To infinity and beyond

Ben · 10-17-2009

Oh no, it's him again!! Well tough tits, dear friends, but I promise you will enjoy this, however before the fun starts, we have to do a bit of homework.

The cardinality of a set is simply its number of elements.

If $S$ is a set, then its powerset $\mathcal{P}(S)$ is simply the set formed from all its subsets. Thus, bearing in mind that $S,\,\,\, \O$ are always subsets of $S$ ( $\O$ is the empty set, btw) then, when $S=\{a,b,c\}$ , then

$\mathcal{P}(S) = \{\{a\},\{b\},\{c\},\{a,b\},\{a,c\}, \{b,c\}, \{a,b,c\}, \O\}$ .

Note that the elements in the powerset are subsets of the set - this will be important.

There is a Theorem of Cantor that says that the cardinality of $\mathcal{P}(S)$ is always strictly greater than the cardinality of $S$ .

In the example above, this is not hard to see; the cardinality of $S$ is 3, that of $\mathcal{P}(S)$ is $8 = 2^3$ . In fact, using all available fingers and toes, and those of our wives and girlfriends (should they be different), it is not hard to see that, in general, for any set $S$ of finite cardinality $n$ , then the powerset $\mathcal{P}(S)$ has cardinality $2^n$

But what if the cardinality of our set is infinite? What do we make of the assertion, say, that $2^{\infty} > \infty$ ? Surely this is madness?

One last thing, of MAJOR importance (sorry for shouting).

A set is said to be countable iff it is isomorphic to a subset of the natural (i.e. "counting") numbers $N$ . Since, from the above, $N$ is always a subset of itself, and is infinite, we have the brain-curdling expression "countably infinite". (That's yet another reason to love mathmen).

Let's assume our set $S$ is countably infinite in this sense.
Let's also assume that $\mathcal{P}(S)$ is countably infinite in the same sense, that is, in accord with intuition, $2^{\infty} = \infty$ . We will find a contradiction

So, since $S$ is countable, we can index each and every element by an element of $N$ (this is due to our isomorphism). Call the $n$ -th element $s_n,\,\,n \in N$ . (Note that the choice of ordering is arbitrary, but, having chosen, we had jolly well better stick with it)

Assuming that $\mathcal{P}(S)$ is also countably infinite, then, to each element here we can also assign an index. So let's write a list of these elements (subsets of $S$ , remember) and call the $n$ -th member of this list as $l_n$ .

Now form the set $D$ by the rule that $s_n \in D$ iff $s_n \in l_n$

Now $D$ is obviously a subset of $S$ , an element in $\mathcal{P}(S)$ , so is eligible to be in our list. Let's call this list element as $l_p$ , so by our rule we have that $s_p \in D$ iff $s_p \in l_p$ .

But hey! $l_p = D$ so we arrive at the breath-taking conclusion:
$s_p \in D$ iff $s_p \in D$ .

Wow! Let's write a paper on that. But wait....

For every subset in $S$ (that is every element in $\mathcal{P}(S)$ ) I can find its complement, that is, there is the element in $\mathcal{P}(S)$ comprising those elements in $S$ that are not in $D$ Let's call it as $D^c$ .

Under the assumption that our powerset is countable, I can add $D^c$ to our list, call it the $q$ -th member, and apply the same rule as before: $s_q \in D$ iff $s_q \in l_q$ .

But $l_q = D^c$ , so we conclude that

$s_q \in D$ iff $s_q \notin D$ . This is clearly nuts, so the assumption that the powerset is countable must be false. Cantor's Theorem is thereby proved.

Way too long, but really sweet, wouldn't you say?

Jay-qu · 10-17-2009

Fixed your title.

Yeah I really liked that, thanks

I went to a lecture on infinities a few years ago - it blew my mind.

I could tell you what infinity is, but it would take me forever..
J

lemit · 10-17-2009

I have been wondering lately, as I wrote elsewhere, if parallel lines meet in infinity, where do they go after that? From the little of your post I could read, Ben, the little that was written in English, I did not see this question addressed. Is it addressed in the part I couldn't read?

Thanks.

--lemit

C1ay · 10-17-2009

Quote:

Originally Posted by lemit

I have been wondering lately, as I wrote elsewhere, if parallel lines meet in infinity, where do they go after that?

Parallel lines by definition do not intersect. If they ever intersected, even in infinite space, they would not be parallel lines.

