to infinity, and then some...

[Cavemonster]

<pedant>A ball is a mathematical object defined as the set of all points within some fixed distance from a given center (≤ if closed, < if open). A sphere is the set of points that are exactly some fixed distance from a given center.</pedant>


But you don't get integers in that case, just infinite strings of digits. And there are indeed as many infinite strings of digits as real numbers.

By george, I'm showing my ignorance all over the place.

 

[Brian-M]

No, what they were saying was that, in an infinite universe - i.e. not curved, just continuing out infinitely far - if you go far enough, the chances of coming across a planet identical to this one are 1 (in fact, there would be an infinite number of them), and, given the number of particles that make up the Earth, you can calculate how far the distance would be to be certain that you encountered a duplicate. I presume they meant within a sphere of that radius, but they weren't very specific, and I'm not convinced by the calculation, although the concept seems sound enough (given that infinities are involved).

So they're only talking about a hypothetical infinite universe, which wouldn't apply if the universe is really finite but unbounded?

I liked the mathematician who said that you can only count so far (though that's further than any number we've got to yet - at least in terms of working out it's exact digit construct) and then the next number is zero ie you go back to the beginning.

Are you sure he's talking about math and not programming? Because if he was talking about programming, he'd be right.

 

[Vorpal]

Thanks to Xeno (or Zeno) Infinity was a forbidden topic in Math for many, many years. Newton and Leibnitz invented Calculus, but this dealt with infinity, so many people thought of Calculus as some sort of "trick" or "gimmick". Yes, it produced results, but after the results were obtained, other methods were used to "prove" the results that Calculus gave.

I don't know about that; some the methods of calculus, including a kind of infinite-limiting method, were used as far back as Archimedes. The innovations of calculus generalized those methods.

It was not until Cantor, and an understanding of infinity, that Calculus became trusted. And Cantor, the first human to understand this concept, spent much of his life in a mental institution.

People like Cauchy and Weierstrass had a lot more to do with that. The epsilon-delta definition of limit, for example, doesn't care about the which orders of infinity we're dealing with. The only thing essential to calculus from that point is the Archimedean and completeness properties of the real numbers (the latter of which can be formally proven, e.g., from defining real numbers as Cauchy sequences of rationals, or by other constructions or just axiomatizations).

 

[JoeTheJuggler]

I liked the mathematician who said that you can only count so far (though that's further than any number we've got to yet - at least in terms of working out it's exact digit construct) and then the next number is zero ie you go back to the beginning.

He seemed to have pulled that theory out of his ass because he didn't like infinity.*


*No doubt I'm shortchanging him and he's done years of research and come up with some proof....

What's strange is I got past that idea when I was about 5 years old. I remember asking my mom what the highest number was. (What I thought I meant was what the highest name for a number is.) She told me there wasn't one. I challenged her on that because I knew I'd learned some numbers, and there other numbers I hadn't yet learned, but they were (in my mind) names for things, and I knew those names were manmade, so I thought there was a limit. She said, you just get to the highest number you can name, and then say "and one more". And you can keep doing that!

5 year old epiphany!

And yet I run into arguments (especially religious) put forth by educated and full-grown adults who start with the premise, we can't conceive of "infinity".

Of course we can!

 

[Dragoonster]

I think I got shut down on this here some time ago, but...

If it were true from all this infinite stuff that there exist infinite identical Earths, wouldn't this mean there are exact worlds as described in all fictional works (including Shakespeare's collected volumes)? At least, the works that are physically possible.

This also means that most Conspiracy Theories (that are physically possible) are actually correct, just not for this Earth. There'd be an Earth where the only difference was Bigfoot really did exist, where 9/11 was an inside job, where there are Nazis underneath Antarctica.

Hm, not sure whether to pack my bags for Middle Earth or Coruscant.

 

[Badly Shaved Monkey]

I withdraw balls 0 and 1, and put ball 1 back. I withdraw balls 1 and 2, and put ball 2 back. I withdraw balls 2 and 3, and put ball 2 back.

Thanks to each of you who has replied to my question.

