1 ( R1 ), guest #2 ( G2 ) in room #2 ( R2 ) and so on.

If there exists a bijection between N and N0, then it follows that we have a spare room for another guest. Let's call that guest G0.

---- R1 R2 R3 ...
G0 G1 G2 G3 ...

There is no longer a bijection between the two sets. G0 is not in any room. And if you try to add it to any room, you will either end up having two guests in a room ( not bijection ) or you will have to kick out one of the guests ( still not bijection. )

There's no way out of this conundrum . . . other than to pretend.

And that's what they do. They pretend.

A bijection does mean that sets can be put into a one-to-one correspondence. — ssu

You're missing the point. What has to be shown is that the fact that one can think of f(n) = n - 1 means that there exists one-to-one correspondence, or bijection, between N and N0. To do that, you have to show that f(n) = n - 1 is not a contradiction in terms.

I can think of the concept of square-circle but that does not mean square-circles exist. You have to show the concept is not a contradiction in terms. And in the case of the concept of square-circle, it very much is ( Taxicab geometry is not a valid counter-argument, it's merely a fashionable response, peddled by people who are not particularly good at logic. )

If you're a superficial thinker -- and most people are -- you will miss the subtleties.

I can very easily show that there is NO bijection between N and N0. But of course, it's not written in the books, so sycophantic ego-driven non-thinkers dismiss it.

No. There are injections and surjections, which aren't bijections (both injection and a surjection) and they are also called functions. — ssu

How exactly does that contradict anything I said?

:lol: — Banno

For a grownup man, that's a pretty childish response.

If the word "function" is defined the way mathematicians define it, namely, as a relation between two sets where each element from the first set is paired with exactly one element from the second, then, if a bijective function that maps N onto N0 is a logical possibility, i.e. if it's not a real oxymoron, then it follows that there's a one-to-one correspondence between N and N0.

Notice the requirement that it must be a logical possibility?

f(n) = n - 1 is a bijective function that maps N onto N0. But, understood through the lens of the above definition, is it a logical possibility? How do you know that? What if it's a real oxymoron that is useful?

You don't know that. And that's why you can't use it as a premise. You can't say, "There's a one-to-one correspondence between N and N0 because there's this function f(n) = n - 1 that maps N onto NO in a bijective way." You don't know if that function is a logical possibility. You have no proof of it.

And it kind of stinks of circular reasoning, doesn't it? "There's a one-to-one correspondence between N and N0 because there is f(n) = n - 1, a one-to-one correspondence between N and N0!"

Nor is your making shit up. — Banno

"Making shit up" is what people confuse with thinking when they know nothing other than to read books and / or be sycophants.

Reading a maths book isn’t just passive; it’s fuel for precise thinking, especially when you’re debating infinite sets. It shows how folk have thought about these issues in the past, and the solutions they came up with that work. — Banno

In your case, it's obviously passive. That you can't see it is your problem. It's pretty clear that you don't know how to think.

Your responses are now a bit too sad to bother with. Thanks for the chat. — Banno

You get what you ask for. But I'm sure you're innocent in your mind.

Well, it's one infinity amongst a few others... — Banno

That's a pretty bad excuse. The dispute was over the definition of the word "infinity". You were supposed to provide a definition that is different from mine. Instead, you provided a definition of a related term that I have no problem with. So what was the point? To show us that you read books? Are you really that pathetic, Banno? Obsessed over how you look in other people's eyes? Even though it's pretty clear that your thinking skills are . . . lacking, to say the least.

Your "definition" of infinity is not a definition of infinity. It's not false, it's just an intuitive approximation. — Banno

You surprise me with the amount of stupidity that you can spew. "It's not a false definition of infinity, it's just not a definition of infinity."

Yep. So I went the step further, presenting one of the standard definitions. — Banno

Of a different term, you imbecile. That I don't even dispute. And that does not go against my definition. How motherfucking stupid do you have to be?

It seems then that you haven't understood Cantor, either. — Banno

I am excused . . . I am not a fanatical book reader. But you're not allowed to make such mistakes. Cantor spoke of infinities that are not equal in size to the number of natural numbers. From that, one can conclude that he didn't believe your idiocy, "There is no last element, therefore nothing is left out."

