@anotherusername said in Stupid things I've seen on Yahoo! Answers (but I repeat myself):

@jaloopa sure, but that requires knowing something additional about the two infinite sets that you're comparing.

For instance, you need to know that "the set of positive numbers greater than 4325632434531" is mathematically defined as such, and doesn't simply look like it but coincidentally also happen to contain all of the numbers between 999 and 999+1 (rational and irrational -- making this now an uncountably infinite set).

Just knowing that they're both infinite doesn't tell you the cardinality of the sets. And really, saying that they're the same "size" is sort of a misnomer, as it implies that you can measure it. You measured the cardinality... the size is still infinite.

"Size" is ambiguous, so I chose a definition that's well known and specific when talking about infinities: cardinality. Underspecified question gets arbitrary valid answer.

@anotherusername said in Stupid things I've seen on Yahoo! Answers (but I repeat myself):

@dreikin said in Stupid things I've seen on Yahoo! Answers (but I repeat myself):

you're going to have to cut it in half

Nobody said that was a requirement. It's not even possible. An infinite object doesn't even technically have a midpoint, because the definition of a midpoint is that there's an equal length on both sides, and to determine equality you need finite, measurable numbers. You can't just take two infinite objects and say that they're the same size because they're both infinite.

Sure it's possible. If it's a rod with finite cross-section extending infinitely in both directions, any point you choose to cut it at will result in two rays with the same cardinality. Cardinalities are comparable. Because of that, I can say two rays with the same cardinality are equal in size. Of course, that they are equal in size is just a bonus: like you said, it's not a requirement. And that requirement is still met, even if we start with a ray extending infinitely in one direction: there was one part. Now there are parts.