The concept of infinity is hard to grasp because it is an abstraction. There are no tangible objects in our lives that are truly infinite in number so we really have nothing to compare it to. The only way to get an infinite number of anything is by invoking infinity elsewhere, which doesn’t really clarify matters much.
For example, the number of elementary particles in our visible universe, although immensely large, is still a finite number. If we assume that the density of particles in the universe is roughly the same everywhere and further postulate that the universe is of infinite size, then we can arrive at an infinite number of particles. But now we have the problem of understanding what a universe of infinite size is like.
If the concept of infinity is not hard enough, it turns out that there is not just one infinity but an infinite number of infinities of different sizes, so to speak.
I thought this video does an excellent job of trying to explain these difficult abstractions.
As I commented on Ed Brayton’s blog, the infinite set of rational numbers is larger hen the infinite set of real integers. This is obvious because there is an infinite number of rational numbers between every pair of integers.
Well, sure, but it’s good to be careful of “obvious” stuff when dealing with infinity.
For example, it is obvious that there are more integers than natural numbers — after all, there are an infinite number of integers that are not natural numbers.
And yet, they are the same size.
Obvious things are often misleading dealing with this stuff.
slc1, it may be “obvious,” but it is wrong.
One problem with your argument is that, although you don’t define what you mean by “larger”, your definition, whatever it is, clearly does not agree with the standard definition, which is that the set A is larger than B if there is no one-to-one map from B into A.
However, this is not the main problem with your argument. The main problem is that it is self-contradictory.
Your argument is not mathematically rigorous, so it’s hard to tell exactly what axioms you are using, but it does strongly suggest that you believe the following statement:
1. If you have an ordered set A, and an ordered set B, and there are infinitely many elements of A between any pair of elements of B, then A is strictly larger than B.
If you don’t believe statement 1, then I don’t see how your argument makes any sense at all, so I am going to assume you believe statement 1.
I am also going to assume you believe the following:
2. No set is strictly larger than itself.
I would argue that anyone who disagrees with statement 2 has got the wrong idea of what “strictly larger” means.
Now, here is the problem: Let A be the rationals, and let B also be the rationals. Note that there are infinitely many elements of A between any pair of elements of B (that is, between any two rational numbers, there are infinitely many rational numbers). Therefore, by statement 1, A is strictly larger than B. However, A = B, so by statement 2, A is not strictly larger than B. This is a contradiction, so either Statement 1 or Statement 2 is false.
Which one would you like?
This is “obvious” but incorrect. The rational numbers are countable, meaning that the set of rationals is the same “size” as the set of naturals.
One fairly easy to understand proof is in the Wikipedia article on countable sets.
To get a constructive intuition for how this can be one can look at something like the Stern-Brocot tree.
Also from 2:24 onward you can see it explained it is not true that the set of rational numbers (Q) is larger than the set of ‘real’ integers (Z) despite the fact that between each and every integer there are an infinite number of ratios of integers. It is shown how to elegantly make a one to one correspondence between all ratios and all integers. It only goes wrong with numbers that cannot be expressed as ratios of integers, which are irrational numbers. All of these numbers are real numbers though.
What you say is true and it is an example of the dense-ordering of the rationals; you can’t conflate dense ordering with countability.
Having the ‘dense’ property does not imply that a set is uncountably infinite. The rationals are a one-to-one and onto map of the integers, therefore both the integers and the rationals are countably infinite.
I’m glad I’m not the only one who noticed the misuse of “real” numbers.
There is a one to one correspondence between the rationals between i and i+1, i being any integer, and the integers. Having exhausted all the integers, we are then left with the rationals between i+1 and i+2, i+2 and i+3, etc. There are no longer any integers to correspond to these rationals as they have been used up in the correspondence with the rationals between i and i+1 so, by definition, the set of rationals is larger then the set of integers.
The example of there being an infinite number of rationals between any two rational numbers and therefore, the set of rationals is larger then the set of rationals is fallacious because there is a 1 to 1 correspondence between every rational and itself.
The problem lies with your statement “Having exhausted all the integers..”
You never exhaust all the integers. That is the chief difficulty with the concept of infinity. It is not simply a very large number that we cannot name but which we can use up. It just keeps going on and on. It is inexhaustible and is unlike anything we encounter in our lives, where everything is finite, which is what makes it so hard to wrap our minds around it.
We have made a 1 to 1 correspondence between every rational between i and i+1 and every integer. Therefore, there are no more integers to correspond to. Period, end of story.
