To infinity and beyond

[Ben]

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

 

[math]\mathcal{P}(S) = \{\{a\},\{b\},\{c\},\{a,b\},\{a,c\}, \{b,c\}, \{a,b,c\}, \O\}[/math].

 

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 [math]\mathcal{P}(S)[/math] is always strictly greater than the cardinality of [math]S[/math].

 

In the example above, this is not hard to see; the cardinality of [math] S[/math] is 3, that of [math]\mathcal{P}(S)[/math] is [math]8 = 2^3[/math]. 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 [math]S[/math] of finite cardinality [math]n[/math], then the powerset [math]\mathcal{P}(S)[/math] has cardinality [math]2^n[/math]

 

But what if the cardinality of our set is infinite? What do we make of the assertion, say, that [math]2^{\infty} > \infty[/math]? 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 [math]N[/math]. Since, from the above, [math]N[/math] 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 [math]S[/math] is countably infinite in this sense.

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

 

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

 

Assuming that [math]\mathcal{P}(S)[/math] 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 [math]S[/math], remember) and call the [math]n[/math]-th member of this list as [math]l_n[/math].

 

Now form the set [math]D[/math] by the rule that [math] s_n \in D[/math] iff [math]s_n \in l_n[/math]

 

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

 

But hey! [math] l_p = D[/math] so we arrive at the breath-taking conclusion:

[math] s_p \in D [/math] iff [math]s_p \in D[/math].

 

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

 

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

 

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

 

But [math]l_q = D^c[/math], so we conclude that

 

[math]s_q \in D[/math] iff [math]s_q \notin D[/math]. 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]

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]

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]

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]

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]

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.

 

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,

 

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]

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.

 

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 :phones: 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]

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]

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 [imath]\varphi[/imath] and of course they will be parallel for [imath]\varphi=0[/imath] you can calculate how far away the point of intersection will be as a function of [imath]\varphi[/imath]. You can call it [imath]d(\varphi)[/imath] and easily calculate the limit:

 

[math]\lim_{\varphi\rightarrow 0}d(\varphi)=\infty[/math]

 

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]

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, "

 

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?

[Jay-qu]

Clay I think you are taking this the wrong way, to say parallel lines intersect at infinity is to say they never intersect (not in any real sense) since you can never get to infinity.

[C1ay]

Clay I think you are taking this the wrong way, to say parallel lines intersect at infinity is to say they never intersect (not in any real sense) since you can never get to infinity.

 

I'm just pointing out what the definition says. I realize you can never get to infinity but that doesn't make it OK to claim that a definition breaks down at infinity either. Definitions are just that, definitions. Parallel lines are defined as lines in a plane that have no point in common and I can not find one reference that states otherwise. I don't imagine anyone else can either or they would have cited it already.

[Jay-qu]

I'm just pointing out what the definition says. I realize you can never get to infinity but that doesn't make it OK to claim that a definition breaks down at infinity either. Definitions are just that, definitions. Parallel lines are defined as lines in a plane that have no point in common and I can not find one reference that states otherwise. I don't imagine anyone else can either or they would have cited it already.

Im sure there is a reference that outlines the argument that Q made, taking the limit of the angle between lines, for it is an old argument. (heres a guy from Stanford talking about it parallel lines 'meeting' Re: two parallel lines will meet in the infinity)

[C1ay]

Im sure there is a reference that outlines the argument that Q made, taking the limit of the angle between lines, for it is an old argument. (heres a guy from Stanford talking about it parallel lines 'meeting' Re: two parallel lines will meet in the infinity)

 

But that tosses aside the whole concept of the term "definition". From Oxford,

 

definition

 

• noun 1 a statement of the exact meaning of a word or the nature or scope of something. 2 the action or process of defining. 3 the degree of distinctness in outline of an object or image.

 

Again, we';re not talking about a theorem or a postulate but a definition of a term. If the term parallel is defined to mean that two entities are forever equidisyant or have no points in common then that is what the term means...always. You do not change a definition to meet your needs because it defeats the whole purpose of making the definition in the first place.

 

If we define for instance that the term "circle" as the term to describe a set of points in a plane that are equidistant from a point on that plane then that is exactly what the term circle will mean.....always. It will never be the correct term for an ellipse or a square or any other set of points that are not equidistant from a point on a plane.

 

A definition is just that, a definition. It is a construct we use to define the exact meaning of a word or phrase used in language to confer what me mean to confer to one another. A definition has nothing to do with math but with language and communication from one to another. In the case of "parallel" the dictionary says,

 

parallel

 

• adjective 1 (of lines, planes, or surfaces) side by side and having the same distance continuously between them.

 

This is what the term parallel means....always. It is what we define the term parallel to mean, in advance, in the construct of language. When we want to communicate to someone else about lines, planes or surfaces that are equidistant from each other it is the term we have defined for that purpose.

 

Yes, you can show that formulae involving parallel lines break down and give inconsistent results when you introduce infinity as a value or undefined values like division by zero but that does not change any definitions we have constructed for language. It just shows the limitations of mathematics with the introduction of such values.

[Qfwfq]

Are you claiming their definition is wrong?

It doesn't mean that the definition you quote is wrong.

it is about the very definition of what the term parallel means.

Definitions are definitions, there isn't always only one way to go.

 

One could use the equivalent definition that all points of one line have the same distance from the other and vice versa. One could also use the angle [imath]\varphi[/imath] in my last post for defining parallel by [imath]\varphi=0[/imath].

 

Statements such as "intersect at infinity" must always be taken with a grain of salt. Similarly for saying that "one over zero is infinity" which is handy in computing limits but in terms of real numbers is really just a shorthand for "the reciprocal of an infinitesimal term is an infinite term". Defining hyperreal numbers gives the simpler statement a meaning as an actual relation between values.

 

Even in terms of limits, in the real numbers, defining those that involve infinity is equivalent to extending the notion of neighborhood to include those of infinity (as well as those of a finite value, which are defined directly by the topology that the metric induces).

 

Having reached infinity, maybe we could all go for a :beer: and then let Ben continue beyond. :)

[Ben]

Despite the fact, or perhaps because of it, I am the son and grandson of professional philosophers, I have little taste for this sort of discussion. Nonetheless, let me try and engage here.

 

It is well known that no "definition" in any subject can be entirely free of circularity - some terms in the definition must of necessity be left undefined, or self-referential.

 

In math, one gets around this encumbrance by stating a set of axioms or postulates that describe the theory at hand. (By theory here I mean things like that of the real numbers, of vectors spaces, of groups etc) These axioms/postulates cannot be derived, but will in general seem reasonable or intuitive to all reasonable men. William of Occam taught us that these had better be kept to a minimum, otherwise a degree of arbitrariness creeps in .

 

Again, in math, definitions extend the axioms in a natural way, and can be regarded as distinct form the axiom set

 

In the case of parallelism, this cannot be done; there seems no way to "define" parallel lines that merely extends the axioms. So no, there is no such definition, parallelism must be grafted on as an additional postulate.

 

Ugh, I seem to be talking rubbish. Oh, stet.

 

Anyway, while you jolly guys go for a beer, and since I am not invited, I'll sit here and wait for Hell to freeze, when my luck is due to change.

 

They say it will be a cold winter this year......

[Qfwfq]

and since I am not invited, I'll sit here and wait for Hell to freeze, when my luck is due to change.

C'mon Ben I suggested we all go...

 

Yeah indeed the very notion of parallelism is so tightly knit with Euclid's fifth, ain't it?