The first trans-finite ordinal is omega (ω), the lower-case Greek letter omega. Aleph-null plus one? That may sound surprising, since fractions appear more numerous on the number line, but as Cantor showed, there’s a way to arrange every single possible rational such that the naturals can be put into a one-to-one correspondence with them. Replacement, and repeated power sets which may or may not line up with the alephs, can keep our climb going forever. Or heck, why not use a bigger ordinal like ω2 to construct aleph-omega-squared? Beyond the infinity known as ℵ 0 (the cardinality of the natural numbers) there is ℵ 1 (which is larger) … ℵ 2 (which is larger still) … and, in fact, an infinite variety of different infinities. Normally 2+1 is the same as 1+2, but ω+1 is not the same as 1+ω. Now using the replacement axiom, we can take any ordinal we already reached—like, say, ω—and jump from aleph to aleph all the way out to aleph-omega. You see, this line doesn’t contribute to the total, but in order to label it according to the order it appeared in, well we need a set of labels of numbers that extends past the naturals. The infinity symbol is written with the Lemniscate symbol: ∞ It represents an infinitely positive big number. This is just called ‘epsilon-naught’ (ε0), and we continue from here. So clearly there’s nothing beyond them, right? Going all the way out to ω+ ω would be to create another infinite set, and the axiom of infinity only guarantees that this one exists. Unlike cardinal numbers (1, 2, 3 and so on), which tell you how many things are in a set, ordinals are defined by their positions (first, second, third, etc. Instead, it’s true because we say it is. Share on Twitter. The whole thing is a wildly-accelerating feedback loop of embiggening. Cantor later went crazy, probably for reasons unrelated to his work on infinity, according to Cooper. Now remember that after ω, ordinals split, and these numbers are no longer cardinals. Well in reality, the biggest number is 40. ), and they were also introduced into mathematics by Cantor, according to the math website Wolfram MathWorld. We need ordinal numbers. Even if we put this new subset back in, diagonalization can still be done. You will receive a verification email shortly. But aleph-null isn’t the end. But some infinities are literally bigger than others. 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. Here’s the subset of just 3, 7 and 12. The symbol ω is defined as the thing that comes after all the other natural numbers — or, as Cantor called it, the first transfinite ordinal. How to count past infinity There is no biggest, last number … except infinity. 10 Ways to Count Past Infinity. Your opponents might require the number you name to be a cardinal that refers to an amount. Both sets have the cardinality aleph-null. Another interesting fact about trans-finite ordinals is that arithmetic with them is a little bit different. But for the purposes of this video, let’s focus on another thing bigger than aleph-null: the power set of aleph-null. If you're counting something and you never reach the end, that's infinity. 20 is the ‘cardinality’ of this set of dots. "What the math makes you ask is, 'What does that really mean?''' Hello! Two to the power of however many members the original set had, to be exact. But what happens when I do this? But you can't treat infinity as a number. No matter how long an arrangement becomes, as long as it’s well-ordered, as long as every part of it contains a beginning element, the whole thing describes a new ordinal number. are also countably infinite. If, even then, there’s a way to keep producing new subsets that are clearly not listed anywhere here, we will know that we’ve got a set with more members than there are natural numbers—a bigger infinity than aleph-null. A googolplex? Thank you for signing up to Live Science. Next, move diagonally-down to 1’s membership in the second subset. I can even add another infinite aleph-null of lines and still not change the quantity. ω+1 isn’t bigger than ω, it just comes after ω. Mathematicians, on the other hand, don't often think of infinity as a concept on its own, he added. Different infinite sets can have different cardinalities, and some are larger than others. As you can see, a power set contains many more members than the original set. If you refuse to accept it, that’s fine—that makes you a finitist, one who believes only finite things exist. 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