this post was submitted on 23 Oct 2024
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[–] [email protected] 5 points 1 week ago (5 children)

Natural numbers being infinite, how it be possible for the values between 1 and 2 to be "more infinite" ?

[–] [email protected] 13 points 1 week ago (4 children)

It's called countable and uncountable infinity. the idea here is that there are uncountably many numbers between 1 and 2, while there are only countably infinite natural numbers. it actually makes sense when you think about it. let's assume for a moment that the numbers between 1 and 2 are the same "size" of infinity as the natural numbers. If that were true, you'd be able to map every number between 1 and 2 to a natural number. but here's the thing, say you map some number "a" to 22 and another number "b" to 23. Now take the average of these two numbers, (a + b)/2 = c the number "c" is still between 1 and 2, but it hasn’t been mapped to any natural number. this means that there are more numbers between 1 and 2 than there are natural numbers proving that the infinity of real numbers is a different, larger kind of infinity than the infinity of the natural numbers

[–] [email protected] 1 points 6 days ago

This reminds me of a one of Zeno's Paradoxes of Motion. The following is from the Stanford Encyclopaedia of Philosophy:

Suppose a very fast runner—such as mythical Atalanta—needs to run for the bus. Clearly before she reaches the bus stop she must run half-way, as Aristotle says. There’s no problem there; supposing a constant motion it will take her 1/2 the time to run half-way there and 1/2 the time to run the rest of the way. Now she must also run half-way to the half-way point—i.e., a 1/4 of the total distance—before she reaches the half-way point, but again she is left with a finite number of finite lengths to run, and plenty of time to do it. And before she reaches 1/4 of the way she must reach 1/2 of 1/4=1/8 of the way; and before that a 1/16; and so on. There is no problem at any finite point in this series, but what if the halving is carried out infinitely many times? The resulting series contains no first distance to run, for any possible first distance could be divided in half, and hence would not be first after all. However it does contain a final distance, namely 1/2 of the way; and a penultimate distance, 1/4 of the way; and a third to last distance, 1/8 of the way; and so on. Thus the series of distances that Atalanta is required to run is: …, then 1/16 of the way, then 1/8 of the way, then 1/4 of the way, and finally 1/2 of the way (for now we are not suggesting that she stops at the end of each segment and then starts running at the beginning of the next—we are thinking of her continuous run being composed of such parts). And now there is a problem, for this description of her run has her travelling an infinite number of finite distances, which, Zeno would have us conclude, must take an infinite time, which is to say it is never completed. And since the argument does not depend on the distance or who or what the mover is, it follows that no finite distance can ever be traveled, which is to say that all motion is impossible. (Note that the paradox could easily be generated in the other direction so that Atalanta must first run half way, then half the remaining way, then half of that and so on, so that she must run the following endless sequence of fractions of the total distance: 1/2, then 1/4, then 1/8, then ….)

[–] [email protected] 5 points 1 week ago

Great explanation by the way.

[–] [email protected] 3 points 1 week ago (3 children)

I get that, but it's kinda the same as saying "I dare you!" ; "I dare you to infinity!" ; "nuh uh, I dare you to double infinity!"

Sure it's more theoretically, but not really functionally more.

[–] [email protected] 5 points 1 week ago* (last edited 1 week ago)

It's like when you say something is full. Double full doesn't mean anything, but there's still a difference between full of marbles and full of sand depending what you're trying to deduce. There's functional applications for this comparison. We could theoretically say there's twice as much sand than marbles in "full" if were interested in "counting".

The same way we have this idea of full, we have the idea of infinity which can affect certain mathematics. Full doesn't tell you the size of the container, it's a concept. A bucket twice as large is still full, so there are different kinds of full like we have different kinds of infinity.

[–] [email protected] 4 points 1 week ago

When talking about infinity, basically everything is theoretical

[–] [email protected] 1 points 1 week ago (1 children)

but not really functionally more.

Please show me a functional infinity

[–] [email protected] 1 points 6 days ago (1 children)

Right, an asymptote I guess, in use, but not a number.

[–] [email protected] 1 points 6 days ago

It's been quite some time since I did pre-calc, but I remember there being equations where it was relevant that one infinity was bigger than another.

[–] [email protected] -2 points 1 week ago (3 children)

Your explanation is wrong. There is no reason to believe that "c" has no mapping.

[–] [email protected] 1 points 6 days ago

because I assumed continuous mapping the number c is between a and b it means if it has to be mapped to a natural number the natural number has to be between 22 and 23 but there is no natural number between 22 and 23 , it means c is not mapped to anything

[–] [email protected] 2 points 1 week ago

Give me an example of a mapping system for the numbers between 1 and 2 where if you take the average of any 2 sequentially mapped numbers, the number in-between is also mapped.

[–] [email protected] 2 points 1 week ago* (last edited 1 week ago)

Yeah, OP seems to be assuming a continuous mapping. It still works if you don't, but the standard way to prove it is the more abstract "diagonal argument".

[–] [email protected] 8 points 1 week ago (1 children)

It's weird but the amount of natural numbers is "countable" if you had infinite time and patience, you could count "1,2,3..." to infinity. It is the countable infinity.

The amount of numbers between 1 and 2 is not countable. No matter what strategies you use, there will always be numbers that you miss. It's like counting the numbers of points in a line, you can always find more even at infinity. It is the uncountable infinity.

I greatly recommand you the hilbert's infinite hotel problem, you can find videos about it on youtube, it covers this question.

[–] [email protected] 1 points 1 week ago

Because the second one is bounded ?

[–] [email protected] 3 points 1 week ago* (last edited 1 week ago)

Basically, if two quantities are the same, you can pair them off. It's possible to prove you cannot pair off all real numbers with all integers. (It works for integers and all rational numbers, though)

How many infinities you accept as meaningful is a matter of preference, really. You don't even have to accept basic infinity or normal really big numbers as real, if you don't want to. Accepting "all of them" tends to lead to contradictions; not accepting, like, 3 is just weird and obtuse.

[–] [email protected] 0 points 1 week ago

I thought the same but there is a good explanation for it which I can't remember

[–] [email protected] -1 points 1 week ago (1 children)

I'm confused as well. Isn't that like saying that there is more sand in a sandbox than on every veach on the planet?

[–] [email protected] 3 points 1 week ago

We're talking about increasingly smaller fractions here. It's more like saying if you ground up all the rocks on earth into sand you would have more individual pieces of sand than individual rocks.