Segements are contained within a line and therefore do not add to the bulk of a line. So to say that the presence of an infinite number of segments on a line would negate that an infinitie number of lines can pass through a point would be false.

Some infinite sets are bigger than other infinite sets.
Sets are assigned a transfinite #.
{1, 2, 3, 4, 5} has a transfinite # 5.
The set of integers has a transfinite number of aleph-null.
The set of even integers has a transfinite number of aleph-null.
2 * aleph-null is aleph null.

The initial set you descibed has a transfinite # "c". c > aleph null.

The second set you described has a transfinite # 2^c, 2^c > c.

I may have the terminology wrong (transfinite order? transfinite #?).

This is such an odd question. Are you taking a class? Or trying to suss out my identity?

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iambigkahunatony.com

nope. i thought of the question in 11th grade (i remember asking my teacher, she brushed the question aside as unrelated to the coursework). for some reason, i remembered it last night during my grad-level math class (which is on ODEs, not at all related to this sort of thing). maybe it was my brain trying to escape the 3 hr suffer-fest? escapism is a beautiful thing sometimes.

thanks for the explanation. i am confused, though. specifically, by your statements:

• 2*aleph-null = aleph-null
• second set you described has a transfinite # 2^c

for the first one, i don't understand how this works unless aleph-null = 0, which i'm pretty sure it doesn't.

for the second one, why 2^c and not c^c?

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http://www.theninjadon.blogspot.com

"The bicycle riders drank much wine, and were burned and browned by the sun. They did not take the race seriously except among themselves." -- Ernest Hemingway, The Sun Also Rises

re: different sized infinities.

Imagine the set of all even integers.
It's infinite in size.
Now imagine the set of even and odd numbers.
It's infinite in size but we know it to be twice the size of the first set.
Another way to think of this is to subdivide an axis in real units of 1.0 out into infinity, creating an infinite series of numbers.
One can split the initial markings in half, making twice as many numbers, but infinite. Repeat the splitting for a bigger infinity, in a way.
Obviously one can always go out "farther" in the lower dimensioned set, but if one "keeps up" in the higher dimension set, it always is bigger.

The other way to think about this, and not being a math guy, but an English and History guy, is to just accept that "infinite" means they go on forever, there is no end, you can't count them or quantify them further than "infinite." There's no such thing as "more infinite" or "twice as infinite." The whole problem with the word "infinite" is that the human mind can't really grasp or imagine what infinite actual means or looks like. We judge everything as a measurement in relation to something else, and infinite just keeps going on forever.



So to answer the question, the infinite segments aren't any more infinite than the infinite number of lines. They are both equally infinite. Such is the nature of the concept of infinity.

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Slowguy

(insert pithy phrase here...)

to infinity and beyond...

--buzz lightyear

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_________________________________
I'll be what I am
A solitary man

Not a math genius here, but it would seem that "infinity" is not effected at all by your comparison of the amounts withing it.

IOW both the line segments and the points are subsets of infinity, no? Neither are actually describing or encompassing infinity, no again? I guess what I'm getting at here is certainly you can say something like 2*infinity....but isn't that infinity? So doing any simple calculation involving infinity is in fact infinity. infinty/x , infinity * Y all equal infinity.

I double triple dog dare you to a million infinities....

~Matt

The mathematical theory to which some of the posters above are referring (using terms like "aleph-null") was developed by Georg Cantor. You might want to google his name. Basically, the theory posits a whole series of infinities with different "cardinalities"--which we can think of as different "sizes" or "orders of magnitude." Some interesting details:

1. Two infinite sets are said to have the same cardinality if it is possible to construct a one-to-one correspondence between them. As a very simple example, you can draw a 1-1 correspondence between the set of positive integers (> 0) and the set of negative integers: 1 ~ -1, 2 ~ -2, etc. So we can say that these sets have the same cardinality, or that they represent the same "size" of "infinity."

2. Don't be confused by the fact that you may sometimes be able to draw a 1-1 correspondence between a set S and some infinite proper subset of S. That does not necessarily mean that S is somehow "larger" than its subset, in the Cantorian sense. For example, let X be the set of positive integers, and let Y be the set of even positive integers. Since Y is a proper subset of X, you might at first blush think that Y had to have a smaller cardinality. But you can easily set up a 1-1 correspondence between X and Y (1 ~ 2, 2 ~ 4, 3 ~ 6, etc.). Because of the 1-1 correspondence, they are considered to have the same cardinality, even though one is a proper subset of the other. In the same way, it is possible to show that the set of rational numbers has the same cardinality as the set of positive integers, even though the latter is clearly a proper subset of the former.

3. On the other hand, in some cases you may be able to show that for a set X and a certain infinite proper subset of X called Y, it is logically impossible to set up a 1-1 correspondence between X and Y. For example, it can be proven that is impossible to set up a 1-1 correspondence between all the real numbers and all the rational numbers (the latter being a proper subset of the former). Consequently, the real numbers enjoy a higher cardinality in Cantor's system than the set of rationals. If you're curious as to the proof, I remember it and can post it.

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Two roads diverged in a wood, and I--
I took the one less traveled by,
Which is probably why I was registering 59.67mi as I rolled into T2.

I cannot help it, - in spite of myself, infinity torments me. ~Alfred de Musset, L'Espoir en Dieu

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Marty Gaal, CSCS
One Step Beyond Coaching
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That's funny; I always thought his tormentor was George Sand. ;)

Obviously, the Romanticists' view of infinity is rather different from the mathematician's no-nonsense view. I should also mention that mathematicians don't treat infinity as a "number," at least in the ordinary sense. That's why it's nonsensical to try to apply arithmetical operations (like multiplication) to infinity. In fact, in most of mathematics (with the possible exception of Cantor's theory) the notion of infinity is regarded as superfluous; at most (as in calculus) the infinity symbol may be used as a shorthand to denote ideas that can be defined perfectly well in terms of finite numbers.

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Two roads diverged in a wood, and I--
I took the one less traveled by,
Which is probably why I was registering 59.67mi as I rolled into T2.