A bijection exists between N and A — Banno

That a bijective function exists, cretin, does not mean that the two sets can be put into a one-to-one correspondence.

Can you please stop quoting books?

This is a philosophy forum, for fuck's sake, not a reading group.

You really should take ↪jgill' advice and read a maths book. — Banno

And you really should take my advice and use your brain for once. Reading isn't thinking.

Matching one to one from the left, the one left out is the 100. :meh: — Banno

Bravo!

With your
A = { 1/2, 1/3, 1/4, ... }
and
N = { 1, 2, 3, .. . }

There isn't last element. Nothing is left out. — Banno

Yikes. That goes against what Cantor said.

And I am pretty sure you won't be able to prove it ( asserting it isn't a proof. )

Yep, it only has every second number, so it must be half the size... Thanks for the giggle! — Banno

You're very clearly a non-thinker, Banno. Just a regular consumer of philosophy with ego issues.

...is not the definition of infinity. “Larger than every integer” is a heuristic, useful for intuition, but the mathematical definitions depend on limits or cardinality. Something like:

S is countably infinite ⟺∃f:N→S that is bijective (one-to-one and onto). — Banno

What you provided is the definition of the countable infinity. That's not the same as infinity. Furthermore, the provided definition does not contradict anything I said.

If you want to prove that my definition is false, you have to either argue that infinity is not a quantity or that it isn't larger than every natural number. You haven't done any of that.

Simply asserting that my definition is a heuristic that is useful for intuition is not an argument. Simply because your favorite books don't define it that way is not an argument. And it's not true anyways.

Sure. Infinities are not integers. — Banno

You're the king of missing the point. Of course they are not. But they are both categories of numbers. The only sense in which "Infinity + 1 = Infinity" is true is the same sense in which "Integer + 1 = Integer" is true. Unfortunately, that does not imply that the resulting number is equal to the one you started with.

The sets {1,2,3,...} and {2,4,6,...} are in one to one correspondence, satisfying the acceptable mathematical notion of "same size". But what happened to the odd integers in the second sequence?

Read a math book or two. — jgill

If you're going to take pride in your book reading skills, even though we're on a forum that is supposedly about thinking and not reading, at least don't conflate sequences with sets.

They aren't the same size. The set of even numbers is two times smaller. It has all of the elements that N has -- except for a half of them, namely, 1, 3, 5, etc. Doesn't matter what Cantor and mathematical establishment say. They aren't reality.

Here's another way one can explain why "Which one is left out?" question is problematic.

Let A be a finite set that is { 1, 2, 3, ..., 100 }.
Let B be a finite set that is { 1, 2, 3, ..., 99 }.

Obviously, these two sets aren't of the same size.

But suppose that someone comes along and makes the claim that they ARE the same because they can be put into one-to-one correspondence.

He shows you this:

1 -> 1
2 -> 2
3 -> 3
...

It might look convincing at first, but on closer inspection, you realize that he hasn't listed all of the pairs. He has listed only a subset of them.

You inform him of this and add that B has one element less than A.

He asks you, "Which one was left out?"

But how can you tell? There are 97 possible answers. He probably hasn't even chosen which one to leave out.

But you answer anyway . . . you say, 4.

He tells you, "Ah, no! I didn't leave that one! That one is paired with 4!"

You gasp and then say . . . You left out 100.

He smiles and says, "Wrong again! 100 is paired 100!"

You keep doing this for a while, failing to prove him each time. After some time, you might get tired, give up and concede. Or you might push him till the very end -- at which point, you win.

But with infinite sets there is no point at which this process can be finished. The person can keep you playing this game for as long as they want.

So there's no point in playing this game.

The correct response is to say that you can't tell which one was left out because there is an infinite number of possibilities. Moreover, in all likelihood, the person didn't even choose which one to leave out, meaning the game is rigged from the very beginning.

Furthermore, it's useful to add that a subset is not the entire set, and that the same applies to functions. If he can't show you the entire set of pairs, he hasn't proved anything.

