Mathematical references to the infinite start with the celebrated paradoxes of another Greek philosopher, Zeno. In one paradox, Achilles challenges the tortoise to a race and gives it a head start.
Achilles is faster, but — Zeno argued — he can never catch the tortoise. Why not? Because by the time he reaches where it was, it has moved a little further on, and this keeps happening indefinitely.
Sorting this out poses deep questions about space and time. Children often wonder what the biggest number is, usually settling for the biggest whose name they know — a hundred, or a thousand. Later they realise that whatever number you choose, a gazillion, say, it can always be trumped by a gazillion and one. So we have to come to grips with it, rather than dismissing it as nonsense. Around AD, the philosophy and mathematics of infinity became entwined with early Christian beliefs.
In AD Eunomius argued that creation as a whole is finite. Consider the most perfect possible being. As you can see, a power set contains many more members than the original set.
Two to the power of however many members the original set had, to be exact. Imagine a list of every natural number. Now the subset of all, say, even numbers would look like this: yes, no , yes, no, yes, no, and so on.
The subset of all odd numbers would look like this. And how about every number—except 5. Or, no number—except 5. Obviously this list of subsets is going to be, well, infinite. But imagine matching them all one-to-one with a natural. The way to do this is to start up here in the first subset and just do the opposite of what we see.
As you can see, we are describing a subset that will be, by definition, different in at least one way from every single other subset on this aleph-null-sized list. Even if we put this new subset back in, diagonalization can still be done. The power set of the naturals will always resist a one-to-one correspondence with the naturals.
The point is, there are more cardinals after aleph-null. Wait … what are we doing? Of course we can. This is math, not science! Its consequences just become what we observe. We are not fitting our theories to some physical universe, whose behaviour and underlying laws would be the same whether we were here or not; we are creating this universe ourselves.
If the axioms we declare to be true lead us to contradictions or paradoxes, we can go back and tweak them, or just abandon them altogether, or we can just refuse to allow ourselves to do the things that cause the paradoxes. The axiom of infinity is simply the declaration that one infinite set exists—the set of all natural numbers.
Are we going to have to add a new axiom every time we describe aleph-null-more numbers? The Axiom of Replacement can help us here. The axiom of replacement allows us to construct new ordinals without end. No problem! But now think about all of these ordinals. There are definitely more real numbers, in a mathematically precise way, than there are natural numbers. Cantor began by assuming that you could create the bijection, listing all real numbers in some order, just like we did for the rational numbers, in order to match them with the natural numbers.
To create this number, he takes the first decimal place of our first number on our list and adds 1 to it, and uses the result as the first decimal place of his new number. If the first decimal place of our first number was 9, he changes it to a 0. He then takes the second decimal place of our second number and adds 1 to it, using the result as the second decimal place of his new number.
Use this interactive to pair up some natural numbers with the beginning of real numbers. We do this by taking one digit from each real number in a diagonal pattern and incrementing it by one.
Is he right? This is in contrast to the decimal expansions of rational numbers, which must eventually repeat themselves. Different infinite sets can have different cardinalities, and some are larger than others. It may seem esoteric, but the understanding of infinity—and set theory—is vital to understanding the very foundations of mathematics.
Beyond infinity Expert reviewers. Does every infinite set have the same size? Can they all be matched up exactly? Despite intuitions, we can count and compare the size of infinite sets. Rational thinking The first way we can expand our number system is by introducing all the negative numbers, forming the set of integers GLOSSARY integers The set of all whole numbers, both positive and negative.
Rather, they employ different ways to think about it in order to get at its many aspects. For instance, there are different sizes of infinity. This was proven by German mathematician Georg Cantor in the late s, according to a history from the University of St Andrews in Scotland. Cantor knew that the natural numbers — that is, whole, positive numbers like 1, 4, 27, 56 and 15, — go on forever.
They are infinite, and they are also what we use to count things, so he defined them as being "countably infinite," according to a helpful site on history, math and other topics from educational cartoonist Charles Fisher Cooper. Groups of countably infinite numbers have some interesting properties.
For instance, the even numbers 2, 4, 6, etc. And while there's technically half as many of them as what's encompassed by the full set of natural numbers, they're still the same kind of infinite.
In other words, you can place all the even numbers and all the natural numbers side by side in two columns and both columns will go to infinity, but they are the same "length" of infinity.
That means that half of countable infinity is still infinity.
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