to infinity, and then some...

[RussDill]

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.

What's even weirder is that if that were to be true, then your consciousness could be the same in the various different worlds. It gives a completely different meaning to there being a 1% chance of something being true. Its both true and not true at the same time from your own point of view. Until an event happens that actually differentiates the two for you, you exist in both.

 

[ddt]

Thanks for taking the time to respond, but I'm afraid it's lost on me (and the amount of time and effort it would take me to research this in order to understand what you're getting is too much just for the sake of discussion--maybe I'll look it up some other time).

It's exactly what other posters have been trying to tell you too.

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?

What kind of number is that? Most real numbers have a never-ending decimal expansion, so you never eliminate all decimals. You might also want to reread my post #27, for it too contains the standard proof that the set of real numbers is really bigger than the set of integers.

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

Just look at the following mapping from even numbers to non-negative integers:

0 --> 0
2 --> 1
4 --> 2
6 --> 3
etc.

Every even number is mapped to a non-negative integer. Every non-negative integer is the image of some even number. So this is a 1-1 correspondence. Right? So there are as many even numbers as there are non-negative integers. Right?

 

[Dorian Gray]

Trite? Check!
Ignorant? Check!
Right-wing? Check!
Racist? Check!

Good job. And a minimum of words too, sort of a haiku.

It's not necessarily any of those things. However:

Kneejerk? Check!

 

[Brian-M]

It's exactly what other posters have been trying to tell you too.

This seems a good time to quote my earlier disclaimer:

If I am wrong in this, I will persist in being wrong until someone convinces me of why I'm wrong. You don't learn very much if you just accept someone else's assertion that you're wrong without understanding why.

Although, having read through that link on infinity, I can see the point being made. But I do think the claim in this thread that the two sets are the same size is slightly misleading... that they are of equivalent size would be more accurate.

 

[jsfisher]

I do think the claim in this thread that the two sets are the same size is slightly misleading... that they are of equivalent size would be more accurate.

I must admit I am at a loss to understand what distinction you are drawing between same and equivalent.

 

[Fredrik]

Regardless of whether you prefer "they have the same size", "they have equivalent sizes", "they have the same number of members" or something else entirely (the appropriate thing to say is that "they have the same cardinality^WP"), the concept needs to be defined, and the definition is...

Two sets X and Y are said to have the same cardinality if there exists a bijective function f:X→Y.

When we say that they have the same number of members, we're being a bit sloppy, but it's still much better than saying that they have the same size. To a math nerd, the "size" (actually "Lebesgue measure") of the set of rational numbers is 0, not infinite.

 

[Myriad]

I must admit I am at a loss to understand what distinction you are drawing between same and equivalent.

Obviously same and equivalent are exactly the same, but are far from equivalent!

Respectfully,
Myriad

 

[jsfisher]

Obviously same and equivalent are exactly the same, but are far from equivalent!

Respectfully,
Myriad

Doh!! It's so obvious now.

 

[69dodge]

But I do think the claim in this thread that the two sets are the same size is slightly misleading... that they are of equivalent size would be more accurate.

Not sure what the difference is. Can you clarify?

It should go without saying, of course, that claiming that two sets have the same size is different from claiming that the two sets are the same. Certainly, the integers and the even integers aren't the same set.

When deciding whether two sets are the same, it matters which elements are in each set. When deciding whether two sets are the same size, the identity of the elements doesn't matter. Imagine replacing them all with question marks.

The set {1, 2, 3} and the set {7, 8, 9} are different, but if we replace each element with a question mark, the two sets look the same: {?, ?, ?}. So they have the same size.

The (positive) integers are {0, 1, 2, 3, ...}. The (positive) even integers are {0, 2, 4, 6, ...}. If we replace each element with a question mark, the two sets looks the same: {?, ?, ?, ?, ...}. So they have the same size. (It's not important that I typed exactly four question marks before the ellipsis. Any number would do, provided it's understood that the ellipsis represents an infinite sequence of question marks.)

 

[69dodge]

Obviously same and equivalent are exactly the same, but are far from equivalent!

Shouldn't it be the other way around? The way he used them, same is more discriminating than equivalent. That is, two things that are not the same might yet be equivalent, but not vice versa.

(Sorry. I really should not take jokes so seriously.)

 

[sphenisc]

...You want proof? Okay. I withdraw balls 0 and 1, and put ball 0 back. I withdraw balls 1 and 2....

I've sat down with my infinite ball bag and having withdrawn ball 1, find I can't withdraw it again. What am I doing wrong?

 

[nathan]

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.

IIRC in another thread you said there were (pairs of) infinite sets whose difference was finite. I thought I understood what you meant, but now find I am confused. Could you clarify here (I think you're safe).

