Either that or there is one continuous blob

The kind of thought that was subject of an excellent 2008 BBC documentary, Dangerous Knowledge.

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.

Cantor showed that some infinities are larger, uncountably infinite. And then there are more than that. It's an interesting, curious area of maths. Check out Cantor's diagonal argument.

↪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.

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 funny. Why do you think that you can line them all up? That seems like an extraordinarily irrational idea to me. You don't honestly believe it, do you?

Do you think anyone can write out all the decimal places to pi? If not, why would you think anyone can line up infinite numbers?

We should take your word for this?

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?

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.

I didn't take Cantor's word for it, I read his diagonal argument.

Consider A = { 0, 1, 2, 3, 4, 5 } and B = { 1, 2, 3, 4, 5 }.

0↔︎1
1↔︎2
2↔︎3
3↔︎4
4↔︎5
5↔︎?

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

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.

It lacks exactly one element. — Magnus Anderson

Which element is missing?

The topic attracts cranks.

See The Enumeration of the Positive Rationals

It should be pretty clear.

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.

I'll leave you to it.

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?

By definition, to add an element X to an existing set of elements S means to increase the size of that set. — Magnus Anderson

Not for infinite sets. For obvious reasons.

ℕ and ℕ ∪ {0} really are the same size
Take:

ℕ = {1,2,3,…}
ℕ₀ = {0,1,2,3,…}

here:

f(n) = n - 1

This is:

• injective (no collisions)
• surjective (every element of ℕ₀ is hit)
• Total

That is a proof of equal cardinality. Nothing is “pretended”.
The fact that this offends finite intuition is exactly what “infinite” means in modern mathematics.

You should get on well with Meta.

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.

If "add" means "increase in size" — Magnus Anderson

But it doesn't.

Adding four to infinity is still infinity.

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.

...a number that is larger than every integer... — Magnus Anderson

...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).

A heuristic for sets is the Infinite means the set never ends; there’s no last element. That allows for sets with transfinite elements.

And adding four to an integer is still an integer. — Magnus Anderson

Sure. Infinities are not integers.

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.

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.

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.

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

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

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

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

They aren't the same size. The set of even numbers has two times smaller. Doesn't matter what Cantor and mathematical establishment say. They aren't reality. — Magnus Anderson

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

...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.

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.

What you provided is the definition of the countable infinity. That's not the same as infinity. — Magnus Anderson

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

If you want to prove that my definition is false — Magnus Anderson

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

Simply asserting that my definition is a heuristic that is useful for intuition is not an argument. — Magnus Anderson

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

That goes against what Cantor said. — Magnus Anderson

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

And I am pretty sure you won't be able to prove it — Magnus Anderson

A bijection exists between N and A — e.g., $f(n) = \frac{1}{n+1}, \quad n \in \mathbb{N}$


You really should take 's advice and read a maths book.

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.

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.

Reading isn't thinking. — Magnus Anderson

Nor is your making shit up.

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.

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

Sorry, I hadn't noticed this:

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

:lol:

Oh, well. :roll: