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I found this video posted by a Professor N. J. Wildberger, which contains the quote in the title. Is this true? I looked up arbitrary unions on Wikipedia but didn't see much there about problems. Could someone explain why we cannot take an arbitrary union of sets? If it is not true, could you go into some detail about 1) why he might say that and 2) why it is wrong; or provide a source doing so?

I'm familiar with the axiom of choice, ordinal numbers, well orderings, etc. and was watching this video for some background.

Thank you so much!

Andrés E. Caicedo
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JohnDoeVsJoeSchmoe
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    Don't listen to anything that Wildberger says about set theory. – Asaf Karagila Oct 21 '16 at 17:16
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    If you search the site for Wildberger, you’ll find some detailed discussions of why you should ignore him when he talks about set theory. – Brian M. Scott Oct 21 '16 at 17:23
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    There is no problem taking the union of an infinite number of sets. The set of the sets you want join has to be a set, but thats all. (This makes it impossible to form the union of "all sets" or something similar, which is a proper class and not a set). But I don't have the patience for the whole video, so I don't know what that guy is actually about. – Simon Oct 21 '16 at 17:24
  • @AsafKaragila As a naughty boy I just did something that I would certainly not have done if I had not read your comment. So watch out for the eventual counterproductivity of comments.:-) – drhab Oct 21 '16 at 17:52
  • @drhab: I have no idea what you had done, but as an expert in procrastination, I'm not easily intimidated by counterproducitivty. :-P – Asaf Karagila Oct 21 '16 at 18:04
  • @AsafKaragila I just listened (for the first time) to someone called Wildberger. This for only one reason: someone told me not to do that. Don't worry about me though. I am not harmed or infected. – drhab Oct 21 '16 at 18:10
  • "which is simply something you cannot do". Nonsense: many people do it, often and with great success. It's simply something he won't do :) – BrianO Oct 21 '16 at 18:39
  • He means it literally. You can not do anything infinitely because you can not have a point of having completed doing everything if there are an infinite number of things to do. Most mathematicians do not have a problem with that. He does, and in the first minute of this video admits he does not accept standard math knowledge i.e. set theory. You have to decide if you agree with him or not. If you accept set theory (or even if you just want to know what it is) you need to consider that set theory believes you CAN. – fleablood Oct 21 '16 at 22:32

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He is a strict constructivist, and does not accept the axioms that most mathematicians do - in particular, he refuses to accept the existence of infinite sets. So when he says that an infinite union is something that you cannot do, he means in his particular axiomatic system. While not technically wrong, nearly everyone disagrees with his choice of axioms.

Michael Biro
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    I will reiterate my view on Wildberger and the finitism that he espouses. http://math.stackexchange.com/questions/527248/refuting-the-anti-cantor-cranks/633490#comment1134435_527865 – Asaf Karagila Oct 21 '16 at 17:30