How many is infinite

And, of course, we can use smaller and smaller intervals, and thus show that the rational numbers have no length at all! Bizarre, right? Like Like. You are commenting using your WordPress. You are commenting using your Google account. You are commenting using your Twitter account. You are commenting using your Facebook account.

Notify me of new comments via email. Notify me of new posts via email. Skip to content In our very first posts we talked about how big infinity is , and how there is more than one size of infinity — countable infinity, which is infinite but you could count it, and larger, uncountable infinities, which are so large that you cannot count them. How do we measure the length of infinity? Great, now for the complicated part.

This shows that the measure of any finite set is zero. What happens if we move to infinite set? The length of countable infinity is always zero! What about larger infinities?

In fact, there are infinitely many different sizes of infinity. For each size of infinity like the size of the real numbers , there is a larger infinity. We talked more about that in the post An even biggerer infinity. But anything times zero is zero, and anything times infinity is infinity.

So what should be? Zero or infinity? Or something else? The fact of the matter is that we need more information. Sometimes it makes sense to say it is zero, sometimes infinity, and sometimes some other number, like 7. We need to carefully use our definition of length in order to know which one is correct for this circumstance.

Thus this sum is 2. Share this: Twitter Facebook Print. Like this: Like Loading Will you be talking about nonmeasurable sets next post? Maybe not next post, but soon, I expect. They proved the two are in fact equal, much to the surprise of mathematicians. Malliaris and Shelah published their proof last year in the Journal of the American Mathematical Society and were honored this past July with one of the top prizes in the field of set theory.

But their work has ramifications far beyond the specific question of how those two infinities are related. It opens an unexpected link between the sizes of infinite sets and a parallel effort to map the complexity of mathematical theories.

The notion of infinity is mind-bending. But the idea that there can be different sizes of infinity? It emerges, however, from a matching game even kids could understand. In the late 19th century, the German mathematician Georg Cantor captured the spirit of this matching strategy in the formal language of mathematics. Perhaps more surprisingly, he showed that this approach works for infinitely large sets as well.

Consider the natural numbers: 1, 2, 3 and so on. The set of natural numbers is infinite. But what about the set of just the even numbers, or just the prime numbers? Each of these sets would at first seem to be a smaller subset of the natural numbers. And indeed, over any finite stretch of the number line, there are about half as many even numbers as natural numbers, and still fewer primes. Yet infinite sets behave differently. Because of this, Cantor concluded that all three sets are the same size.

After he established that the sizes of infinite sets can be compared by putting them into one-to-one correspondence with each other, Cantor made an even bigger leap: He proved that some infinite sets are even larger than the set of natural numbers.

Consider the real numbers, which are all the points on the number line. Because of this, he concluded that the set of real numbers is larger than the set of natural numbers. Thus, a second kind of infinity was born: the uncountably infinite. He guessed not, a conjecture now known as the continuum hypothesis. In , the German mathematician David Hilbert made a list of 23 of the most important problems in mathematics. He put the continuum hypothesis at the top. Do in-between infinities exist?

We may never know. Throughout the first half of the 20th century, mathematicians tried to resolve the continuum hypothesis by studying various infinite sets that appeared in many areas of mathematics.

But written as a decimal number the digit 3 repeats forever we say "0. So, when we see a number like "0. You cannot say "but what happens if it ends in an 8? This is why 0. A Googol is already bigger than the number of elementary particles in the known Universe, but then there is the Googolplex.

It is 1 followed by Googol zeros. I can't even write down the number, because there is not enough matter in the known universe to form all the zeros:. For example, a Googolplex can be written as this power tower: That is ten to the power of 10 to the power of ,.

But imagine an even bigger number like which is a Googolplexian. But none of these numbers are even close to infinity. Because they are finite, and infinity is We can sometimes use infinity like it is a number, but infinity does not behave like a real number.

Which is mathematical shorthand for " negative infinity is less than any real number, and infinity is greater than any real number".