 

[Vorpal]

There are no pairs of infinite sets such that the size (cardinality) of one is a finite amount more than the other. In general, if the axiom of choice is assumed, then a + b = a whenever a is an infinite cardinal number and b≤a. So he was probably talking about something else, e.g., a set difference (A-B = {x in A: x not in B}) having finite cardinality or measure or some other notion of size.

 

[jsfisher]

IIRC in another thread you said there were (pairs of) infinite sets whose difference was finite. I thought I understood what you meant, but now find I am confused. Could you clarif

I am not sure what statement you are referring to, but it is certainly possible to have the set difference between two infinite sets be a finite set.

Let A = {1, 2, 3, 4, ...} and B = {0, 1, 2, 3, ...}, for example.

Both sets are clearly infinite, and both sets have the same cardinality. However, the set difference, B - A, is the finite set {0}.

 

[Brian-M]

Not sure what the difference is. Can you clarify?

I'm not sure, but I'll try.

Compare "even numbers" with "even numbers and the numbers 3 and 5". Logically, by including two more numbers, the size should be two more, so they're not the same size. But because the cardinality remains unchanged, they're mathematically equivalent, and can be used interchangeably.

(Probably not the clearest explanation, but the best I can do at the time of night it is over here.)

 

[Vorpal]

Compare "even numbers" with "even numbers and the numbers 3 and 5". Logically, by including two more numbers, the size should be two more, so they're not the same size.

That only follows if two more gives a size different from the initial one. But there's nothing illogical about the statement x + 2 = x. It's only problematic if one makes the further claim that x is (for example) a real number, but we're not doing that.

You're looking for some notion of "size" that conforms to those intuitions, but there's no way of doing so in general, for arbitrary sets.

You could define a notion of "size" tailor-made for the natural numbers that may distinguish at least some of those cases. For example, for a set A of naturals, let A be the number of elements of A that are less than n. Then the asymptotic density is the limit of A/n, and is a number between 0 and 1 that successfully distinguishes the "sizes" of the odds or evens (both 1/2) and the naturals (1). But one pays for this by having any finite set have "size" 0.

 

[nathan]

I am not sure what statement you are referring to, but it is certainly possible to have the set difference between two infinite sets be a finite set.

Let A = {1, 2, 3, 4, ...} and B = {0, 1, 2, 3, ...}, for example.

Both sets are clearly infinite, and both sets have the same cardinality. However, the set difference, B - A, is the finite set {0}.

Ok, thanks. That's probably what you meant -- I couldn't find the posting I'm half remembering.

 

[readme.txt]

Thanks for taking the time to respond, but I'm afraid it's lost on me (and the amount of time and effort it would take me to research this in order to understand what you're getting is too much just for the sake of discussion--maybe I'll look it up some other time).

Bijection is a simple way to verifiy if 2 sets have the same amount of elements or not, even with infinite sets.

 

[W.D.Clinger]

I've sat down with my infinite ball bag and having withdrawn ball 1, find I can't withdraw it again. What am I doing wrong?

Sorry, my mistake again. Third try:

I withdraw balls 0 and 1, putting ball 0 back. With ball 1 beside me, I withdraw balls 2 and 3, putting ball 1 back. With balls 2 and 3 beside me, I withdraw balls 4 and 5, putting ball 2 back. I now have balls 3, 4, and 5 beside me.

Starting with n=6 and balls 3 through n-1 beside me, I withdraw balls n and n+1, putting ball n+1 back. That increases the number of balls sitting beside me by 1. Then I increment n and repeat.

At "the end" of this process, there are three balls in the bag (0, 1, and 2) and infinitely many balls beside me (all the others).

A countable infinity of steps in which two balls are removed and one put back at each step left us with 3 balls in the bag. Hence a countable infinity (the original contents of the bag), minus two countable infinities (the balls that were removed), plus another countable infinity (the balls that were put back), added up to 3.

The point of this example (and of my previous failed attempts) is that subtraction of infinite cardinals is not always well-defined. When subtracting a countable infinity from a countable infinity, you can get any cardinal you want, from 0 through countable infinity, by changing details of the subtraction process.

If you accept the axiom of choice, then addition of cardinals is well-defined (and, for infinite cardinals, is the same as taking the maximum). Subtraction of cardinals is not well-defined in general, although it is well-defined for finite cardinals and when subtracting a smaller cardinal from a larger one.

 

[dlorde]

I have seen this many times, but why should it be so? The space of all real numbers is as infinite as infinity gets, as far as I know. Yet there is one and only one of each number. Why should the infinity of an infinite universe be any different?

Because physically possible arrangements of particles are not unique, unlike numbers. From there, I suppose it's just a matter of what is possible. It may be that the existence of our planet in it's current state depends on the way our solar system developed, and so it would need a very similar history in a very similar solar system. The idea is that if it has happened once, it has a non-zero probability, so in an infinite universe you would expect it to happen infinitely often. I suppose this assumes that the local conditions in our neck of the galaxy are infinitely common in the universe (does this imply homogeneity?).