I went when I was really young. I think I mainly visited resorts and amusement parks.
Infinity….
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Assume the following:
You and me are in a non-physical universe (that is important later)
I have a bag, you have a bag. In my bag there is the complete set of natural numbers, all natural numbers from 1, 2, 3, etc. etc. … it’s an infinite number.
Your bag is empty.
Now it’s 1 minute to noon. I give you two numbers, you give me a number back.
Whenever the time left to noon halves, i give you two numbers, and you give me one back.
When the clock strikes noon, we have a look at our bags.
(The clock will strike noon, and we will have been incredibly fast in our number exchanges (faster than light, therefor unphysical)).So, it is noon.
How many numbers are in your bag, how many in mine?PS: I would like to see you answers, and after some ppl (hopefully cryptic, city, yb, Moses and FinsterniS … but everyone else is welcome :) ) have posted that, i would like to see their explanation and reasoning for their result.
After that i will post what i think (and what i think is right :) ) -
Ooh, that is an infinite geometric series, right?
We would be trading numbers for infinity, so we would both have an infinte amount of numbers in our bag.
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haha, infinity my old friend, we can’t live with it, and we can’t live without it. Don’t worry i will answer, but i’ll wait a little…
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:)
the moment i saw the question i smiled.has to be an infinite number (with all appropriate/impossible assumptions) in each bag.
In theory, the time would never become noon as we only move half-way closer to it at a time, but unlike matter (atoms) time is infinitely divisible. Therefore, there would be an infinite number of “halfway points” to t=noon. With this infinite number of “number exchanges” then both bags, by noon, supposing you ever reached it, would have an infinite number of numbers in them.It’s like the other questions: If you were 10 (or 1) meter from a door, and every minute you went half-way to the door, when would you (if ever) leave the room? simply never.
With matter, things are a little different. If i have avogadro’s number of carbon atoms in a lump of coal, and i keep removing half of it, eventually i would come to that last atom. . .
but this rant was really unwarrented . . . apologies everyone.
So F_alk, i’m looking forwards to your take on this. I beg your indulgence in my ignorance - i have little physics/quantum mechanics background, and no philosophy (as is obvious to FinsterniS by now, i am sure). -
@cystic:
:)
the moment i saw the question i smiled.(snip)
In theory, the time would never become noon as we only move half-way closer to it at a time, but unlike matter (atoms) time is infinitely divisible.
(snip)It’s like the other questions: If you were 10 (or 1) meter from a door, and every minute you went half-way to the door, when would you (if ever) leave the room? simply never.
That’s an “old” paradox called “Achill and the turtle”.
The point is: we know that it will be noon, the point is we just get infinitelty fast, that’s why i included the non-physical notion:).
And anyway, it’s just a picture for “after an infinite number of steps” :) -
@F_alk:
@cystic:
:)
the moment i saw the question i smiled.(snip)
In theory, the time would never become noon as we only move half-way closer to it at a time, but unlike matter (atoms) time is infinitely divisible.
(snip)It’s like the other questions: If you were 10 (or 1) meter from a door, and every minute you went half-way to the door, when would you (if ever) leave the room? simply never.
That’s an “old” paradox called “Achill and the turtle”.
The point is: we know that it will be noon, the point is we just get infinitelty fast, that’s why i included the non-physical notion:).
And anyway, it’s just a picture for “after an infinite number of steps” :)it’s been 10 years, but here goes . . .
your bag will have an infinite amount of numbers in it,
the other bag will have a number “approaching infinity” at noon.
is that “vague” enough? -
Ooh, that is an infinite geometric series, right?
It can be seen as an infinite series, but it’s not at all geometric.
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@F_alk:
Ooh, that is an infinite geometric series, right?
It can be seen as an infinite series, but it’s not at all geometric.
It can a problem in geometry; but in this question it was only an arithmetic question; related to the paradox of the arithmetic continuum.
An other strange one is when you look at two series…
The N numbers, 1, 2, 3, 4, 5, 6, 7…
The “pair”* number; 2, 4, 6, 8, 10…There is an infinity of N numbers; give any N number X, there is a X + 1 number.