Ben · 4 Weeks Ago

Clay: You make a bold assertion, one that, as far as I am aware, has never been proven (It's usually called Euclid's 5th postulate, one with which he himself was unhappy, it seems). In general, I think, in mathematics it is rash to use the term "never", since it is not easy to be sure what this means.

Like.....

When I was a little younger, I asked a girl in a bar if she would like to share my bed for the night: "Sure, like when Hell freezes over".

So. Is this never, or that I might get lucky after an infinite amount of time had passed?

lemmit: Your question is a legitimate one, and I cannot answer it - sorry

C1ay · 4 Weeks Ago

Quote:

Originally Posted by Ben

Clay: You make a bold assertion, one that, as far as I am aware, has never been proven (It's usually called Euclid's 5th postulate, one with which he himself was unhappy, it seems). In general, I think, in mathematics it is rash to use the term "never", since it is not easy to be sure what this means.

I made no assertion. I merely provided a link to the definition of parallel lines and clarified what it says, i.e.

Quote:

Originally Posted by Wolfram MathWorld

Parallel Lines

Two lines in two-dimensional Euclidean space are said to be parallel if they do not intersect.

In three-dimensional Euclidean space, parallel lines not only fail to intersect, but also maintain a constant separation between points closest to each other on the two lines. Therefore, parallel lines in three-space lie in a single plane (Kern and Blank 1948, p. 9).

Is it your assertion that this definition of parallel lines is false? It is after all a definition and not the parallel postulate that you are referring to. As a matter of fact I'm not particularly aware of the practice of proving or disproving definitions. Take the definition of "line" for instance,

Quote:

Originally Posted by MathWorld

Line

A line is a straight one-dimensional figure having no thickness and extending infinitely in both directions. A line is sometimes called a straight line or, more archaically, a right line (Casey 1893), to emphasize that it has no "wiggles" anywhere along its length. While lines are intrinsically one-dimensional objects, they may be embedded in higher dimensional spaces.

Is there a "proof" somewhere that this is in fact what a line is? A proof that it's not something else? That this is in fact the correct definition of a line?

Qfwfq · 4 Weeks Ago

Slow down C1ay, the matter is highly subtle and besides, there can be different but equivalent ways of defining and constructing some things in the Edifice of mathematics. Remember too, that the girl in Ben's story was to all effect slapping a big, fat "never!" into his face.

Quote:

Originally Posted by lemit

I have been wondering lately, as I wrote elsewhere, if parallel lines meet in infinity, where do they go after that?

No, he did not address that in any way.

I think they go for a

together. Or maybe instead they talk of forming a government together despite being enemies during the Cold War, but then the Red Brigades bump off the one that suggested the idea.

I dunno, infinity means unending and they never get there anyway...

When times are mysterious
Serious numbers will always be heard
And after all is said and done
And the numbers all come home
The four rolls into three
The three turns into two
And the two becomes a
One

C1ay · 4 Weeks Ago

Quote:

Originally Posted by Qfwfq

Slow down C1ay, the matter is highly subtle and besides, there can be different but equivalent ways of defining and constructing some things in the Edifice of mathematics.

Are you suggesting that that exists some possibility, any possibility, where parallel lines can intersect? To me the definition is clear, parallel lines are lines that do not intersect. Is there some other definition, compatible with this one, where they may intersect? When is a definition not a definition?

Qfwfq · 4 Weeks Ago

They do not intersect at any finite distance.

If you consider one straight line, and a variable one through a fixed external point and in a same plane as the first line, call the angle between the two lines $\varphi$ and of course they will be parallel for $\varphi=0$ you can calculate how far away the point of intersection will be as a function of $\varphi$ . You can call it $d(\varphi)$ and easily calculate the limit:

$\lim_{\varphi\rightarrow 0}d(\varphi)=\infty$

This is pretty much like saying: When they are parallel, the intersection is at infinite distance. That's why sometimes it is put in these terms.

C1ay · 4 Weeks Ago

Quote:

Originally Posted by Qfwfq

They do not intersect at any finite distance.

The definition quoted from MathWorld says nothing about finite distance, only that they do not intersect. Are you claiming their definition is wrong?

I also have a refernce I keep on the shelf here, The Handbook Of Mathematics, Bronshtein · Semendyayev, 1985, Verlag Harri Deutch (publisher) which states, "

Quote:

Two lines lying in one plane have either one or no point in common. In the second case they are parallel.

Is this reference wrong as well? Again, I will point out this side discussion is not about a theorem or postulate, it is about the very definition of what the term parallel means. Is it your assertion that two lines can intersect, can have a point in common and can still be parallel?

To infinity and beyond

Hypography?

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