What do you mean?

I mean, what is information - a question directed at Devan, to help clean up his comment that there is twice as much of it in four as in two.

I'm playing with words in order to show that there is nothing more going on here than wordplay.

Zeno might wonder how you traverse between discrete links in a "chain of moments" .

But every time his wife told him to take out the trash, he'd reply. "How possibly could I traverse that infinite distance."

No, it doesn't. 4 might be twice two, but what could it mean to say it has twice the information? — Banno

The interval 4 is twice the length of 2 so it should contain twice as many real numbers. In our minds both intervals contain an equal and infinite number of real numbers . But I believe that could not be the case for any real life interval - the larger interval would contain more numbers/information.

So your mind is not in the real world? Infinity is not a thing like my cat or last Tuesday? What's going on here? Is infinity a thing like my mortgage? Like a unicorn? — Banno

Infinity is an illogical concept. Illogical concepts can exist in our minds but not reality. Faeries, talking trees, infinity we can all imagine but they do not exist in reality IMO.

The interval 4 is twice the length of 2 so it should contain twice as many real numbers. — Devans99

That is not how real numbers work. By such (il)logic, there should be twice as many integers as even numbers, which is also not the case. A discrete collection of four items obviously does contain twice as many objects as a discrete collection of two items, but a continuum (such as space-time) does not consist of discrete items at all. The unwarranted axiom here is that reality consists entirely of discrete items and collections thereof.

Now, I do not dispute that actuality consists entirely of discrete items and collections thereof; but I deny that reality--that which is as it is, regardless of what anyone thinks about it--is limited to actuality. Specifically, there are real continua of potential items that are not even remotely exhausted by their discrete instantiations. Being and existence are not coextensive.

By such (il)logic, there should be twice as many integers as even numbers, which is also not the case. — aletheist

Twice as many integers as even numbers makes sense. We can count them:

1, 2, 3, 4, 5, 6, ...
Vs
2, 4, 6, ...

Appears to be twice as many integers by counting up-to 6. We can us our knowledge of extrapolation to conclude that there are twice as many integers than even numbers.

Again, that is not how it works. There are twice as many integers as even numbers within any finite (and even) interval, but neither the set of all integers nor the set of all even numbers is finite. There is no greatest integer, and there is no greatest even number; so for every integer, no matter how large, there is a corresponding even number.

Infinity is an illogical concept. — Devans99

Not so much.



There's a lot of cute maths around infinity. But your OP depends on continuity. Check out

Incidentally, recognizing this is the key to dissolving Zeno's famous paradoxes. — aletheist

Some of them. They do nothing to resolve the paradox of the arrow, so far as I can tell.

pointless. Devan will just keep saying infinity is contradictory even when you ask him to show the exact formal contradiction as opposed to a non-intuitive conclusion.

There are twice as many integers as even numbers within any finite (and even) interval, but neither the set of all integers nor the set of all even numbers is finite. — aletheist

You are claiming:

- Twice as many integers as even numbers within a finite interval
- An equal amount of integers and even numbers in an infinite interval

This is nonsense.

Perhaps. @Devans99 is trying to make sense of infinity. There is a vast background of material they are missing, that sets out ways of talking reasonably about infinity. Away from this thread, they may learn some of these approaches.

You are claiming ... An equal amount of integers and even numbers in an infinite interval — Devans99

I said nothing whatsoever about "equal amount" or "infinite interval," concepts that mistakenly treat infinity as if it were extremely large, but still finite. How many integers are there? Infinitely many. How many even numbers are there? Infinitely many. If we paired up each integer with an even number, when would we run out of even numbers, but still have integers left? Never.

Note that whether an actual infinity is possible or impossible is completely irrelevant here. This is mathematics, which is the science of drawing necessary conclusions from formal hypotheses. The definitions of "integer" and "even number" constrain us to recognize these somewhat counterintuitive relations between them.

They do nothing to resolve the paradox of the arrow, so far as I can tell. — MindForged

As summarized by Wikipedia, the arrow paradox states, "If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible." If motion is a more fundamental reality than position, and space-time is a true continuum, then both premises here are false--nothing is ever completely motionless, and time is not composed of discrete instants.

I hope so. Previous threads on the matter leave me cynical about them accepting that material.