There is an infinity of even number, give any “pair” number X, there is a X + 2 “pair” numbers.
So the two series constitutes an infinity…
It seem to be evident there is more Natural numbers, because all “pair” Number are natural, but all natural numbers are not “pair”, also if you take more than 3 following number, there will always be more Natural numbers… If i ask the question, how many Natural numbers there is ? An infinity… and how many “pair” number ? and infinity.
It does’nt make sence because it implie there is a “larger” infinity, while infinity is the maximum.
- Sorry about the French word, but i really don’t know how “pair” can be translate, par ? even ?
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No no no! I don’t want to know!
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@F_alk:
Assume the following:
You and me are in a non-physical universe (that is important later)
I have a bag, you have a bag. In my bag there is the complete set of natural numbers, all natural numbers from 1, 2, 3, etc. etc. … it’s an infinite number.
Your bag is empty.
Now it’s 1 minute to noon. I give you two numbers, you give me a number back.
Whenever the time left to noon halves, i give you two numbers, and you give me one back.
When the clock strikes noon, we have a look at our bags.
(The clock will strike noon, and we will have been incredibly fast in our number exchanges (faster than light, therefor unphysical)).So, it is noon.
How many numbers are in your bag, how many in mine?This is a weird one. I do not live in non-physical universe so it is hard for me to “see” actual infinite, only potentially infinite. (Can Space be Euclidean?). FinsterniS bring up good point. I might imply infinite in both bags, but there are varying “levels” of infinity? Take the set of all integers - it’s infinite. So is the set of real numbers. But there are more real numbers than integers - the “infinite number of reals” is greater than
the “infinite number of integers.” Sister knows more of this than I do.Of course if you want an answer, I like to call upon a little friend I like to call, “Mr. Zero.” Case closed! :roll:
@F_alk:
what i think is right
Well, I wouldn’t want to dispute that – you know what happened last time we pissed off the Germans. :wink:
“I can’t help it, the idea of the infinite torments me.”
Alfred de Musset (1810-1857) -
@TG:
@F_alk:
what i think is right
Well, I wouldn’t want to dispute that – you know what happened last time we pissed off the Germans. :wink:
Hm, Germany won the quarterfinal ?
:D -
So, let’s have a look again, and let’s take tis look from a “game theory” point of view in the sense that there exist different strategies:
(but first some words about infinity that already have been mentioned here:
there are as many natural numbers, as there are even or odd numbers (even though even plus odd makes up the naturals). There are as many integer numbers as there are natural numbers, even though the naturals are “only” the positive branch of the integers. There are as many fractions (rational numbers) as there are naturals. This all comes from Cantor’s diagonal argument. These sets are called “countable infinite” sets, because you can map them one-on-one onto the natural numbers which you could count (if you have infinite time that is). The “real numbers”, which are the rationals plus irrationals (like Pi, e, 2^(1/2) which cannot be expressed by fractions), cannot be mapped in such a way. This is an “uncountable infinite” set, and there are more reell numbers than naturals.)So, let’s look at our two players (A and B) and different strategies (for simplicity i assume both A and B be male)
(1) A gives B ascending numbers (starting with one, not giving back a number that B already had), B gives A his lowest number:
So, it will look like that:
A -> B: 1,2
B -> A: 1
A -> B: 3,4
B -> A: 2
etc.
B will give back all numbers back to A, having an empty bag at the end (2*“amount of naturals”=“amount of naturals”!), A will have all numbers.(2)A gives B his lowest numbers (inlcuding those given beforeand handed back), B gives A his highest number.
A -> B: 1,2
B -> A: 2
A -> B: 2,3
B -> A: 3
etc.
So, A will have given all numbers to B, and end up with an empty bag, while B has all natural numbers.(3)A gives B his lowest numbers (inlcuding those given beforeand handed back), B gives A his lowest number.
A -> B: 1,2
B -> A: 1
A -> B: 1,3
B -> A: 1
etc.
