[QUOTE=Flameon;48353411]Right but why can't you just keep shifting?[/QUOTE]
The difference is that one of the infinities are "countable", as in you can keep counting "0, 1, 2, 3, 4 (...)"
The other infinity is uncountable, this means you can't even BEGIN to count it.
"0, 0.0000000000(...)" you're already stuck with an infinitely long number on the very first step!
Infinity is infinity though, I don't understand how one can be assumed bigger or smaller, as they're both infinitely large so things like that should not apply.
[QUOTE=paul simon;48354040]The difference is that one of the infinities are "countable", as in you can keep counting "0, 1, 2, 3, 4 (...)"
The other infinity is uncountable, this means you can't even BEGIN to count it.
"0, 0.0000000000(...)" you're already stuck with an infinitely long number on the very first step!
Infinity is infinity though, I don't understand how one can be assumed bigger or smaller, as they're both infinitely large so things like that should not apply.[/QUOTE]
The human mind fundamentally has a lot of trouble with understanding infinity, the concept that something is truly endless is so bizarre that we still try to think about a "last number" in an infinite sequence.
Just because a sequence is endless does not make it equal to all other endless sequences, the grand hotel analogy is a good example as the infinite occupants will always be able to fit into the infinite rooms even if you were to add [I]infinitely[/I] many new guests thus showing that the infinity of the rooms is larger than that of the occupants.
So does infinity have infinity inside itself therefore infinity is bigger than itself?
[QUOTE=Laserbeams;48348890]No, but the real world definition and the mathematical definition have to be different, because mathematical spheres are impossible in the real world[/QUOTE]
Just as unjustified as the first assertion
[QUOTE=Laserbeams;48348890]No matter how precise your measuring of distance is, it will still be off to some degree because every measuring tool has an observational error. Whatever you do, you will still end up with an "almost perfect" sphere in the real world[/QUOTE]
But the point still exists. It's irrelevant how well we can measure.
[QUOTE=Laserbeams;48348890]You can't take a random point of the cloud and say for sure that the electron will be there at some point in time. It [i]might[/i] be, but it won't (assuming we even know where the electron actually is, which is impossible)[/QUOTE]
Right. Which is the point.The cloud itself (at least for certain configurations, i.e. certain hydrogen states) is spherically symmetric.
[QUOTE=Novangel;48354386]So does infinity have infinity inside itself therefore infinity is bigger than itself?[/QUOTE]
Not sure if this will answer your question but I'll try:
Infinity is not a number, it is a size/limit/cardinality.
Think of the integers and the even integers - the even integers are contained within the integers but both are the same size.
So a set of infinite size can contain another set of infinite size, but this doesn't mean that "infinity is bigger than itself".
[QUOTE=Flameon;48352659]Question for folks.
The claim is made that the infinity between 0 an 1 is larger, infinitely so, than the infinity of all natural numbers. My question is, why can't the work that is done with the Infinte-Hotels example and/or the sphere example of 'shifting one over' prove that they are all equally infinite? So, for example, if you do the diagonal test, why couldn't you just shift all your numberes back one to cover your bases?[/QUOTE]
Because you can only shift natural numbers over to accommodate a countable infinity. But there are an uncountable infinity of numbers left off of your list. You can shift one number over. And then two over. And then three. Notice that even if you keep doing that forever, you're only enumerating the natural numbers again.
I don't get it. The entire video seems to be saying uncountable infinity is equal to uncountable infinity, even when you take different paths to it. e.g you can measure an infinite set of points on/within a sphere multiple different ways then say they're all equal to eachother because they're all sets of infinite points on/within a sphere. I mean, it just seems we're dealing with multiple sets of infinite that seem to have been arbitrarily 'split up' into multiple infinites when that's not necessary. noduh when you 'split' up a system of multiple infinites it still equals infinity when put back together.
I might look into this a bit more, but what's bothering me is that you're adding infinities together.
I don't know. It doesn't seem that much like a paradox.
Countable Infinity: infinite one story buildings
Uncountable Infinity: infinite infinity story buildings
at least that's how I understand it.
[QUOTE=Agent 47;48355740]Countable Infinity: infinite one story buildings
Uncountable Infinity: infinite infinity story buildings
at least that's how I understand it.[/QUOTE]
That's a bit oversimplified. Think about the rational numbers. There are infinitely many of them between 0 and 1. And there are infinitely many between 1 and 2, etc. But there are still only countably many rational numbers.
so basically an infinitely complex object has enough complexity to spare to define infinite copies of itself? Is that the idea?
[QUOTE=CoixNiro;48356054]so basically an infinitely complex object has enough complexity to spare to define infinite copies of itself? Is that the idea?[/QUOTE]
The idea with countability is, an infinite amount of things is countable if you can figure out how to iterate over them in such a way, that after infinite time, you'd get all of them... For example, there is an uncountable infinity of real numbers, because no matter how you order them, you can't point at one, then at another, and get to the end without skipping any numbers.
Another way to look at it is that if you zoom in on a countable infinity of anything, you will see descrete, individual objects, whereas an uncountable infinity is continuous, i.e. there are no "indiivisible" parts to it.
At least thats my intuition. I don't know a more rigorous definition.