I get the flavour of your argument, but are you sure you've got the detail right? You said you end up with balls 0, 1 and 2 in the bag at the end, but here you just had ball 0 leaving the bag and not being put back.

 

[Badly Shaved Monkey]

I think I got shut down on this here some time ago, but...

If it were true from all this infinite stuff that there exist infinite identical Earths, wouldn't this mean there are exact worlds as described in all fictional works (including Shakespeare's collected volumes)? At least, the works that are physically possible.

This also means that most Conspiracy Theories (that are physically possible) are actually correct, just not for this Earth. There'd be an Earth where the only difference was Bigfoot really did exist, where 9/11 was an inside job, where there are Nazis underneath Antarctica.

Hm, not sure whether to pack my bags for Middle Earth or Coruscant.

I find fascinating and convincing the argument that in an infinite universes there will be infinite copies of identical Earths. There will also be infinite copies of an Earth where just one atom is a nanometer further to the left.

What I wonder about is the Earth where there is a T. rex living in Hyde Park for no readily identifiable reason.

What I mean is that if you treat the state of the local universe as a finite set of balls picked from an infinite bag then all arrangements are possible, but with many of these the causal connection with their own past would require such an accumulation of near-zero probability events to have occurred that its inhabitants would not be able to generate a coherent explanation of their past even though their present is not explicitly disallowed- their world would look like causality doesn't work properly. I suspect the phrase 'Anthropic Principle' may feature in any response to this, but I'm not sure that would satisfy the inhabitants of the infinite Earths where everything makes sense except for the trillion tonne diamond moon that hovers for no clear reason, other than the random combination of motions of all its constituent atoms, six feet off the ground in the centre of their capital city.

 

[Badly Shaved Monkey]

(I suppose the residents of Diamond City might come to accept as fact Douglas Adams' idea that beauty trumps physics, but let's not go down that route)

 

[realpaladin]

I can't believe you invited doron to this thread.

Come on, you know he won't come. Too much math going on here. Too little word-salad

 

[realpaladin]

Hm, not sure whether to pack my bags for Middle Earth or Coruscant.

Stay away from MY world. The one with the infinite amount of bikini-clad amazonian looking damsels that... oh... wait... I am typing this aloud...

 

[Michael C]

You can make the same kind of 1:1 correspondence between integers and any set of numbers with n decimal places by multiplying the set by 10ⁿ. As real numbers are numbers with infinite decimal places, couldn't you make a 1:1 correspondence by multiplying the set of real numbers by 10^infinite?

Yes, but 10^infinite is not part of the set of countable integers. If you raise any integer (other than 0 or 1) to an infinite power, it can be proved that the result will be cardinally larger than the original infinity.

I found this page which gives a good introduction to the idea of different "sizes" of infinity.

 

[Fredrik]

You can make the same kind of 1:1 correspondence between integers and any set of numbers with n decimal places by multiplying the set by 10ⁿ. As real numbers are numbers with infinite decimal places, couldn't you make a 1:1 correspondence by multiplying the set of real numbers by 10^infinite?

I don't know what that would mean, but I posted the standard proof that there are more real numbers than integers in #47. (Ignore the first paragraph about 2^(10^118) which I misread as (2^10)^118).

Edit: You can also find that proof on the page that Michael C linked to in #91, with more details explained.

 

[The Man]

In this thread there's also a lot of fun about infinity and cardinality.

Am going to invite a few of them over

Thanks realpaladin, I don’t have anything to add to the discussion quite yet, but at least I’ve got this thread on my “subscribed to” list now.

 

[69dodge]

You can make the same kind of 1:1 correspondence between integers and any set of numbers with n decimal places by multiplying the set by 10ⁿ.

Yes.

As real numbers are numbers with infinite decimal places, couldn't you make a 1:1 correspondence by multiplying the set of real numbers by 10^infinite?

I'm not sure what it might mean to multiply a real number by 10^infinity.

For example, 1/7 is 0.142857 142857 142857 ... (the same six digits repeat endessly). Multiplying it by 10^infinity is presumably supposed to give a number with no digits to the right of the decimal point. Yes? Ok. What's the last digit of that number, i.e., the one just to the left of the decimal point?

