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riddles >> hard >> Infinity
(Message started by: BenVitale on Feb 5th, 2008, 3:00pm)

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Title: Infinity
Post by BenVitale on Feb 5th, 2008, 3:00pm
I came across an article in the Scientific American,

I've been reading Penrose, and i have few questions: The diagonalization method showed that the continuum 2^R was could have different senses of the "sizes" of infinity. Thus we still cannot prove Cantor's intuition that 2^aleph 0 = aleph 1.

But if we have successively larger indexes of the transfinite numbers, aleph 0, 1, 2... from the viewpoint of what is the continuum on a number line (as perhaps the idea of compactness) how is it said that the larger indices's represent a smaller infinity?

Also, the idea of dimension Cantor did not find a way to define is such that the dimensions can be reduced to some point as interlaced numbers: .xyxyxy... for a point rather than two dimensions. But it is also clear that we cannot do this for a sequence of 0's and 1s with out encountering numbers that eventually end in all zeros and ones (do these algorithms in a sense halt?). Yet, in some region we can so represent them as a single number if we interlace say 3 between them .31303031... Now, as all numbers are on the Riemann sphere how might this relate to some unique representation of some such number (say the 1 as prime and the 0 as not)? Clearly such spheres although perhaps not unique really have to project some idea of a plane with quite distinct ordering from diagonalization ideas.

Title: Re: Infinity
Post by Icarus on Feb 5th, 2008, 6:47pm
??? Alright, you seem to be thoroughly confused. Diagonalization shows only that for any cardinal n (infinite or not), 2n http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/ne.gifn. All it tells us about Cantor's guess is that 2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varaleph.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sub0.gif  http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/ge.gif http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varaleph.gif1.

That the two are actually equal is called "The Continuum Hypothesis", and the reason we can't prove it is that it turns out to be independent of the normal axioms of set theory. I.e., if you were to assume it is not true, you wouldn't ever encounter a contradiction. However, showing this is much more difficult than just using diagonalization (though I am sure some variation was used).

I have no idea what you mean about larger indices representing a smaller infinity. The definition of the http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varaleph.gifnumbers is such that the size of the infinity represented goes up as the index goes up. Where did you see anything different?

And I have no clue what you are asking about in the last paragraph. Interweaving digits to show that http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif2 ~ http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif is something we talk about only to give the general idea. The actual proof must be done with more care exactly for the reason you describe. However, the proof does work. Dimensionality is something that is built into the other structures we put on the set (in particular, it first appears in the topology, and in the algebraic operations). It is not found in unstructured sets.

Every Complex number is found on the Riemann Sphere. There are other types of numbers that do not appear there. In particular, since the Riemann sphere is provided with a single infinite point, you cannot have more than http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/varaleph.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sub0.gif on the Riemann sphere. And even that is not a good identification. It is far better to just consider the http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif of the Riemann Sphere a different beast altogether than the infinite cardinals and ordinals of Cantor, even if it gets called by the same name.

Title: Re: Infinity
Post by BenVitale on Feb 11th, 2008, 4:16pm
Yes, I got confused. Thanks for clarifying it.

There 2 infinities on the extended real line and only 1 on the extended complex plane. I imagine that it has been done before - I mean  to have a system with infinitely many infinities added to the complex plane. Perhaps it was found that it wasn't as useful as it is for the reals.

Title: Re: Infinity
Post by Hippo on Feb 12th, 2008, 12:57am
Don't be confused by the difference in dimension and the "order" of infinity.

Size of the set is compared by injections. ... The map f:A->B where x<>y => f(x)<>f(y). If such a map exists |A|<=|B|. Often |A|<=|B| and |B|<=|A| ... it can be proved that in that case there exists bijection f:A->B and g:B->A such that for any u from A g(f(u))=u, and from any v from B f(g(v))=v. In that case |A|=|B|.

There are only trivial problems to encode finite number of infinite sequences in one infinite sequence so the 'final' dimension "does not count".

Title: Re: Infinity
Post by Icarus on Feb 12th, 2008, 4:09pm
I tried to post this yesterday, but my ISP crashed while I was typing it in:

First of all, infinities added to http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif or http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbc.gif really have nothing to do with cardinal or ordinal infinities. The Real and Complex infinities are (mostly) topological creatures, resulting from a process called compactification. Compactification can be thought of as either filling in the holes in a space, or else providing them with sharp well defined edges. There are many ways a space can be compactified.

Topologically, the infinite regions of both http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif and http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbc.gif are really nothing more than holes. To see this for http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif, consider it to sit as the x-axis in the plane. Then draw the unit circle about the origin, but leave out the point (0,1). We can map http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif onto this punctured circle by using projection: for a real number r, draw the line through (r, 0) and (0,1). The point where the line intersects the circle is the map of r. In particular, this maps r --> (2r, 1 - r2) / (1 + r2). We see that the near-infinite regions of  http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif map around the point (0,1), which itself is missing: the hole at http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif. This hole need not consist of a "single missing point". If I had used (0, 0.9) as my projection point, I would have had a larger hole at the top of my circle. Topologically, any hole is expandable. The same thing can be done with http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbc.gif, but now we have a hole in a sphere.

