balderdash9

@balderdash9@lemmy.zip

I’m mostly half-serious.

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balderdash9,

Might make your boss do a double-take lol

balderdash9,

You’re the boss

balderdash9,

To establish whether one set is of a larger cardinality, we try to establish a one-to-one correspondence between the members of the set.

For example, I have a very large dinner party and I don’t want to count up all the forks and spoons that I’ll need for the guests. So, instead of counting, everytime I place a fork on the table I also place a spoon. If I can match the two, they must be an equal number (whatever that number is).

So let’s start with one $1 bill. We’ll match it with one $100 bill. Let’s add a second $1 bill and match it with another $100 bill. Ad infinitum. For each $1 bill there is a corresponding $100 bill. So there is the same number of bills (the two infinite sets have the same cardinality).

You likely can see the point I’m making now; there are just as many $1 bills as there are $100 bills, but each $100 bill is worth more.

balderdash9, (edited )

You’re right that we don’t need to, but mathematicians can use this method to prove that two infinite sets are the same size. This is how we know that the infinite set of whole numbers is the same size as the infinite set of integers. We can also prove that the set of real numbers is larger than the set of whole numbers.

https://lemmy.zip/pictrs/image/d69ea099-c7d8-4ef8-b20f-ab73e5b7c5ab.webp

I’m not quite sure how else to explain it, so I’ll link a Numberphile video where they do the demonstration on paper: www.youtube.com/watch?v=elvOZm0d4H0&t=19s . Here you can see why it’s useful to try to establish this 1-1 correspondence. If you can’t do so, then the size of the two infinite sets are not equal.

balderdash9,

You can map 1000 one dollar bills to every single 100 dollar bill and never run out, even in the limit, and therefore conclude (equally incorrectly) that the infinite $1 bills are worth more, because infinity isn’t a number.

Yes, the mapping that you’re describing isn’t useful; but that doesn’t invalidate the simple test that we can do to prove that two infinite sets are the same. And the test is pretty intuitive in some cases. If for every fork there is a spoon, then they must be an equal number. And if for every positive whole number (e.g “2”) there is a negative whole number (e.g. “-2”), then the set of positive whole numbers and the set of negative whole numbers must be the same.

Another thing that seems unintuitive but might make the concept in general make more sense is that you cannot add or do any other arithmetic on infinity.

I agree with you that adding infinity to itself just gets you the same number. But now we’re starting to ask: just what is infinity? We could think of infinity as a collection of things or we could think of infinity like we do numbers (yes, I know infinity is not technically among the real numbers). In the latter sense we could have a negative infinity {-1…-2…-3…etc.} that is quantitatively “worth less” than a countable positive infinity. On the other hand, if we think of infinity as a collection of stuff, then you couldn’t have a negative infinity because you can’t have less than zero members in the set.

I’ll admit that I’m getting out of my depth here since I know this stuff from philosophical study rather than maths proper. As with anything, I’m happy to be proven wrong, but I’m quite sure about the 1-1 correspondence bit.

balderdash9,

Good catch, I’ll edit that sentence

balderdash9, (edited )

I understand that both sets are equally infinite (same cardinality). And I do see the plausibility in your argument that, in each case, since there’s an infinite number of bills their value should be equally infinite. If your argument is correct, then I should revise my understanding of infinity. So maybe you can help me make sense of the following two examples.

First, the number of rational numbers between 0 and 1 is countably infinite. That is, we can establish a 1-1 correspondence between the infinite set of fractions between 0-1 and the infinite set of positive integers. So the number of numbers is the same. But clearly, if we add up all the infinite fractions between 0 and 1, they would add up to 1. Whereas, adding up the set of positive integers will get us infinity.

Second, there are equally many positive integers as there are negative integers. If we add up the positive integers we get positive infinity and if we add up all the negative integers we get negative infinity. Clearly, the positive is greater than the negative.

In these two cases, we see that a distinction needs to be made between the infinite number of members in the set and the value of each member. The same arguably applies in the case of the dollar bills.

EDIT: I see now that I was mistaken.

balderdash9, (edited )

~~I was thinking that, bill for bill, the $100 bill will always be greater value. But I can see the plausibility in your argument that, when we’re counting both the value of the members of each set, the value of the $100 bill pile can always be found somewhere in the series of $1 bills. The latter will always “catch up” so to speak. But, if this line of reasoning is true, it should apply to other countably infinite sets as well. Consider the following two examples.

First, the number of rational numbers between 0 and 1 is countably infinite. That is, we can establish a 1-1 correspondence between the infinite set of fractions between 0-1 and the infinite set of positive integers. So the number of numbers is the same. But clearly, if we add up all the infinite fractions between 0 and 1, they would add up to 1. Whereas, adding up the set of positive integers will get us infinity.

Second, there are equally many positive integers as there are negative integers. There is a 1-1 correspondence such that the number of numbers is the same. However, if we add up the positive integers we get positive infinity and if we add up all the negative integers we get negative infinity. Clearly, the positive is quantitatively greater than the negative.

In these two cases, we see that a distinction needs to be made between the infinite number of members in the set and the value of each member. The same arguably applies in the case of the dollar bills.~~

EDIT: I see now that I was mistaken.

balderdash9, (edited )

I meant to say that we can infinitely divide the numbers between 0 and 1 and then match each with an integer. But I realized that the former wouldn’t be rational numbers, they would be real numbers.

That aside, I see now that the original idea behind the meme was mistaken.

balderdash9, (edited )

I meant to say that we can infinitely divide the numbers between 0 and 1 and then match each with an integer. But I realized that the former wouldn’t be rational numbers, they would be real numbers.

That aside, I see now that the original idea behind the meme was mistaken.

balderdash9,

For what it’s worth, people actually taking the time to explain helped me see the error in my reasoning.

balderdash9,

This is a good moment to remind the Lemmy devs to add more fucking mod tools

balderdash9,

I love that someone edited Tammy’s outfit for this lol.

balderdash9,

How about instead of cheating on my mom, teach me how to talk to girls… and don’t cheat on my mom lol

balderdash9,

https://lemmy.zip/pictrs/image/9044c6f9-74e5-4bb5-b2ce-3361973203d8.webp

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