Adding four to infinity is still infinity. — Banno

And adding four to an integer is still an integer.

The resulting category is the same. If you add four to a number that is larger than every integer, you get a number that is larger than every integer.

But the resulting number isn't the same.

Not for infinite sets. For obvious reasons. — Banno

Not quite. Definitions are prior. Nothing can invalidate them. If "add" means "increase in size", nothing can make it change its meaning. And the word "add" means "increase in size" for all quantities -- not merely for finite ones. The rest is ad hoc rationalization.

That is a proof of equal cardinality. Nothing is “pretended”. — Banno

And that is precisely what's being disputed. Your "function proof" is no proof at all. It's smoke and mirrors. The fact that f(n) = n - 1 exists merely means that you can use it on any natural number. That's all. It does not mean there's a one-to-one correspondence between N = { 1, 2, 3, ... } and N0 = { 0, 1, 2, 3, ... }. If you were to sit down and use every natural number starting at 1 as the input of that function, you won't end up producing all of the numbers from N0.

I can't think instead of you, Banno. If you can't do it, that's fine. But don't make it look like it's the other person's problem.

By definition, to add an element X to an existing set of elements S means to increase the size of that set. If you take a set N = { 1, 2, 3, ... } and you add 0 to it, you get a larger set. They are not the same merely because someone can pretend that they can be put into one-to-one correspondence. You can't list all of the pairs, can you? You can't. You can only list a subset. So that's not a proof you can put the two sets in one-to-one correspondence. On the other hand, the definition of the word "add" is indisputable. Nothing about infinities can change that.

Your question "Which element is left out?" is silly because I can't answer it, not because no element was left out, but because I can't tell which one you left out.

It's answerable if we're working with finite sets, e.g. A = { 1, 2, 3, 4 } and B = { 1, 2, 3 }. If you claim that A and B can be put into one-to-one correspondence and do the following:

1 -> 1
2 -> 2
3 -> 3
...

I can easily tell that you left out 4.

But with infinite sets, there's an infinite number of candidates. So how can I answer which one you left out? In fact, you didn't even CHOOSE which one to leave out. Yet, you want me to tell you which one you left out.

And you call that philosophy?

Which element is missing? — Banno

Silly question. The point is that you can't put them into one-to-one correspondence. In other words, one element must be left unpaired. Which one? You can pick any one. The ellipsis allows you to hide that.

There are not enough items in your second set to map one-to-one to the first set. Hence the cardinality of the firs tis larger than that of the second. Looks pretty convincing to me. — Banno

And there are not enough elements in the set A = { 1/2, 1/3, 1/4, ... } to put it into one-to-one correspondence with the set of natural numbers N = { 1, 2, 3, .. . }. It lacks exactly one element. Looks pretty convincing to me.

But that does not stop people from tricking themselves into believing that it's possible to do so by using the following "proof".

f(n) = 1/n - 1
1/2 -> 1
1/3 -> 2
1/4 -> 3
...

The fact that they can't list all of the elements is what makes it easy for them to trick themselves.

We should take your word for this? — Banno

Why not? If you can take Cantor's, you can take mine.

I gave an argument - albeit briefly. Fractions can be placed in a sequence, and so are no more than countably infinite.

Were did I go wrong? — Banno

You didn't do that. You merely asserted that you did it.

I can do the same for finite sets. Consider A = { 0, 1, 2, 3, 4, 5 } and B = { 1, 2, 3, 4, 5 }.

Here's me using the kind of proof that you're using to prove that A and B have the same cardinality.

0 -> 1
1 -> 2
2 -> 3
...

The fact that you can't list all of the pairs when working with infinite sets is what makes it easy to fall for that trick.

↪an-salad You are right that there are infinite infinities, but even with all those fractions, there are still only the same number as there are integers - ℵ₀, the smallest infinity - countably many. You can list them in a sequence, 1/1,1/2, 1/3, 2/3, 1/4, and so on, and so you can count them - line them up one-to-one with the integers. — Banno

That's not true.

