Pardon my ignorance, but I wasn’t aware that the “strange” results dealing with infinite sets of various cardinality were related to the “strange” results related to accepting the axiom of choice. Is this a limitation of my mathematics education, or are the infinite set “paradoxical” results independent of the axiom-of-choice sphere cutting “paradoxical” results?
To really push my understanding of the terminology, I thought that definitions of equivalent size for infinite sets based on one-to-one and onto correspondence did not require reference to the axiom of choice.
Alternatively, I’m not understanding the implications I’m supposed to get from:
How about this? Take the set of all natural numbers. Divide it into two sets: the set of even naturals, and the set of odd naturals. Now you have two infinite sets, the set {0, 2, 4, 6, 8, …}, and the set {1, 3, 5, 7, 9, …}. The size of both of those sets is the ω - which is also the size of the original set you started with.
Now take the set of even numbers, and map it so that for any given value i, f(i) = i/2. Now you’ve got a copy of the set of natural numbers. Take the set of odd naturals, and map them with g(i) = (i-1)/2. Now you’ve got a second copy of the set of natural numbers. So you’ve created two identical copies of the set of natural numbers out of the original set of natural numbers.
Pardon my ignorance, but I wasn’t aware that the “strange” results dealing with infinite sets of various cardinality were related to the “strange” results related to accepting the axiom of choice. Is this a limitation of my mathematics education, or are the infinite set “paradoxical” results independent of the axiom-of-choice sphere cutting “paradoxical” results?
To really push my understanding of the terminology, I thought that definitions of equivalent size for infinite sets based on one-to-one and onto correspondence did not require reference to the axiom of choice.
Alternatively, I’m not understanding the implications I’m supposed to get from:
You are correct. As several commenters have already pointed out, the provided explanation of the Banach-Tarski paradox is just bad.