[QUOTE=Nikita;48370156]The idea with countability is, an infinite amount of things is countable if you can figure out how to iterate over them in such a way, that after infinite time, you'd get all of them... For example, there is an uncountable infinity of real numbers, because no matter how you order them, you can't point at one, then at another, and get to the end without skipping any numbers.
Another way to look at it is that if you zoom in on a countable infinity of anything, you will see descrete, individual objects, whereas an uncountable infinity is continuous, i.e. there are no "indiivisible" parts to it.
At least thats my intuition. I don't know a more rigorous definition.[/QUOTE]
This is correct apart from the discrete/continuous thing. Set theory has nothing to say about those properties. For that you need to step into topology. You can put the trivial topology on a countable set or the discrete topology on an uncountable set as counter examples to what you said.
[editline]4th August 2015[/editline]
In fact it gets worse: what you said is not even true for subsets of R in the standard topology! The Cantor set is uncountable, but it is not dense in any interval.
[editline]4th August 2015[/editline]
But of course the rationals are dense so maybe that's not the right context to state what you're saying.
If you want to feel better about this weirdness, you should know that the paradox relies on the [URL="https://en.wikipedia.org/wiki/Axiom_of_choice"]axiom of choice[/URL]. The axiom of choice [I]is[/I] accepted by most mathematicians, but it often gets put in a sort of "Here be dragons" category because using it leads to all kinds of crazy ass results.
I believe the part where he uses the axiom is after he does all the UDLR stuff the first time and then says to repeat that for every point you missed until you get the whole sphere.
[QUOTE=Larikang;48375059]If you want to feel better about this weirdness, you should know that the paradox relies on the [URL="https://en.wikipedia.org/wiki/Axiom_of_choice"]axiom of choice[/URL]. The axiom of choice [I]is[/I] accepted by most mathematicians, but it often gets put in a sort of "Here be dragons" category because using it leads to all kinds of crazy ass results.
I believe the part where he uses the axiom is after he does all the UDLR stuff the first time and then says to repeat that for every point you missed until you get the whole sphere.[/QUOTE]
To be fair, taking the rejection of the axiom of choice leads to even weirder results. Like you can break up a set into nonempty pieces so that there are more pieces than there were elements in the set to begin with. And you can have vector spaces with no basis.
You can count the miles you've walked, but the road will never stop giving you distance.
My mind is sludge after that video
Could you even do the hotel thing to the sphere? I'd you could percive a point as missing, couldn't you always percive a point as missing as they're moved around?
[QUOTE=JohnnyMo1;48375736]To be fair, taking the rejection of the axiom of choice leads to even weirder results. Like you can break up a set into nonempty pieces so that there are more pieces than there were elements in the set to begin with. And you can have vector spaces with no basis.[/QUOTE]
I don't like to assume AC to be false just as much as I don't like to assume it to be true. The ZF axioms seem perfectly intuitive to me, and relying on them alone you get intuitive results. When you start considering AC (whether you accept it or negate it) you get weirdness. So I think for any mathematical theorem it's always worth keeping in mind whether the proof relies on AC, or its negation, or is independent of it.
[QUOTE=Larikang;48381232]I don't like to assume AC to be false just as much as I don't like to assume it to be true. The ZF axioms seem perfectly intuitive to me, and relying on them alone you get intuitive results. When you start considering AC (whether you accept it or negate it) you get weirdness. So I think for any mathematical theorem it's always worth keeping in mind whether the proof relies on AC, or its negation, or is independent of it.[/QUOTE]
Assuming it to be anything is silly though. It's independent of ZF. It has no truth or falsehood in relation to ZF.
But it's also totally equivalent to this: An arbitrary cartesian product of non-empty sets is non-empty. That's far too "obviously true" for me to reject just because it produces results that are not very intuitive at first. Tracking whether a theorem requires AC (or dependent choice or what have you) is interesting, but I and most mathematicians have few qualms using it anymore. Imo the weirdness of Banach-Tarski is just something you have to get over eventually if you want to do interesting math.
[editline]5th August 2015[/editline]
Interesting enough, though, is that the axiom of choice is obviously false in certain categories. The categorical statement is that every epimorphism splits in [B]Set[/B], the category of sets. But if we replace [B]Set[/B] with [B]Top[/B] (topological spaces and continuous functions) for instance, not every epimorphism splits.
[editline]5th August 2015[/editline]
Personally though, I think it's obvious enough even with its normal formulation. "If you have a collection of bags with things in them, you can choose one thing from each." That's all it says.
I partially agree. I think the axiom of choice is intuitive when applied to countable sets (which is what I think most people imagine when they hear the "collection of bags" analogy). But for something like Banach-Tarsky it really jumps out as something deeply weird.
[QUOTE=Larikang;48384194]I partially agree. I think the axiom of choice is intuitive when applied to countable sets (which is what I think most people imagine when they hear the "collection of bags" analogy). But for something like Banach-Tarsky it really jumps out as something deeply weird.[/QUOTE]
I can't quite see how adding more sets to the product would make it less obvious that producting non-empty sets together gets you a non-empty product.
but of course Banach-Tarski doesn't quite require the full axiom of choice (but you can't get away with just countable choice either). You can get away with the ultrafilter lemma.
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