Hi @Tristan L,

If the discovery-process is deterministic, the concrete instance of the solution exists from the start, although it only becomes “seeable” at the time that it manifests in a direct shape. Therefore, this is only creation in the broad sense, not in the strict sense. For example, the concrete software solutions that my algorithm will find already exist now, although not in a recognizable shape, so they can’t yet be used right now. They only become usable once the algorithm actually finds them, and that is the moment at which they are created (in the not-strict way).

If the discovery-process is deterministic, the discovered solution will necessarily be a solution that existed in a number of spaces long before it was discovered. Two of those spaces are:

1) the space of all possible solutions to all possible problems that can be discovered by following an existing algorithm (this is the space you're referring to)

2) the space of all possible solutions to all possible problems (this is where even solutions that were discovered through a random process existed long before they were discovered)

However, the discovered solution does not necessarily exist in the space of all possible solutions to all possible problems hitherto actualized (by humans, other living beings or machines.) And it is this space that ultimately matters.

One can very easily write a computer program that outputs every possible 32-bit 1920x1080 bitmap. The moment someone does so is the moment the first space (the one you're referring to) becomes filled with EVERY possible painting. If creativity is measured in relation to that set, that would make every subsequent painter an uncreative painter (even if they came up with a painting that depicts something of value that wasn't previously visually depicted.)

That is also what I think if what you call “possible idea” is what I call “idea” and what you call “actual idea” is what I call “concrete mental instance of an idea”.

I'd say so.

There is the set of everything someone can think of (possible ideas) and the set of everything people thought of (actual ideas.)

I agree with you if you mean the following: A mental instance of an idea in Alice’s mind has been invented by Alice, unless it was first invented by Bob’s mind, in which case Alice’s mind only discovers that instance of the idea.

I wouldn't say that Alice discovered it. I would say she reinvented it.

@Pffhorest

Creativity seems to be popularly held to be some kind of non-deterministic, random process of some kind of magical, metaphysically free will, but I hold that that is not the case at all. — Pfhorrest

Creativity is simply the ability to discover previously undisocvered solutions to problems. How you're going to discover such solutions is completely irrelevant. In other words, you can use a deterministic process but you can also use a random process. It does not matter.

On the other hand, I do agree with you that most people discover such ideas by following a deterministic process. (Most are merely not aware that what happens under the hood is largely, if not entirely, deterministic.)

I hold that there really isn't a clear distinction between invention and discovery of ideas: there is a figurative space of all possible ideas, what in mathematics is called a configuration space or phase space, and any idea that anyone might "invent", any act of abstract "creation" (prior to the act of realizing the idea in some concrete medium), is really just the identification of some idea in that space of possibilities.

I disagree with the bolded.

I will repeat what @Luke said.

"Discovery" implies that the thing that is discovered existed before discovery whereas "invention" implies that the thing that is invented did not exist before.

If you are talking about the set of all possible ideas, these can't be invented, since they already exist; they can only be discovered.

But that's because we're talking about the set of all possible ideas. The set contains all ideas that are possible -- there is absolutely no room for new ideas. If we're talking about the set of all actual ideas, however, one can introduce new ideas to it so as long it does not contain all possible ideas. An actual idea, that one that either existed within someone's brain at some point in time or did not, can be invented, provided there was no brain within which it existed previously.

That aside, compare this "fact" of infinity with other concepts that are considered uncountable. How about love, courage, joy? These concepts are categorized as uncountable i.e. unquantifiable and fall under the category of quality. So, it doesn't seem wrong to say that infinity is not actually a quantity, a number, but rather a quality like love or courage, etc. — TheMadFool

But love, courage and joy are neither greater than nor less than any sort of number.

I'd say anything that can be said to be greater than or less than something else is a quantity (and therefore a number.) Infinity > every integer. Therefore, infinity is a number.

Depends on whether we're talking about infinity in the general sense of the word (as in, any number greater than every integer) or in the specific sense of the word (as in, specific number greater than every integer.) It's not like there is only one number greater than every integer. There are many such numbers. And infinity in the general sense of the word does not refer to a specific number of that sort (hence why infinity + 1 = infinity holds true.)

Infinity is a number greater than every integer.

