tag:blogger.com,1999:blog-9633767.post3466340874619491838..comments2024-06-19T18:23:57.007-07:00Comments on The Curious Wavefunction: An eternity of infinities: the power and beauty of mathematicsWavefunctionhttp://www.blogger.com/profile/14993805391653267639noreply@blogger.comBlogger6125tag:blogger.com,1999:blog-9633767.post-82945918336922663812010-12-15T12:25:49.527-08:002010-12-15T12:25:49.527-08:00Brilliant post! I was actually similarly introduce...Brilliant post! I was actually similarly introduced to the beauty of Cantor's arguments, in my freshmen year of high school. Needless to say, we were quite impressed.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-9633767.post-74678496764301381752010-12-07T22:22:14.028-08:002010-12-07T22:22:14.028-08:00"I would be interested in knowing why mathema..."I would be interested in knowing why mathematicians believe the consistency of ZF axiomatic set theory."<br /><br />Because it's convenient? You try coming up with a different set theory. It's exhausting!DNAuticshttps://www.blogger.com/profile/05575969140780530027noreply@blogger.comtag:blogger.com,1999:blog-9633767.post-37867358454757570902010-12-07T05:50:39.790-08:002010-12-07T05:50:39.790-08:00Thanks for the comments. I will check out the link...Thanks for the comments. I will check out the links. I would be interested in knowing why mathematicians believe the consistency of ZF axiomatic set theory. About Hilbert's Hotel, there should not be a problem with countably infinite rooms and countably infinite guests. It's amazing that you can get such fantastic results from relatively simple arguments.Wavefunctionhttps://www.blogger.com/profile/14993805391653267639noreply@blogger.comtag:blogger.com,1999:blog-9633767.post-10281894835893459972010-12-06T13:00:47.159-08:002010-12-06T13:00:47.159-08:00The diagonalization argument works in general for ...The diagonalization argument works in general for any set and its power set (set of all subsets), and it's possible to prove that the real numbers have the same size as the power set of the natural numbers.<br /><br />The tower of power sets of the natural numbers are the "beth" (beis if you're ashkenazi) numbers; and so the continuum hypothesis is the assertion that aleph-one = beth-one.<br /><br />There are some very interesting results, including a famous one proving that continuum cannot be aleph-sub-omega-six.<br /><br />As to what omega six is... I'd suggest reading about "ordinal" infinites, you get strange things like inifnity plus one not being equal to one plus infinity.DNAuticshttps://www.blogger.com/profile/05575969140780530027noreply@blogger.comtag:blogger.com,1999:blog-9633767.post-69062832470231716142010-12-06T12:50:34.360-08:002010-12-06T12:50:34.360-08:00Nice site, nice article.
Hilton's Hotel is no...Nice site, nice article.<br /><br />Hilton's Hotel is not a paradox. Empty rooms are created because there will be guests moving around looking for an empty room. In a hotel with infinite rooms and infinite guests occupying all rooms you cannot create empty rooms by moving guests.<br /> <br />PS I am aware that some travel agencies think the opposite regarding bookings.EKhttps://www.blogger.com/profile/05025141274204991267noreply@blogger.comtag:blogger.com,1999:blog-9633767.post-44094399355856039292010-12-06T08:50:58.883-08:002010-12-06T08:50:58.883-08:00Great post! I was a double major in applied analy...Great post! I was a double major in applied analysis and organic chemistry, and even though I went to grad school for chemistry, I miss the math. I always found the concept of differing infinities to be fascinating, and I had dreams of solving the continuum hypothesis. Too bad that many mathematicians think that it might be undecidable -- ah, the incompleteness theorem ...Anonymousnoreply@blogger.com