@slc1:
This argument is incorrect. You appear to be relying on the following premise (and if you think you are relying on a different premise, then it is not obvious from your post, so please make it explicit):
1. If there is a 1-to-1 correspondence between sets A and B, and C isa set that strictly contains A, (in your argument A = rationals between 0 and 1, B = integers, C = rationals), then there can be no 1-to-1 correspondence between C and B.
However, Premise 1 is false, so the argument collapses.
A simple counterexample is the following: Let A be the even natural numbers (that is, the integers 2, 4, 6, …). Let B be the natural numbers (1, 2, 3, …). Let C = B. We are all familiar with a one-to-one correspondence between A and B; namely, the function f: A -> B given by f(n) = n/2. However, there is also a 1-to-1 correspondence between C and B (namely, the identity). So this is a counterexample to Premise 1.
But but but -- god is “infinite love!”
I thought my dogs were infinite love, too, but they died. And that’s the problem with finite beings like christians saying god is infinite love. Unless they can make an argument from love-induction it seems to me that god’s love might end just around the corner beyond mortal ken.
slc1, we can map each rational number m/n to the integer 2^m*3^n. Obviously this is a mapping from all rationals to a subset of integers, which means that there are at least as many integers as rationals.
One of the characteristics of infinite sets is that they can be put into one-to-one correspondence with a proper subset of themselves.
For instance, one can construct a one-to-one correspondence between the rational numbers and a proper subset of the integers as follows: Write each rational number as (-1)^n * p/q, where n=0,1 and where p,q are relatively prime. It’s not too difficult to show that the map taking (-1)^n *p/q to 2^p 3^q 5^n is a bijection, and it only gets a proper subset of the integers; for instance, it misses all the positive multiples of every prime > 5. Were we to use your mode of interpretation, slc1, we would conclude there are more integers than there are rational numbers, this despite the fact that the integers are themselves a subset of the rational numbers.
“only gets”…I mean only maps to a proper subset of the integers. Aargh.
Given that god doesn’t exist, Christians can say god is infinite love and be completely truthful. Just not for the reasons they think.
Heh. And both your and slc1’s observations allow us to use the Schroder-Bernstein Theorem to conclude that the set of rationals and the set of natural numbers are the same “size.”
Sorry, but this topic is one of the things that originally got me interested in abstract mathematics.
Another way to refute slc1’s error is to employ the Grand Hotel as an analogy:
Suppose the rational numbers between 0 and 1 are all comfortably ensconced in their hotel rooms (which are numbered by the natural numbers 1,2,…). The hotel is full (every room is occupied).
Now, the rational numbers between 1 and 2 arrive in a bus; the hotel manager can immediately free up space by moving every current occupant of room N into room 2*N, leaving rooms 1,3,5,… free for the new arrivals.
This works even if a (countably) infinite group of buses (each holding a countably infinite set of guests) arrives at once, since the new arrivals can be counted off diagonally to give a room number, then assigned to the odd-numbered rooms as above.
For some reason, Christians are generally comfortable with the idea of eternity, a future without an end, but a great many insist that a past without a beginning is inconceivable.
According to WP (and their sources), when Cantor discovered the hierarchy of infinite cardinals, some theologians spent a lot of energy on the issue of whether or not it was heretical pantheism (presumably, the argument is that if God is the “Absolute Infinite,” then all the lesser infinities must be lesser deities or something). Cantor himself didn’t help matters by polluting his mathematical work with theology, thereby opening the door.
@SLC1 -- you seem to be confused. Just because you can make a 1-1 correspondence between a subset of set A and B does not mean you cannot also create a 1-1 correspondence between A and B.
For example, there is a 1-1 correspondence between the even natural numbers and the integers. By your argument, then “all the integers would be used up”, and so clearly a 1-1 correspondence between ALL the natural numbers and the integers would be impossible, because we have an infinite number of natural numbers left even after using up all the integers on the even ones.
But this is false, because the natural numbers ARE in 1-1 correspondence with the integers. So just because a subset of the rationals is in 1-1 correspondence with the integers doesn’t mean ALL the rationals aren’t.
That would be a claim you’d have to prove, and you won’t be able to, because the 1-1 correspondences the have been discussed exist.
Sorry, eigenperson, I should learn to read.
It had never struck me that the idea of a hierarchy of infinities had theological implications but now that you point it out, I can see why there was concern.
Yes, there is a one to one correspondence between the rational numbers and the positive integers.
It is entirely possible for A to be a proper subset of B and for there to be a one to one correspondence between A and B.
This is impossible, of course, if A and B are of finite size.