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How to explain infinity to kids

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Re: How to explain infinity to kids

#111

Post by n4r9 »

SideburnsOfDoom wrote: post #101
sandov wrote: post #77

As a kid, I asked my mom if all numbers have a name, she said yes, and then I asked the obvious follow-up question: If there are infinite numbers, and all numbers have a name, then there must exist a number called door, another one named airplane, because at some point you would run out of numbery names, I don't remember what she said, but she didn't answer the question. That question was unresolved for me until rece…

> If there are infinite numbers, and all numbers have a name, then there must exist a number called door, another one named airplane, because at some point you would run out of numbery names, I don't remember what she said, but she didn't answer the question. Ah, this is the philosophical distinction between every possible thing, and every conceivable thing. The set of possible number names may be very large or infin…

I think the distinction is really between "A given infinite set" and "every conceivable thing". It's a common fallacy to equate one infinity with any other infinity, and therefore any given infinity must contain all things. This fallacy is very neatly encapsulated by the phrase

>If there are infinite numbers, and all numbers have a name, then there must exist a number called door

Re: How to explain infinity to kids

#112

Post by shoo »

wwweston wrote: post #85
jpulgarin wrote: post #78

Earlier quoted context omitted.

If you're referring to real numbers, then virtually no real numbers have a name. If all real numbers had a name then you could order them alphabetically and put them in one-to-one correspondence with the natural numbers, which we know is impossible: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

Does this apply if names are infinitely long?

[edit: for clarity i'm not going to use the term "name"]

the proof crucially relies upon each real number being represented as an infinitely long (perhaps ending in an infinite sequence of repeated digits for less interesting numbers) sequence of digits. i.e. the representation that Cantor uses for reals is infinite sequences -- effectively representing each real number infinitely long (in the countable sense) strings.

Cantor represents each real number as an infinite sequence of digits. apart from technical details`+` you can think of this as the base 10 or base 2 expansion of the number.

for simplicity, you need only consider counting the real numbers between 0 and 1. there are plenty enough of those. each such number can be addressed in base 2 as some infinite sequence of 1s and 0s after the binary point. Cantor's argument shows that if you try to enumerate all such numbers (i.e. count them) by their series expansion, then you can generate a new number that doesnt apppear in the enumerated list, which is a proof by contradiction, refuting the assumption that you could enumerate them all in the first place.

`+` :

technical details include that some numbers have non-unique representations as infinite sequences of digits.

For example, in base 10, 1 can be represented as 1.000... (where the 0s keep going) or 0.999... (where the 9s keep going).

if this irritates you, let z = 0.999...

then 10 * z = 9.999... then 10 * z - z = 9 then 9 * z = 9 so z = 1

Re: How to explain infinity to kids

#113

Post by n4r9 »

mhneu wrote: post #67

A professor of mine once said to me that all teaching was a process of lying, and then of replacing the lies with successively better approximations of the truth. This is the opposite of what Feynman thought. He said, and I agree, something like "the hard part about teaching is making concepts simple without saying things that are false." The whole challenge of teaching is NOT lying, but instead saying simple things…

Feynman might have been sympathetic to the idea that all of science is a process of "replacing the lies with successively better approximations of the truth". Certainly Asimov was:

https://chem.tufts.edu/answersinscience/relativityofwrong.ht...

Re: How to explain infinity to kids

#114

Post by Retric »

hrnnnnnn wrote: post #90
cyborgx7 wrote: post #83

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.

Physically it’s possible for a ice cube to show up in the center of the sun. Each nucleus in the center of the sun is stripped it’s electrons but that’s the average view in shorter time frames they can have all of those electrons just very briefly. Three such atoms could collide just right and form a water molecule in the center of the sun. From their it’s just increasingly unlikely situations but that’s not enough in the face of unlimited random variations.

Re: How to explain infinity to kids

#116

Post by SideburnsOfDoom »

azeirah wrote: post #110
SideburnsOfDoom wrote: post #101

Earlier quoted context omitted.

> If there are infinite numbers, and all numbers have a name, then there must exist a number called door, another one named airplane, because at some point you would run out of numbery names, I don't remember what she said, but she didn't answer the question. Ah, this is the philosophical distinction between every possible thing, and every conceivable thing. The set of possible number names may be very large or infin…

> A number called "door" is conceivable (like superman) but it is not a possible under the rules of maths (or physics in superman's case) because "door" is not in that set of symbols. Certainly you are able to represent "door" in a base-64 numbering system? (or base-36, or even base-28)

Ah, but "a base-64 numbering system" is a different set of rules ;)

And it's rules in which "door" is a "numbery name". And "Достоевский" is not.

You could still ask "If there are infinite base-64 numbers, and all base-64 numbers have a name, then must there exist a number called "Достоевский" ?" and the same argument against would apply.

Re: How to explain infinity to kids

#117

Post by udkyo »

cyborgx7 wrote: post #84
kirubakaran wrote: post #80

Earlier quoted context omitted.

A million seconds is over 11 days straight without stopping for sleep, food, or anything! I very much doubt that they actually did it.

If you count one a second, you're counting very slowly. Also, you can take breaks and pick it back up, making a note where you left off. I did it this way once, growing up.

If you count one a second, you're counting very slowly.

That stops being true long before you reach the finish line. Try timing yourself counting from 147,895 to 147,900 and see how long it takes.

Re: How to explain infinity to kids

#118

Post by cyborgx7 »

udkyo wrote: post #117
cyborgx7 wrote: post #84

Earlier quoted context omitted.

If you count one a second, you're counting very slowly. Also, you can take breaks and pick it back up, making a note where you left off. I did it this way once, growing up.

If you count one a second, you're counting very slowly. That stops being true long before you reach the finish line. Try timing yourself counting from 147,895 to 147,900 and see how long it takes.

fair enough

Re: How to explain infinity to kids

#119

Post by cobbal »

navane wrote: post #96
dnate wrote: post #95

> 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.

It's two different processes. Normal counting is saying, "take a thing and add one" (a successor ordinal). It's how we get from 42 to 43. If a number can be reached from zero using just this kind of counting, we call it finite.

The new type of counting also allows you to take a different step and say, "take a collection of things and add a new thing at the end" (a limit ordinal). It's how you get from the set of finite numbers to ω. There's nothing that's "one before" ω, but all the finite numbers are before it.

Repeat these steps as much as you want (making sure none of your sets are circularly defined), and you're counting with ordinals.

Re: How to explain infinity to kids

#120

Post by injb »

From the article: “Imagine taking all the numbers that you could reach by counting,” I said. “Then add one more, after all of them. That is infinity.”

How is 'adding one more' not the same thing as counting?

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