So nine years later, out of nowhere, a high school geometry question pops into the old noodle. Namely:
This ray {----->} can be said to have a starting point, but no end point. It stretches on out into infinity.
This line {<----->} continues for infinity into both directions.
The question(s): which one is longer? Are both examples of infinity? (I mean, how much bigger can you get than infinity? Should it matter that one has a defined starting point?) Or is this like when a little kid screams “infinity plus one!” to counter the infinite forcefield power claimed by his sibling in the war of finding the exact middle in the back seat of the family station wagon?
The ray and the line are the same length. Infinite items are compared by looking for one-to-one onto mappings. The length of a line segment is the number of segments of a unit length that can be mapped onto it (eg. a 10-inch line segment is one in which ten 1-inch line segments can be placed end-to-end without overlapping and without leaving any of the original line segment uncovered). Both the ray and the line can hold an unlimited number of one-inch line segments, so their length is infinite. To determine whether it is the same ‘degree’ of infinity, we would look to see if there is a way to pair up the line segments on the ray with those on the line in such a way that every segment on one is matched to exactly one segment on the other, with no duplication & none left over. One matchup for the given case: mark an arbitrary ‘origin’ point on the line. Match up the first inch of the ray to the first inch to the right of the origin on the line; match up the second inch of the ray to the first inch to the left of the origin on the line; match up the third inch of the ray to the second inch to the right of the origin on the line; and so on endlessly. You will never run out of inches on the ray to match up, and no inches on the line will be left out; so both have the same number of inches of length.
There are ‘bigger’ infinities, defined by this method; for instance, there are more points on either line (or on one inch line segment) than there are inches. But I think that this will stretch your brain enough to start with.
Yep. This is an example of the “bigger infinity” paradox, which demonstrated why you can’t treat infinities as a number.
Another example:
The set of positive integers starts at one and increases by one until infinity.
The set of odd positive integers starts at one and increases by two until infinity.
Both sets, therefore, are infinite.
But:
All odd positive integers are positive integers.
Not all positive integers are odd positive integers.
There are positive integers that are not odd positive integers.
So is the set of positive integers “bigger” than the set of odd positive integers? The answer is, nope. They’re both infinite, which means they are beyond our usual conceptions of bigness.
Is the above mentioned paradox something that was formally presented within the scientific community at one point? Did it cause a whole bunch of debate? More importantly, is there someplace good I can read up on it, or is it too boring for the lay to delve into?
And a follow-up: what is the term for infinity as represented by a regular line segment’s ability to be divided infinitely without ever reaching zero? Ex: half of a half of a half, etc…
The early work on this was done by Georg Cantor (1845-1918). You might try a web search on terms like ‘transfinite numbers’ or ‘infinite sets’ for some references … there’s a good introduction to the basic principles in Isaac Asimov’s On Numbers. I haven’t personally read anything else that dealt with it at any length that wasn’t aimed at mathematics graduate students, which I doubt you’re interested in at this point.
Yeah, it was a big debate among the mathematical community around the turn of the 20th century. Cantor discovered that if you take any set (even if it’s infinite), you can always find a larger set–namely, its power set, which is the collection of all subsets of the given subset.
This met with a good deal controversy at the time, particularly from the mathematician Kronecker (Cantor had been a former student of Kronecker). They had a big falling out;Cantor had paranoid and depressive tendencies to begin with, and the rejection of his ideas certainly didn’t help matters; it’s difficult to say just how much of an impact the rejection had on him, but he ended up spending a good deal of time in an sanitarium.
One thing I forgot to mention here, when I say larger, I don’t mean by just one or two elements, or any finite number of elements–that set would still be the same size as the original infinite set (adding on finitely much stuff ain’t gonna really make a difference to an infinite set). Even if you “double” the size of the set, you’d still get the same size of set. When I say larger I mean unimaginably larger–in fact we basically have no idea just how much larger the power set will be than the original set.
It may be counterintuitive and downright weird, but it’s not a paradox. It is not self-contradictory.
smoke wrote:
A lot of the formalization of the infinite is due to Georg Cantor. And yes, his work was very controversial and eventually the criticisms of it led to his numerous nervous breakdowns that finally landed him in the looney bin.