So, A will have given all numbers to B, and end up with an empty bag except for the “1”, while B has all natural numbers, except for the “1”.(4) A gives even numbers, B does anything:
A will have all odd numbers at least, regardless of B’s strategy. So, A has an infinite amount of numbers. The exact amount of B depends on what B does.So, the answer:
you can’t tell, unless you know how they choose to give and give back numbers. It can be anything from nothing over any finite number (by changing strategies halfway) to an infinite amount. -
“So, the answer:
you can’t tell, unless you know how they choose to give and give back numbers. It can be anything from nothing over any finite number (by changing strategies halfway) to an infinite amount.”Greaaaattt… I hate questions without any right “answers,” but I wasn’t that off with “Mr. Zero.” :wink:
“Hm, Germany won the quarterfinal ?”
Well, football is European sport (well South America, too!) so we expected to lose. But next time, we will be out for blood! :wink:
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@F_alk:
So, let’s have a look again, and let’s take tis look from a “game theory” point of view in the sense that there exist different strategies:
(but first some words about infinity that already have been mentioned here:
there are as many natural numbers, as there are even or odd numbers (even though even plus odd makes up the naturals). There are as many integer numbers as there are natural numbers, even though the naturals are “only” the positive branch of the integers. There are as many fractions (rational numbers) as there are naturals. This all comes from Cantor’s diagonal argument. These sets are called “countable infinite” sets, because you can map them one-on-one onto the natural numbers which you could count (if you have infinite time that is). The “real numbers”, which are the rationals plus irrationals (like Pi, e, 2^(1/2) which cannot be expressed by fractions), cannot be mapped in such a way. This is an “uncountable infinite” set, and there are more reell numbers than naturals.)So, let’s look at our two players (A and B) and different strategies (for simplicity i assume both A and B be male)
(1) A gives B ascending numbers (starting with one, not giving back a number that B already had), B gives A his lowest number:
So, it will look like that:
A -> B: 1,2
B -> A: 1
A -> B: 3,4
B -> A: 2
etc.
B will give back all numbers back to A, having an empty bag at the end (2*“amount of naturals”=“amount of naturals”!), A will have all numbers.(2)A gives B his lowest numbers (inlcuding those given beforeand handed back), B gives A his highest number.
A -> B: 1,2
B -> A: 2
A -> B: 2,3
B -> A: 3
etc.
So, A will have given all numbers to B, and end up with an empty bag, while B has all natural numbers.(3)A gives B his lowest numbers (inlcuding those given beforeand handed back), B gives A his lowest number.
A -> B: 1,2
B -> A: 1
A -> B: 1,3
B -> A: 1
etc.
So, A will have given all numbers to B, and end up with an empty bag except for the “1”, while B has all natural numbers, except for the “1”.(4) A gives even numbers, B does anything:
A will have all odd numbers at least, regardless of B’s strategy. So, A has an infinite amount of numbers. The exact amount of B depends on what B does.So, the answer:
you can’t tell, unless you know how they choose to give and give back numbers. It can be anything from nothing over any finite number (by changing strategies halfway) to an infinite amount.see, the problem with this question is that it is in essence non-sensical. If you have an infinite amount of activities, one taking place with every division before noon, then in theory you could never reach noon (i know that you said that we can in this example, but it still makes little sense that we could reach noon assuming an infinite number of time-divisions).
Anyway, this got me to thinking - assuming there is an infinite number of divisions of time before noon, how do we ever reach it? Or is noon still a theoretical idea, and i choose to eat my sandwich at some arbitrary (increasingly longer) fraction of the way towards noon? Or is it that our non-spatial, yet linear travel through time have an increasing velocity that approaches infinity relative to noon as we approach noon?
These are some of the things i think about when i’m jogging or biking. -
Greaaaattt… I hate questions without any right “answers,” but I wasn’t that off with “Mr. Zero.”
Did somebody say my name? 8)
Of course if you want an answer, I like to call upon a little friend I like to call, “Mr. Zero.” Case closed!
I’m glad someone believes in my abilities. :oops:
Sorry, couldn’t resist the play on my name.
