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That's the clearest, most concise explanation I've seen for how different infinite sets can have different cardinalities. Thanks!
by lukasb 11y ago
That's the clearest, most concise explanation I've seen for how different infinite sets can have different cardinalities. Thanks!
- yongjik 11y agoWait, but between n and n+1 there are also an infinite number of rational numbers, but the cardinality of all rational numbers is the same as that of integers (or that of all rational numbers between a fixed n and n+1). So, while what the gp wrote is correct, it's not an explanation of why there are more real numbers than integers.
- mikeash 11y agoI don't think the summary in the comment is correct, though. There are also an infinite number of rational numbers between any given integer n and n+1, and indeed an infinite number of rational numbers between any two rational numbers, yet the cardinality of the rationals is still the same as the cardinality of the integers. Diagonalization is brilliant, though!
- buzzdenver 11y agoIt's a little confusing though, at least for me. There are infinite integers, and "doubly-infinite" reals. But there are also only "doubly-infinite" complex numbers. IMO the easiest way to illustrate different, and infinite number of, infinities is to think of sets. Suppose you have N-infinite number of things, say triple-infinite to use the grandparent's lingo. Now the number of sets that you can create from these triple-infinite objects is greater than triple-infinite, and it's called quad-infinite. You can prove that indirectly. Assume that there is a one to one mapping of the original object to the sets. That means that some of the objects are mapped to sets that the objects are members of (let's call these red objects), others not (let's call them blue objects). Does the set of blue objects get paired with a blue or a red object ? Cannot be either, so we've reached a contradiction, ergo there are always more subsets of things than things.