However, the technique of comparing infinite sets that Cantor discovered, Cantor diagonalization, is arguably one of the major contributions to 19th century mathematics. SCSimmons gave a good explanation of Cantor diagonalization.
Check out Morris Kline’s ‘Mathematics: The Loss of Certainty’ for this and other mathematical controversies.
Using Cantor’s notation, it would be aleph-1. Cantor labeled the size of various infinite sets as follows:
aleph-0 = the size of the set of integers (or rational numbers or any set that can be completely enumerated by the integers)
aleph-1 = the size of the set of real numbers or irrational numbers
aleph-2 = the size of the set of all sets of real numbers.
and so on.
For more mind-bending fun, contemplate the Continuum Hypothesis with a detour to the undecidable and Gödel’s Incompleteness Theorem, indulge in set theory, behold Russell’s Paradox and peruse the Axiom of Choice and its paradoxical progeny.
Actually, that’s not right; that’s what I was getting at when I said we have no idea how big the power set is.
Aleph-0 is the size of the integers, aleph-1 is the next smallest infinity, aleph-2 the next smallest, and so on. But we don’t know how big the set of reals is, or how big the set of all sets of reals is. It’s consistent with everything else in standard set theory to say that the reals have size aleph-1, on the other hand it’s consistent to say that they’re just about any other size, too; we simply don’t know.
Not to be Clintonian, but it depends on how you define ‘know.’ Cantor’s work was comparing the relative size of sets and when you’re dealing with infinite sets that means the cardinality of the sets is infinite.
But we do know that the set of reals is ‘bigger’ than the set integers: any one-to-one mapping of integers to real numbers will exhaust all integers but not the reals (in fact leaving aleph-1 reals left over).
Not according to Cantor’s Theorem. Actually, Cantor’s theorem says that the cardinality of the power set of any set is greater than the cardinality of set itself.
I may have erred in claiming that the cardinality of the power set of reals is aleph-2 as that might fall into the generalized Continuum Hypothesis trp, but it’s at least aleph-n where n > 1.
Aleph-1 is the next smallest infinity. The transfinite cardinals go aleph-0, aleph-1, aleph-2, and so on–there is no cardinal between aleph-0 and aleph-1. True, there are ordinals between aleph-0 and aleph-1, but only cardinals are used to describe the sizes of sets.
It’s true that we do know that for any set, its power set must have a greater cardinality, but once we start getting into the infinite cardinals, that’s about as much as we know of how big the power set is.
The cardinality of the reals is the same as the cardinality of the power set of the natural numbers–we do know that it’s strictly bigger than aleph-0, but that’s about it. It could be aleph-1 (which is what the continuum hypothesis (CH) says, but unfortunately CH is undecidable under the standard set theory axioms), it could be aleph-2, -3, -4,…, aleph-3443654585574, or there could even be infinitely many alephs before it. Any one-to-one mapping of integers to real numbers will have reals left out, but it won’t necessarily be a set of cardinality aleph-1 left out; the cardinality of the set of left out reals will be whatever the reals were to begin with, and we don’t know what that is–only that it’s bigger than aleph-0. The reals could be huge compared to the integers.
Same case with the power set of any infinite set–we know it has strictly larger cardinality, but we have no idea how much larger.
A mathematician named Ian Stewart gives an examplein one of his books that I liked, describing the problems of infinitely large numbers; (this is paraphrased from memory):
There’s a hotel with an infinite number of rooms; on Monday, an infinitely large coach arrives and an infinite number of guests check in for the week (it takes quite a while to hand out all of the keys).
On Tuesday, only one guest checks in, but all of the infinte number of rooms are full, do the hotel manager asks the guest in room 1 to move to room 2, the guest in room 2 to move to 3 and so on, so room 1 is free for the new guest.
On Wednesday, another infinitely large party arrives, so thnking on his feet this time, the manager asks all of the existing guests to move to the room number that is double the number of their current room, there are an infinite number of rooms, so this is not a problem; now, only the even-numbered rooms are occupied and the new guests can be put in the infinite number of vacant odd-numbered rooms.
