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How to explain infinity to kids

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Re: How to explain infinity to kids

#121

Post by octosphere »

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.

Yes but there could be a universe with weird physics that could permit an ice-cube to be in the center of the sun, and not just briefly, but for a substantial amount of time.

Re: How to explain infinity to kids

#122

Post by Izmaki »

TuringTest wrote: post #92
Izmaki wrote: post #28

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…

I believe that approach comes naturally to us software developers, as we tend to think about mathematics in procedural terms (which is how programs run in the execution environment) rather than in string rewriting (which is how mathematical proofs are made). I also consider infinity as a process that never finds its ending condition, my mental model is the "infinite loop". "ω" is not almost 0, that would be epsilon "…

This cleared up a bit. Thanks!

Re: How to explain infinity to kids

#123

Post by SideburnsOfDoom »

octosphere wrote: post #121
hrnnnnnn wrote: post #90

Earlier quoted context omitted.

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.

Yes but there could be a universe with weird physics that could permit an ice-cube to be in the center of the sun, and not just briefly, but for a substantial amount of time.

> there could be a universe with weird physics that could permit an ice-cube to be in the center of the sun

Could there be? This assumes that

1) different universes have different physics

2) there is at least 1 self-consistent rules of physics that allows this. I suspect that this is a harder problem.

It's fair to say that if our universe has uniform physics, then even if it is infinite, there are no solar ice-cubes anywhere in it.

Re: How to explain infinity to kids

#124

Post by octosphere »

SideburnsOfDoom wrote: post #123
octosphere wrote: post #121

Earlier quoted context omitted.

Yes but there could be a universe with weird physics that could permit an ice-cube to be in the center of the sun, and not just briefly, but for a substantial amount of time.

> there could be a universe with weird physics that could permit an ice-cube to be in the center of the sun Could there be? This assumes that 1) different universes have different physics 2) there is at least 1 self-consistent rules of physics that allows this. I suspect that this is a harder problem. It's fair to say that if our universe has uniform physics, then even if it is infinite, there are no solar ice-cubes…

> This assumes that

> 1) different universes have different physics

Yes this is the presumption. Remember infinity allows all possible skewing of our current physics and all possible permutations of it. Further reading: https://en.wikipedia.org/wiki/Multiverse

Re: How to explain infinity to kids

#125

Post by SideburnsOfDoom »

octosphere wrote: post #124
SideburnsOfDoom wrote: post #123

Earlier quoted context omitted.

> there could be a universe with weird physics that could permit an ice-cube to be in the center of the sun Could there be? This assumes that 1) different universes have different physics 2) there is at least 1 self-consistent rules of physics that allows this. I suspect that this is a harder problem. It's fair to say that if our universe has uniform physics, then even if it is infinite, there are no solar ice-cubes…

> This assumes that > 1) different universes have different physics Yes this is the presumption. Remember infinity allows all possible skewing of our current physics and all possible permutations of it. Further reading: https://en.wikipedia.org/wiki/Multiverse

> Remember infinity allows all possible skewing of our current physics and all possible permutations of it.

No, it does not. Compare:

> If there are infinite numbers, and all numbers have a name, then there must exist a number with a different kind of name to the usual ones.

> If there are infinite universes, and all universes have laws of physics, then there must be universes with different laws of physics.

It's the same basic fallacy. There might universes with different physics but it does not follow from infinity.

Re: How to explain infinity to kids

#126

Post by enedil »

navane wrote: post #102
romwell wrote: post #100

Earlier quoted context omitted.

You are correct. Counting here changes its meaning. 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 le…

What I hear you say is that you can count to finite numbers in finite time, and infinite numbers in infinite time. That doesn't help me in understanding infinity.

The key point here is that you shouldn't be thinking of counting as a process that happens in time. How is infinity (here, omega) defined? We just say that it comes after any other natural number. So how to think about it? You shouldn't tell yourself:

- ok, take number n, it's smaller than omega, so what about n+1, n+2, etc. ?

You should be telling yourself - ok, take number n. It's smaller than omega. Now check any other number m. If this property holds for any given number, then omega is greater than any given number, and thus infinite.

Re: How to explain infinity to kids

#127

Post by enedil »

peter30744 wrote: post #57

I am not a mathematician, but I know that at least in standard mathematics it is fairly ill-advised to treat infinity implicitly as a number. Doing so can result in various contradictions (two different seemingly valid solutions to a problem). It should be thought of as a property of a process, i.e. for any number x you can change the process so that it results in a number greater than x. One example of infinity migh…

Now you're conflating two different notions. You're (and your quotes) are talking about the first bullet point from the article. In turn, when taking about the last one, the one chosen by article author, from my experience, I'd say that it's beneficial not to even consider it in time, i.e. don't think about it as process, don't think of it as something completed. There's no completion, since there was no process to begin with. The object just is, and we can manipulate it according to the rules.

Re: How to explain infinity to kids

#129

Post by enedil »

romwell wrote: post #25
hutzlibu wrote: post #8

Why so complicated? Just ask them to do i++ and never stop. That is also infinity. (or in simpler words, imagine you have a table than you put one apple to it, then one again, amd again and again and never stop. And yes, they would also soon understand, that the table needs to be infinitly big) edit: of course it is about the sum of the process. they should imagine the pile of things or the number if you never stop a…

>Why so complicated? Because ifinity is complicated. >Just ask them to do i++ and never stop. That is also infinity. What, specifically, is infinity here? You are being vague here. Is it the process of counting up and never stopping? (that's the calculus infinity) The number you get in the end? (that's the infinite ordinal, but that is, as you noted, complicated ) The number of numbers ? (that's the cardinality of th…

Answer to your question number 3 is: well, if I have just one apple, with label 3.1415926535 then obviously it will fit. Notice that no two apples have the same labels.
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