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He has a pretty solid understanding of basic addition, subtraction and multiplication.
He was asking about infinity and although he knows it's very big he was still asking about infinity plus one. Maybe it's not too soon to explain the concept.
I don't understand this explanation, and I'm an adult with an interest in mathematics and a degree in software development (although not mathematical per se, they usually go hand-in-hand). I like how "w" is the smallest number you cannot count to - if I understand it right it's almost 0 but not quite - but I don't understand the bonus questions and answers. If this is the best "explain it like I'm 5" explanation of i…
"ω" is not almost 0, that would be epsilon "ε". Omega ω is a number higher than any other natural number, i.e. "to the right" of the infinite line of numbers.
Mathematicians "will" this number into existence, so to say, starting from a contradiction. It's the same that they do with irrational numbers ("imagine there's this number "i" that, when multiplied by itself, it gives you -1"). With infinity, it's like: "you know this process that never ends? Well, imagine that it finishes, and let's call the result "ω".
Once they have this new number defined, they do lots of mathematical operations with it, trying to find its properties. What they never remember again after that is that the number dit not appear as the result of following the initial process to completion; they had to assume that it existed independently from the process.
* BTW, this is also why they have different kinds of infinities. They are using different never-ending processes in their respective definitions, and using the same name for all of them.
Earlier quoted context omitted.
The difference between "infinite" and "all possible" seems to be really unintuitive. The example I got to is that there are infinite even numbers, but not all natural numbers are even. This also maniests itself when people talk about infinite universes. "If there are infinite universes, there has to be one in which X". Not necessarialy. Maybe only even universes exist and universes in which X is the case might all be…
I heard a really good interpretation of this recently. In all these infinite universes, surely there's one where there's an ice cube in the middle of the sun, since that's one possible configuration of matter. But no! There's no story that ends with an ice-cube in the middle of the sun. There's no sequence of events that could have resulted in that so it's not in the space of possible universes.
One scenario I would not rule out is a large icy comet moving at large speed straight for a sun. Is there no combination of size and speed for which that would result in a small ice cube in the centre of a sun?
Earlier quoted context omitted.
The difference between "infinite" and "all possible" seems to be really unintuitive. The example I got to is that there are infinite even numbers, but not all natural numbers are even. This also maniests itself when people talk about infinite universes. "If there are infinite universes, there has to be one in which X". Not necessarialy. Maybe only even universes exist and universes in which X is the case might all be…
I heard a really good interpretation of this recently. In all these infinite universes, surely there's one where there's an ice cube in the middle of the sun, since that's one possible configuration of matter. But no! There's no story that ends with an ice-cube in the middle of the sun. There's no sequence of events that could have resulted in that so it's not in the space of possible universes.
If you teleport a block of ice to coordinates which happen to be smack dab inside a star, then maybe for an instant, like a single "tick" of the processing engine, there would be an ice cube in the middle of the sun.
I see a problem with saying that. One of my earliest troubles when dealing with infinity in algebra was understanding, that you cannot add or subtract real numbers from infinity to make it something else.
e.g. inifnity - infinity is not 0. Suddenly saying that infinity + 1 \neq inifinity would just make it more confusing.
The article is a nice mind exercise but IMO not really helpful in explaining infinity to a child.
> Instead we can decisively say that there is another number after infinity, which is called “infinity plus one”. I see a problem with saying that. One of my earliest troubles when dealing with infinity in algebra was understanding, that you cannot add or subtract real numbers from infinity to make it something else. e.g. inifnity - infinity is not 0. Suddenly saying that infinity + 1 \neq inifinity would just make i…
Disclaimer: I'm an adult that doesn't understand infinity. If you "add one more" you're still counting.
Picking the first infinite ordinal as the infinity to explain to kids might not be the best choice for every kid, though. Oridnals are tricky. I am not too comfortable with them myself, and I say that as an adult with a degree in math! The thing is, this kind of infinity just doesn't come up that often when dealing with other objects in math. Even though, as Tom Lehrer sang, one can count up to infinity - or somewher…
> The other kinds of infinity - cardinals, for example - are encountered early on, and there are things you can do with them. There's plenty you can do with ordinals too! Being able to interate a function transfinitely many times can be quite useful. > So, I might be biased in that, but I think that the cardinals are the most playful type of infinity. Definitely disagree. Once you know the basics, doing things with c…
>There's plenty you can do with ordinals too
Any examples that you could introduce to a kid who just asked you "What is infinity?" - genuinely curious.
If we're going for useful lies to tide kids over, I quite like the explanation in the Postgres docs: > infinity (date, timestamp) later than all other time stamps I think you could tell a kid who wasn't quite ready for Aleph numbers that: > infinity is a useful made-up number that's bigger than all other numbers Which is useful when kids first hear about it, I guess. I think my first "practical" introduction to infin…
> infinity is a useful made-up number that's bigger than all other numbers I love the emphasis on this being a choice, and I wish this kind of thinking was taught more in math. So many math explanations act like these things are immutable facts of the universe, rather than human constructions. We lose some of the history and character of math when we teach it as law rather than invention. Zero is also a useful made-u…
but that's much the same as "It's the smallest number you can't count to."
> Instead we can decisively say that there is another number after infinity, which is called “infinity plus one”. I see a problem with saying that. One of my earliest troubles when dealing with infinity in algebra was understanding, that you cannot add or subtract real numbers from infinity to make it something else. e.g. inifnity - infinity is not 0. Suddenly saying that infinity + 1 \neq inifinity would just make i…
“Imagine taking all the numbers that you could reach by counting,” I said. “Then add one more, after all of them. That is infinity.” Disclaimer: I'm an adult that doesn't understand infinity. If you "add one more" you're still counting.
The correct form should be "all the numbers reachable by counting in finite time".
E.g. you can count up to 100 in a minute. Up to a million in a month[1]. Up to a billion in quite a long, but finite time. You take all such numbers, and you say that a number named Omega comes just after all of them (just like million and one comes just after all numbers that are less than or equal to ine million).
Omega is your first infinte ordinal - or, simply, infinity.
And now you're counting in a new way.