To compactify a space, we need to get rid of these intangible drop offs. There are two simple approaches to handle them. You can fill the hole in, or you can put up a fence (there also numerous more complex approaches). The easiest way to fill the hole is to just add a single point, and glue the edges of the hole to it: Take that original projection onto the circle less (0,1), and add in (0,1) as a single http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif. This is the approach used by the Riemann Sphere. But the same trick works for http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif. To put up a fence, you stick boundary points all around the hole. This time, use the projection from (0, 0.9), and add in the boundary points (for http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif) or circle (for http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbc.gif). This is the approach used by the extended real line.

The important difference between these two methods is that when the hole is filled, every point is the same. The environs of http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif in the Riemann Sphere look just like the environs of 0 (and in fact they are mapped onto each other by the 1/x function). For analysis, this is just what you want. Special points mean extra headaches and exceptions to all your results. Complex analysis on the Riemann Sphere is the paragon of mathematical theories. Its the place where everything works with exquisite perfection. The only reason we don't bother with this for http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif is that analysis on the reals naturally extends to the complex plane anyway, so we just go all the way to the Riemann Sphere. But, on http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif we have a very useful property that is lost with the transferal to http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbc.gif, or even with the circle compactification of http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif: order. In situations where sign is important, having a single infinity loses critical distinctions. So instead of filling in the hole, we prefer to fence it off at each end of the real line. This way we can keep our order, though we now have to treat the infinite points different from the rest.

Title: Re: Infinity
Post by Eigenray on Feb 13th, 2008, 2:29am

on 02/11/08 at 16:16:39, BenVitale wrote:
 I imagine that it has been done before - I mean  to have a system with infinitely many infinities added to the complex plane.

There's another way to add an infinity for each 'direction' in the plane, other than just making a disk.

Let RP2 (the 'real projective plane') denote the set of lines through the origin in http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif3.  Any such line is determined by a non-zero point (x,y,z) that it passes through, and two such points determine the same line iff one is a scalar multiple of the other.  So if we take the set of non-zero points http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif3 - 0, and identify the points (x,y,z) ~ (tx,ty,tz), thttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/in.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif*, we get RP2 = (http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif3-0)/~.  An element of RP2 is denoted [x,y,z] = {(tx, ty, tz) | t http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/in.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif*}.

[Two other ways of thinking of RP2 are as the sphere S2 with antipodal points identified, or as a disk with antipodal boundary points identified.]

Now for each point (x,y) http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/in.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif2, we can think of it as the point (x,y,1) http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/in.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif3, and that determines the line [x,y,1] http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/in.gifRP2.  So we can think of http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif2 as embedded in RP2.  The only points of RP2 that don't come from http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif2 are those of the form [x,y,0].

Now suppose we go to infinity along the line y=ax+b in http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif2.  Each point (x, ax+b) corresponds to [x, ax+b, 1] = [1, a+b/x, 1/x] http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/in.gifRP2.  As xhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/to.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif, this point converges to [1,a,0].  In this way, any point of RP2 is a limit of points of http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif2, and we say http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif2 is dense in RP2.  Since RP2 is compact, it is therefore a [link=http://en.wikipedia.org/wiki/Compactification_%28mathematics%29]compactification[/link] of http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbr.gif2.

But note that we've added a whole circle of 'points at infinity'.  If we move along two lines of slopes a,a', we end up at two points [1,a,0] and [1,a',0].  Now here's the kicker: if two lines intersect in the plane, they have different slopes, so they don't intersect 'at infinity'.  On the other hand, if they have the same slope, they are parallel, so they don't intersect in the plane, but they do intersect 'at infinity'.  So in RP2, we have that any two (distinct) lines intersect in exactly one point, whether they are parallel or not.  It's properties like this one that make projective space a more natural place to do algebraic geometry.

Title: Re: Infinity
Post by BenVitale on Mar 8th, 2008, 9:24pm
I kinda took the concept of infinity for granted, because I've realized that the word "infinity" can mean different things in different contexts, and whether or not a certain concept exists can depend on the context in which you ask the question.
For example, in the context of a number system in which infinity would mean something one can treat like a number. In this context, infinity would not exist. However, in the context of a topological space, in which infinity would mean something that certain sequences of numbers converge to -- in this context, infinity exists. And in the context of measuring size of sets, infinity does exist.

I cannot help myself but to ask, what does "exist" or "existence" mean in mathematics?

Title: Re: Infinity
Post by Hippo on Mar 9th, 2008, 12:14am

on 03/08/08 at 21:24:37, BenVitale wrote:
 I cannot help myself but to ask, what does "exist" or "existence" mean in mathematics?

