Professor Hamman and Class,
I forgot to get clarification on my question from class yesterday. I asked, "at what point does a set become too large to count, making it not a set?" From the reading, I can see that the answer is Russell's Paradox, but I'm not sure I understand when/at what point/what value a set turns in to NOT a set. I may have missed the answer to this in class. Help!
When you can no longer say it is the same size as N, then it is considered uncountable. So all you do is check and see if you can find a one-to-one correspondence between the two.
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ReplyDeleteI definitely grasp that concept. I don't know if what I'm asking is important or necessary. I guess I need to define when a paradox starts. I get that infinity has a 1 to 1 correspondence and that the real numbers do not have a 1 to 1 correspondence, making it uncountable. At what point does a (P(R)) or even larger power set become a paradox? Or does it? I am going to stop now because I feel a strong urge to ramble on... Julian, you're a smart dude - HELP ME!!
ReplyDeleteA set does not become a paradox by repeatedly taking power sets of itself. You can do this as much as you want and still have a working set. However, you cannot have a set be one of its own elements. This is what makes Russell's paradox a paradox.
ReplyDeleteSorry about my original comment. I just re-read your question and realized I was reading it as a different question at first.
ReplyDeleteGreat question, but a little tricky to answer so I thought I would add my comments here. This question is akin to the description I gave in class a few weeks ago. Imagine you have a pile of sand and you remove a grain of sand. Clearly what you have left is a pile of sand. However, if you keep repeating this process you are left with a few grains of sand (clearly not a pile). It would be impossible to try to determine exactly when that pile stopped being a pile, even though it clearly did.
ReplyDeleteWe face a similar problem when working with collections that are really big. I know all the power sets are really sets, but the collection in Russell's paradox is a class (in fact a proper class = NOT a set). To truly show that other collections are classes we resort to our old friend, the one-to-one correspondence. If we can find a corsepondence to a known proper class, then the new collection is also a class. This sort of work is beyond the scope of this class, but the idea is relevant. I know this is not as direct an answer as you were hoping for but I hope it gives some insight.
Aha!! My light bulb just flickered! I get it now. Thank you guys for clearing that up for me.
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