It’s also worth noting that there are subsets of any line segment which have no length, but have the same number of points as the whole line segment. I can’t describe one here, cause they’re very complicated, but they do exist (as long as you accept the axiom of choice).
Another similar question, for interest:
What percentage of all integers contains at least one instance of the digit three? ( For example, 13, 31, 33 and 103 all contain the digit “three” at least once. )
Answer:
100% of all integers contain at least one three.
The solution is surprising. It is difficult to believe that 100% of integers contain the digit three at least once. The simple fact that the number 7, for example, has exactly zero threes in it seems to dispute this.
The percentage of numbers with threes in them can be expressed as 1 - (.9)^n, where n is the number of digits. It reaches 99% at about the point where n has 42 digits.
The ratio of “threed” to “three-less” numbers at infinity would be 1 - (.9)^(Infinity), or 1.
It is interesting to note that there are also an infinite number of integers which do not contain the digit three. The simple progression "1, 11, 111, 1111… " illustrates this fact.
This seeming paradox illustrates one of the many “problems” associated with trying to apply concepts (like percentages) used for regular sets on the infinite.
Muad’Dib,
What about the square root of -1?
(Sorry, just couldn’t leave that hanging)
That’s imaginary. You want sqrt(2).
Time to help out this thread. Too much confusion, false statements and poor use of terminology.
What is under discussion is cardinality of sets. How many items are in the set. Cantor put cardinality of sets on firm logical foundations. Dodging Cantor is a great way to be illogical. Many amateurs on the Net have tried, every single one has failed. Two sets have the same cardinality iff they can be put in 1-1 correspondence.
Let me give some categorization.
-
Finite sized sets. Sets of size 0,1,2,3 etc. Obviously any two sets of the same size can be put in 1-1 correspondence but two sets of different sizes can’t.
-
Countably infinite sets. The most well known of course are the Natural numbers. But it also includes the Integers, the even Integers, the Rationals, and pairs (and n-tuples in general) of countably infinite sets. Hence the “square” of a countably infinite set is also countably infinite. (The opposite of what someone posted above!) The 1-1 mappings are quite trivial for anyone who knows such things. The cardinality of countable sets is called “Aleph_0” in the literature, where “Aleph” is the first letter of the Hebrew alphabet.
-
The size of the continuum, the set of reals. The same cardinality as the set of functions from integers to integers or the powerset of Natural numbers. Note that the cardinality of the reals in any finite (non-trivial) range is the same as any other range, finite or infinite. (Again the opposite of what has been stated.) Whether the continuum has the same cardinality as “Aleph_1” is called “The Continuum Hypothesis” and taking TCH as an axiom or not gives you different set theories.
-
The cardinality of the set of functions from reals to reals, or the powerset of reals gives the next higher level. The functions over those or their powerset gives the next higher, etc. “To infinity and beyond.” There may also be “in between” cardinalities, depending on your starting axioms.
Hence there are “a lot” of infinities and most posters are not indicating which one they are discussing.
Note that you are talking about size of sets. Most sets do not have “length” and thus “length” is an utterly useless when comparing the size of sets. After all, what is the “length” of the powerset of US Presidents? of the set of all functions from reals to integers? etc. Use length when comparing lengths, use size when comparing sizes. Don’t ever think the two are the same.
:smack:
Dang!
That’s what happens when I try to be witty.
The one that blew my mind was this:
Between every two rational numbers is an irrational number.
Between every two irrational numbers is a rational number.
For every rational number, there are an infinite number of irrationals that can be mapped to it (not 1-1).
It’s been awhile since I’ve had classes so cannot remember the terminology but I remember seeing the proofs done by the professor. Totally mind blowing.
How about the object that has a finite volume but infinite surface area? Holds 2 gallons of paint but could never paint it.
Common sense breaks down at infinities
Sure you can paint it, you just have to use very thin paint.
The reason for this is, of course, that we’re still trying to establish whether you can compare infinities. It seems a bit premature to talk about cardinalities, the definitions of which implicitly rely on comparing infinite sets.
Ok, so, what, exactly, are the different types of infinities?
I suggest you read the thread that DarrenS linked to above to answer that question. However, that thread has the convenient property of getting more and more technical as you go along, so I suggest you quit reading when you’re satisfied (or not satisfied). But at least give it a shot.
The basic problem here is that you have no answer to “how long is a point”? In mathematics a point has no length, thus you cannot say how many points when put together can be a certain length.
Thus you cannot measure a line and then say how many points are in it.
Thus you cannot say the length of two lines are proportional to how many points there are in them.