And infinity + 1 = infinity is true only in the sense that if you take an infinite number and add one to it, you get an infinite number. Basically, only if infinity does not refer to a specific infinity (i.e. only if infinity - infinity =/= 0.) Otherwise, if you're working with specific infinity, infinity + 1 > infinity.

Thank you very much.

As you say, an infinite sum may not have a limit. If you say that the concept of infinite sum and the concept of limit are one and the same concept then how is it possible for an infinite sum to have no limit?

Furthermore, the difference between the two concepts is bigger than that. For example, two infinite sums that approach the same value can represent different quantities. For example, $0.9 + 0.99 + 0.999 + \dots$ represents a number greater than $0.5 + 0.25 + 0.125 + \dots$ even though they both aproach but never reach $1$. Indeed, there are numbers greater than $0.999\dots$ but lower than $1$. Hexadecimal $0.FFF\dots$, for example, lies somewhere between the two numbers.

↪Magnus Anderson I can't even hazard a guess as to how you think "most people" define "0.333~" (I am more accustomed to the ... notation, but I assume you mean the same thing). — SophistiCat

I avoid the "..." notation because it looks ugly when used in forums without LaTeX support. But yes, that's what I mean.

"0.333~" represents the infinite sum 3 x 1 / 10^1 + 3 x 1 / 10^2 + 3 x 1 / 10^3 + ... + 3 x 1 / 10^inf. It does not represent its limit.

Some perfectly sensible and familiar rational numbers, such as 1/3 = .3333333..., have infinitely-long decimal representations. — fishfry

This depends on the meaning of the symbol "0.333~". According to the way most people define "0.333~", it is not true that "1/3 = 0.333~". By standard definition, 0.333~ is a number smaller than 1/3 (in the same way that 0.999~ is a number smaller than 1.) It's not a decimal representation of it. "0.333~" does not mean "the limit of 0.333~".

You said points have no size. I do not see how any part of time could have no size. If it has no size, then no time is passing at that "point", therefore it is not part of time. The same principle holds for space. If it has no size, then it cannot be part of spatial existence, because there is no space there. It is very clear to me, that if points have no size, then they are excluded from space and time, because things existing in space and time have size. Having size is what makes them spatial-temporal. Do you not understand this? — Metaphysician Undercover

I do not understand why you think that things that exist in space and/or time must have size. Why is it impossible for something to exist (in space and/or time) and not have size?

Points don't exist in physical space. According to the description they are non-spatial. — Metaphysician Undercover

I am not sure why you think so, Points do exist (both in time and space.) Consider that at any point in time, you occupy certain point in space. So there exist at least some of the points that we can imagine. Points that do not exist cannot be occupied by anything under any set of circumstances.

Physical space is made out of points. The fact that physical space is made out of things that have no size (points) does not mean that it has no size itself. Not sure why you think so.

We've already placed the no-length point as right out of the category of things to be measured, so how can a point appear on a line to be measured? — Metaphysician Undercover

Though you cannot measure how long a point is (since it has no length, as per definition) you can identify a point. And we do so through a complex process that involves the movement of our bodies (if we're talking about identifying points in physical space, that is.)

There are many things that have no size but that nonetheless exist (and are not logically contradictory, illogical or otherwise irrational.) The word "existence" does not imply size. For example, colors and feelings exist, and yet, they have no size. A typical counter-argument is that colors are light waves and that light waves have size (their wavelength.) But light waves are not colors. Rather, light waves are things that cause colors. (This is evident in the fact that light waves can exist without conscious beings whereas colors can't.)

If they had the same positions relative to all other objects, they would have the same identity, which means they would be one apple, not two. — litewave

My point is, not necessarily.

In general, history is very important. Have you heard of Richard Gregory's top-down theory of perception?

When it comes to identity, matters are different and space-time properties are critical to its meaning. One object cannot occupy two locations in space at the same time and it is this impossibility that gives objects their identity. So, two/more objects can be identical because they share all properties except space-time properties but they all have different identities because one object can't occupy two locations in space at the same time. — TheMadFool

Identity can also be (and it mostly is) established by history. So two objects can occupy the same position in space at the same time and still be identified as two different objects simply because they have different histories.