Mathematics is a tool. It's a case study. It sometimes answers given questions if given assumptions hold.

There are attempts to minimize assumptions wich are required to answer given question. And vice versa to formulate as most conclusions from given assumptions.

So we study a lot of different models, and the existence depends on the model (and may not to be determined by it).

Title: Re: Infinity
Post by rmsgrey on Mar 9th, 2008, 8:38am
In mathematics, there are only so many good words/symbols to use, so mathematicians tend to reuse labels when they come across related or similar concepts - so, for example, "1" gets used for multiplicative identities in all sorts of contexts, including integers, rationals, reals, complex numbers - they all behave in pretty much the same way, so it works out pretty well - things that apply to one 1 usually apply to the other 1s.

With "infinity", there's a range of more loosely related concepts, all of which get that same label applied to them, so things that apply to one infinity often don't apply to all infinities...

Title: Re: Infinity
Post by BenVitale on Mar 11th, 2008, 12:35pm
What is Cantor's argument for different sized infinities and uncountable infinities? Could someone explain it?
Did Cantor run some encryption through the idea that when you count up using natural numbers, every new symbol contains all the other symbols below it -1 until you hit 1?

Title: Re: Infinity
Post by Hippo on Mar 11th, 2008, 4:00pm
Diagonalisation:
If you make a list of sequences numbered by natural numbers on alphabet with more than one letter, you can construct a sequence which is not in the list.

n-th element of the sequence is choosen such that it differs from n-th element of n-th sequence.

So you cannot number all 0-1 sequences by natural numbers.

There are technical details with identifying reals with 0-1 sequences as there are ambiguities when for a natural k the number is multiple of 2-k.

Title: Re: Infinity
Post by BenVitale on Mar 12th, 2008, 8:18am
let's consider the proof that there is no injection between [0,1] and N='0,1,2,3, etc.'
Cantor derived a real number A of [0,1] that can not be in a series Ui=alpha(i), with the alphas in [0,1]. This number has its nth decimal different of that of alpha(n) : A is built from the "names" of the alphas, by building a scriptural difference between A and the aphas. It is more a proof by the SHAPE of the number than by its content. If correct, are there any implications?

Title: Re: Infinity
Post by Eigenray on Mar 12th, 2008, 9:49am

on 03/11/08 at 12:35:50, BenVitale wrote:
 What is Cantor's argument for different sized infinities and uncountable infinities? Could someone explain it?

Cantor's argument that there is no bijection between S and http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/scrp.gif(S) works exactly the same for any set, finite, countable, or uncountable: if f : S http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/to.gif http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/scrp.gif(S) is any function, then {x : x http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/notin.gif f(x)} is not the in image of f.

In the case where S=http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbn.gif, there is no bijection between http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbn.gif and http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/scrp.gif(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbn.gif), which we can think of as the set of all countably infinite {0,1}-sequences, or functions from http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/bbn.gif to {0,1}: Given a sequence x=(x(1), x(2), ...), we think of the set {n : x(n)=1}; conversely, given a set we have its characteristic function.  Then Cantor's argument becomes: given any countable list of infinite {0,1}-sequences xn, the sequence given by x(n)=1-xn(n) is not any of the xn's, so the list cannot be complete.

If we forget for a moment that real numbers in [0,1] don't have a unique binary representation, then this becomes: given any countable list of real numbers in [0,1], form a new number by taking as its n-th bit the complement of the n-th bit of the n-th number.  This new number is not the n-th number for any n, so it's not on the list.

Quote:
 Did Cantor run some encryption through the idea that when you count up using natural numbers, every new symbol contains all the other symbols below it -1 until you hit 1?

I don't know what you mean here.

on 03/12/08 at 08:18:00, BenVitale wrote:
 It is more a proof by the SHAPE of the number than by its content.

Cantor's argument was about cardinalities of sets.  In the category of sets, two sets are isomorphic iff they have the same cardinality.  The individual elements don't matter; they could be numbers, functions, or duck-billed platypi.

Title: Re: Infinity
Post by BenVitale on Apr 4th, 2008, 10:35am
Russell found it necessary to assume that the number of real objects in the universe is itself infinite. He used his axiom of infinity.

An axiom is self evident, whereas a postulate is something that is assumed for convenience even though it is not self evident.

Why did Russell not refer to the principle as his Postulate of Infinity?

Title: Re: Infinity
Post by towr on Apr 4th, 2008, 12:20pm

on 04/04/08 at 10:35:18, BenVitale wrote:
 An axiom is self evident
I can't say that's my experience with axioms, so I've looked it up:

http://dictionary.reference.com/search?q=axiom

Quote:
 ax·i·om    –noun1.a self-evident truth that requires no proof.2.a universally accepted principle or rule.3.Logic, Mathematics. a proposition that is assumed without proof for the sake of studying the consequences that follow from it.