The problem is that the decimal expansion of 1/7 (and most other real numbers) has no last digit, so there's no way to find the last digit, regardless of how many places you shift the number left by.

 

[W.D.Clinger]

I get the flavour of your argument, but are you sure you've got the detail right? You said you end up with balls 0, 1 and 2 in the bag at the end, but here you just had ball 0 leaving the bag and not being put back.

Good catch, thanks!


Here's what I meant to say (I think):

Ah, but how could we end up with exactly 3 balls in the bag? I withdraw balls 0 and 1, and put ball 0 back. I withdraw balls 1 and 2, and put ball 1 back. I withdraw balls 2 and 3, and put ball 2 back. From then on I proceed as follows. I withdraw balls 3 and 4, and put ball 3 back. I withdraw balls 4 and 5, and put ball 4 back. I withdraw balls 5 and 6, and put ball 5 back. When I'm "done", the bag will contain balls 0, 1, and 2. Balls 3, 4, 5, 6, 7, 8, et cetera will have been removed without being put back.

 

[Brian-M]

The problem is that the decimal expansion of 1/7 (and most other real numbers) has no last digit, so there's no way to find the last digit, regardless of how many places you shift the number left by.

Good point, you're right.
(Although, that still doesn't convince me that there are just as many even numbers as integers.)

I found this page which gives a good introduction to the idea of different "sizes" of infinity.

Thanks for that. I've bookmarked the page, and will look at it later.

 

[Modified]

Good point, you're right.
(Although, that still doesn't convince me that there are just as many even numbers as integers.)

You need to think of each element as a "thing" and not a number. If you can label the things in each set using unique labels so that both sets contain the same labels, then the sets must be the same size. The fact that their "natural" labels do not satisfy this is irrelevant.

 

[jsfisher]

Good point, you're right.
(Although, that still doesn't convince me that there are just as many even numbers as integers.)

For finite sets, it is very easy to tell if they are the same size, even without counting the members. You can pair up members from each, and if no members are left unpaired, the sets are the same size. If one set has members left over, its the larger set. It doesn't how you do the pairing with finite sets.

With infinite sets, different pairings can give different results. That's just the nature of infinite sets. However, if there exists even just one pairing that matches each member of one set with a unique member of the other set, then the two are equal in size.

A simple pairing of integers with the even integers is any integer N can be paired with an even integer 2N (which also means any even integer M is paired with integer M/2). It doesn't matter that other pairings end up with unpaired integers (or unpaired even integers), one pairing that works is sufficient.

Does that help?

 

[Ferguson]

Good point, you're right.
(Although, that still doesn't convince me that there are just as many even numbers as integers.)

Try thinking about it like any ol' function and set of outputs, for example in:
y = 5x
or
y = x/2
Are there more y's or x's? You can put anything in x and get y, or vice-versa. You can scroll your graph forever without running out of x or y values. If the set of even integers is y and all integers is x, then these sets are just the output of:
y = 2x

For every value of y, there is a value of x, the sets match 1:1.

 

[69dodge]

Although, that still doesn't convince me that there are just as many even numbers as integers.

Ultimately, it's just a matter of definition, of course.

You can dislike the standard definition of "same number", which implies that there are the same number of integers as even integers, because that seems strange to you, but then think about what definition of "same number" you might like better.

The standard definition, involving pairing up the elements of the two sets being compared, is very general, in that the sets need not be related in any way. It is not necessary that one be a subset of the other, nor need they have any elements in common at all. The definition is still able to tell us whether they're the same size or not.

You think that a proper superset^* of a set ought to be considered bigger than it, but what about unrelated sets? How do you propose to compare their sizes?

For example, consider these two sets of finite-length strings of letters: One set contains all strings that consist of only the letters a or b (e.g., a, aa, b, bb, abaaba, etc.). The other set contains all strings that consist of only the letters c, d, or e (e.g., c, ccddc, edd, ecdce, etc.). Which set is bigger? Are they the same size? How can we decide? Or are they simply incomparable, in your opinion?


^*definition: A proper superset of a set S is a set that contains everything that S does, plus some additional stuff besides, the way that the integers contain all the even integers, plus the odd ones too.