Why do some infinities seem bigger than others, and how do mathematicians compare their sizes?
In: Mathematics
Because our mathematical definitions give rise to these implications.
I would recommend VSauce’s excellent (https://youtu.be/SrU9YDoXE88?t=100) on infinite cardinals and ordinals.
TL;DW:
Cardinality describes the “size” of a set, how many things are in the set. Two sets have the same cardinality if they can be put into one-to-one correspondence, if there exists a bijection between then. For finite sets, the cardinality of the set is just a natural number (“The set of primary colors has 3 members.” `3` is the cardinal). But for infinite sets, we also want a way to talk about their sizes (and their size can’t be any finite number), so we use other symbols to represent infinite cardinals.
Now why are we justified in talking about infinite sets? Who says infinite sets exist? Well, we do. Math is an invention of the human mind. We invent axiom systems of our choice, picking and choosing which axioms we want in our mathematical framework, and then we explore what consequences must necessarily follow. The most popular and commonly accepted axiom system within which we “do math” is called ZFC, and in it, we define “infinity” into existence via the axiom of infinity, which simply declares the existence of an infinite set, one with the cardinality of the natural numbers (`ℕ`), which we name `ℵ₀` (“aleph null”). We say this is the first and smallest infinity, a countable infinity. We call it “countable” because it’s the cardinality of the “counting numbers,” as we use the natural numbers for counting.
In asserting an infinite set exists, we’re not saying this concept corresponds with anything in the real physical world (i.e., formal math doesn’t care if there are actual infinities in the real world), we’re simply saying “Let’s say there is a set with infinite members. Let’s just say there is one, and see where that takes us, how the rest of our mathematical framework reacts to that, and see if we don’t end up contradicting ourselves.”
Using other axioms, we can show there are larger infinities than `ℵ₀`. The axiom of power set instantly guarantees there is a set of things with cardinality larger than that of ℕ: it’s the power set of ℕ, the set of all subsets of the natural numbers. This is also the cardinality of the reals. This is what you would call an “uncountable infinity,” because sets of this size cannot be put into one-to-one correspondence with the naturals. I.e., you can’t “count” its members by lining all members up, going down the line, and pointing at each, saying “You’re #1, you’re #2, you’re #3,” etc.
This difference in sizes of infinities leads to some really interesting and deeply profound consequences that get right to the heart of mathematics. The undecidability and incompleteness of ZFC (and pretty much any formal system), the fact that there are real numbers that exist in a formal sense but cannot be computed (uncomputable numbers), even some that cannot be defined (undefinable numbers) are all indirectly related to this difference in size of infinites. In short, the set of all strings is countable, while the set of all languages is uncountable.
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