Your opponents might require the number you name to be a cardinal that refers to an amount. These numbers refer to the same amount of stuff, just arranged differently. Well, because it can be shown that there are infinities bigger than aleph-null that literally contain more things.
In my episode on the Banach-Tarski paradox, I used it to show that the number of real numbers is larger than the number of natural numbers. The power set of a set is the set of all the different subsets you can make from it. For example, from the set of 1 and 2, I can make a set of nothing, or 1, or 2, or 1 and 2.
The power set of 1,2,3 is: the empty set, 1, and 2, and 3, and 1 and 2, and 1 and 3, and 2 and 3, and 1,2,3. 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. All the different ways to arrange aleph-null things. Well these are well-ordered, so they have an order type—some ordinal that comes after all of them.
It could be equal to aleph-one—that belief is called the Continuum Hypothesis. The Continuum Hypothesis, by the way, is probably the greatest unanswered question in this entire subject, and today, in this video, I will not be solving it—but I will be going higher and higher, to bigger and bigger infinities. Wherever I land will be a place of even bigger numbers, allowing me to make even bigger and more numerous jumps than before. The whole thing is a wildly-accelerating feedback loop of embiggening.
We can keep going like this, reaching bigger and bigger infinities from below. Replacement, and repeated power sets which may or may not line up with the alephs, can keep our climb going forever. Not so fast. Why not accept as an axiom that there exists some next number so big, no amount of replacement or power-setting on anything smaller could ever get you there. Not even close to aleph-null, the first smallest infinity. For this reason, aleph-null is often considered an inaccessible number.
We will have to do the same for inaccessible cardinals. Set theorists have described numbers bigger than inaccessibles, each one requiring a new large-cardinal axiom asserting its existence, expanding the height of our universe of numbers.
Will there ever come a point where we devise an axiom implying the existence of so many things that it implies contradictory things? Will we someday answer the Continuum Hypothesis? That would mean that we have, with this brain, a tiny thing, a septillion times smaller than the tiny planet it lives on, discovered something true outside of the physical realm.
Something that applies to the real world, but is also strong enough to go further, past what even the universe itself can contain, or show us, or be. Another interesting fact about trans-finite ordinals is that arithmetic with them is a little bit different. One plus omega is actually just omega. While other Vsauce contributors came and went, Kevin kept creating.
Slowly the video game comedy theme of Vsauce matured into science and technology. Kevin gauged the audience's interests, interacted with fans and fine-tuned his videos to fit a wider audience. After three years of deep-frying chicken wings and cleaning bars while making videos in his spare time, Kevin began working on Vsauce2 full-time in Kevin is based in New York, where he makes episodes that inspire and entertain by examining the everyday amazing… and the not-so-everyday amazing.
From mind-blowing technology to the origins of the things around us, Vsauce2 is a hub for utmost awesomeness in human creation. Jake sneakily worked his way into editing a DONG video and the rest is internet history.
Jake is based out of New York where he writes, hosts and edits his episodes. Vsauce3 covers the real world science of fictional worlds: books, video games, and movies. Jake believes everything is interesting. There are amazing lessons to learn and facts to find in all aspects of life, even virtual ones.
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