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Ha, well you won’t know how many many times I “play” on my name :wink:
Yep, I parted the Red Sea 8)
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I agree with your answer, but you will have to admit it is a little strange… We are not suppose, in arithmetic, to consider those thing like the order in wich we distribute numbers… x(x - 1) = x^2 - 1x and x + 2x = 3x, whatever the manner you distribute the x + 2x, it would give 3x… Also the “Infinity is not equal to Infinity” thing is a problem that can only be very, very partialy resolve with Cantor theory of different type of infinity… the resolution also make us ask some strage question, like “is there is other infinity between Aleph 0 and Aleph 1 ?” Anyway, i don’t know if infinity really does exist, maybe, like Poincaré, Gauss & Kronecker thinking, infinity does not exist, it is just potential… But it seem it is not the case…
Damn Cantor, Gauss said infinity was just problem ! :)
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“I agree with your answer, but you will have to admit it is a little strange… We are not suppose, in arithmetic, to consider those thing like the order in wich we distribute numbers… x(x - 1) = x^2 - 1x and x + 2x = 3x, whatever the manner you distribute the x + 2x, it would give 3x… Also the “Infinity is not equal to Infinity” thing is a problem that can only be very, very partialy resolve with Cantor theory of different type of infinity… the resolution also make us ask some strage question, like “is there is other infinity between Aleph 0 and Aleph 1 ?” Anyway, i don’t know if infinity really does exist, maybe, like Poincaré, Gauss & Kronecker thinking, infinity does not exist, it is just potential… But it seem it is not the case…
Gauss said infinity was just problem”Well, I am sure that Cantor believed in actual infinite numbers transfinite numbers. However, I have not seen an example of actual infinity thus far, except if you believe space is continuous. For Aleph 0 and Aleph 1, Cantor believed in the Continuum hypothesis or Generalised Continuum hypothesis, which would seem to imply no. But Cantor’s hypothesis was independent of the other set-theory axioms, so who knows? :roll: I might use the Axiom of Choice, though this might lead up to some strange paradoxes. Perhaps, these cardinalities just ‘float around in an annoying way.’ Maybe Monsieur FinsterniS is right on Gauss, ‘just keep infinity out of mathamatics!’ :D
PS: I find it amusing, yet appalling, to see learned mathematicians fight among each other, especially the terrible war waged between Georg Cantor and Leopold Kronecker.
‘I see it but I don’t believe it.’ ~ Georg Cantor
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@TM:
Well, I am sure that Cantor believed in actual infinite numbers transfinite numbers. However, I have not seen an example of actual infinity thus far, except if you believe space is continuous.
In fact there is a lot of exemple of infinity. You won’t see a lot of Aleph 0, but a lots of Aleph 1 !
For Aleph 0 and Aleph 1, Cantor believed in the Continuum hypothesis or Generalised Continuum hypothesis, which would seem to imply no.
Continuum Hypothesis is true when you look only at it as an axiom to base other sets, but false when you look at math as only abstaction… (it is often the inverse). Anyway i don’t trust infinity, as one of my teacher said; we would need a new logic to understand infinity, often you come up with the conclusion it is False, then you look at another way and it is True… Just like the continuum hypothesis…
Maybe Monsieur FinsterniS is right on Gauss, ‘just keep infinity out of mathamatics!’ :D
Well no, i don’t agree with Gauss this time, i think it was just too frustrating for him… Like it is now for us anyway, if nobody around me were talking about Infinity i would never have think of it; too strange. But Gauss was right that infinity is just a lots of problem.
PS: I find it amusing, yet appalling, to see learned mathematicians fight among each other, especially the terrible war waged between Georg Cantor and Leopold Kronecker.
Don’t forget Cantor’s allies; Hilbert, Dedekind, Godel…
And also don’t forget the “potential infinity” fidels; Poincaré, Weyl, Brouwer
… -
“Don’t forget Cantor’s allies; Hilbert, Dedekind, Godel…”
Lets not forget Mittag-Leffler and Weierstrass. But I favor Dedekind, really stuck by him to the end when Cantor was at his worst.
“In fact there is a lot of exemple of infinity. You won’t see a lot of Aleph 0, but a lots of Aleph 1 !”
She probably means in real life. I’m am still unsure if there can be an actual infinity in real life? Potential more like it.
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