Ah. I just saw this.

You may be right.

You may be right. — Banno

He is right. We talk about the magic of infinity and thats just it; its magic not maths. Galileo's paradox I've already mentioned but its pretty central to why infinity is contradictory:

https://en.wikipedia.org/wiki/Galileo%27s_paradox

So there are clearly more numbers than squares in all intervals. Except an infinite interval when there are the same amount. So we have assumed actual infinity exists and derived a contradiction - Galileo's paradox is proof by contradiction that actual infinity does not exist.

One more time: Mathematical infinity is not an actual infinity, but it is a real infinity. If we paired up every number with its square, when would we run out of one or the other? Never. How is this a contradiction?

One more time: Mathematical infinity is not an actual infinity, but it is a real infinity. If we paired up every number with its square, when would we run out of one or the other? Never. How is this a contradiction? — aletheist

There are less squares than numbers in the following intervals:
0-10
0-1000
0-1000000
So via extrapolation, there must be less squares than numbers in ALL intervals.

The fact that actual infinity contradicts the above means we can induce that actual infinity does not exist.

Mathematical infinity is not actual infinity. Which part of this do you still not understand?

Mathematical infinity is not actual infinity. Which part of this do you still not understand? — aletheist

Mathematical infinity in set theory is actual infinity.
Mathematical infinity in calculus is potential infinity.

?

Mathematical infinity in set theory is actual infinity. — Devans99

No, it is not. Like all mathematical theories, set theory--especially as applied to infinite sets--is based on certain hypothetical formalizations that may or may not correspond to anything actual.

How many integers are there? Infinitely many. How many even numbers are there? Infinitely many. If we paired up each integer with an even number, when would we run out of even numbers, but still have integers left? — aletheist

The problem being that we cannot pair them up because there is an infinite number of either one of them. So your conditional "if we paired up each integer with an even number", is a statement of an impossibility, and therefore must be dismissed as a false premise.

If we paired up every number with its square, when would we run out of one or the other? Never. — aletheist

Again, an impossible conditional. "If we did X" when X is impossible.

Yes I think so. It was all motivated by misplaced belief I think: Cantor an Co thought God was infinite so infinity was shoe-horned into mathematics for that reason.

Nothing wrong with having a finite-sized God IMO. — Devans99

Infinite means in-finite, not finite or not discrete. It is only finite things which have a size. If God is infinite that does not mean God is infinitely large. In a sense even finite objects are infinite because they are not composed of discrete points, whether those points are of zero dimension or some dimension.

As others have said you seem to be confusing yourself by reifying mathematical concepts.

The problem being that we cannot pair them up because there is an infinite number of either one of them. — Metaphysician Undercover

It's called a bijection.
(Fairly basic high school mathematics, if memory serves.)
Kind of odd to just deny something without really knowing about it. :brow:

I already know about it. The fact remains that we cannot actually pair them because there is an infinite number of them. So the proposition states something which is, by definition, impossible (i.e.it is contradictory). Therefore we ought to reject it as a falsity, as is customary for propositions which are recognized as self-contradictory. Whether or not it's basic high school mathematics is irrelevant. If it's a falsity it ought to be rejected.

The fact remains that we cannot actually pair them because there is an infinite number of them. So the proposition states something which is, by definition, impossible (i.e. it is contradictory). — Metaphysician Undercover

A proposition is not contradictory merely by virtue of stating something that is actually impossible, only if it states something that is logically impossible--which is certainly not the case here. Mathematics has to do with the hypothetical, not the actual.

A proposition is not contradictory merely by virtue of stating something that is actually impossible, only if it states something that is logically impossible--which is certainly not the case here. — aletheist

As I said, it is impossible "by definition". This means that it is logically impossible, contradictory. "Infinite" is commonly defined in such a way that it is impossible, by definition, to pair up infinite things, because the task would never be complete. It is only by changing the definition of "infinite" to something else, that this might become possible. But then "infinite" loses it's meaning, so what's the point? You simply change the definition of "infinite" to create the illusion that the logically impossible is actually possible.

Not sure it's worthwhile mentioning the obvious, but that's what a bijection does, @Metaphysician Undercover. Feel free to derive the contradiction you mention.