tag:blogger.com,1999:blog-1522013117369989006.comments2013-05-15T05:05:54.994+03:00Turing CompleteHarry Zisopouloshttps://plus.google.com/103086974555362960954noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-1522013117369989006.post-4153778560686706312013-05-15T05:05:54.994+03:002013-05-15T05:05:54.994+03:00Hello and thanks for your comment! When I was writ...Hello and thanks for your comment! When I was writing this the proofs I could find online either proofs that simply gave the rational spiral (thus informal) or proofs that either handwaved the uniqueness of the canonical form or went into quite some trouble to prove it.<br /><br />The point I want to make is that you can form a bijection with a subset of a countable set and still have a valid proof. This is similar to reducing a problem to another problem, where the instances you get from the reduction have a specific structure and do not cover all possible instances of the problem you reduce to.<br /><br />Perhaps the above is a trivial example, but the same proof idea could be used extensively.<br /><br />I find very interesting what you mention about an infinite cardinality smaller than aleph-naught. I would assume this is the smallest one by definition. Any pointers to such a proof or a discussion of this issue would be very interesting!Harry Zisopouloshttp://www.blogger.com/profile/09364120444779754928noreply@blogger.comtag:blogger.com,1999:blog-1522013117369989006.post-43829447386408732322013-05-13T10:22:31.619+03:002013-05-13T10:22:31.619+03:00Am I missing something, or did you just replace a ...Am I missing something, or did you just replace a method based on constructing a set S of fractions followed by taking a subset of S to get countability with a method based on constructing a set S of fractions followed by taking a subset of S to argue countability?<br /><br />I don't really see what this accomplishes. In both cases, you're hand-waving away the argument that there is no infinite cardinality smaller than aleph-naught.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-1522013117369989006.post-45059242560247705392011-03-29T11:30:57.284+03:002011-03-29T11:30:57.284+03:00Thank you for your answer. I appreciate your attit...Thank you for your answer. I appreciate your attitude that you give anyone the chance to read the work. I will read it :)<br /><br />I wish you success for your diploma thesis.Marchttp://cstheory.stackexchange.com/users/2517/marc-gillenoreply@blogger.comtag:blogger.com,1999:blog-1522013117369989006.post-88441275456177788492011-03-29T09:27:49.190+03:002011-03-29T09:27:49.190+03:00Hello Marc and thank you for your interest. Greek ...Hello Marc and thank you for your interest. Greek legislation requires that all papers that grant a degree must be in Greek.<br /><br />However, I believe that any work should be able to be read by anyone. Therefore, especially if I am successful in producing a new result, I will translate the diploma thesis to English.chazisophttp://www.blogger.com/profile/09364120444779754928noreply@blogger.comtag:blogger.com,1999:blog-1522013117369989006.post-46540328074509455052011-03-28T17:30:38.161+03:002011-03-28T17:30:38.161+03:00Hi,
I know you from TCS.SE where I find the link ...Hi,<br /><br />I know you from TCS.SE where I find the link to your blog, too. In your profile you say that the topic of your diploma thesis is the time hierarchy theorem. Do you write your diploma thesis in english or greek?<br />I find the topic very interesting. If you write it in english or publish some results I would be happy if I can read it :)<br /><br />Best regards<br />MarcMarchttp://cstheory.stackexchange.com/users/2517/marc-gillenoreply@blogger.comtag:blogger.com,1999:blog-1522013117369989006.post-88748243039463689492011-03-08T19:57:15.235+02:002011-03-08T19:57:15.235+02:00Well, reflecting on proofs, it is good to have man...Well, reflecting on proofs, it is good to have many as long as they use different approaches, so that we have a better understanding of the statement proven, why it is true etc. On textbooks of course, criteria for selecting a specific proof is its comprehension and how it relates to the rest of the material on the book.<br /><br />However, algorithms must also be practical to use in order to be useful. So a new protocol or algorithm, in my opinion, should justify its existence and how it offers something different (or more efficient) than what already exists.<br /><br />In the reverse direction, we have to trust in the good intentions and integrity of the author, so that he doesn't advertise one protocol over the other, especially when the given technologies are new. An author gains a lot of respect in my eyes when he justifies why his pedagogical approach, especially when multiple choices are available.chazisophttp://www.blogger.com/profile/09364120444779754928noreply@blogger.comtag:blogger.com,1999:blog-1522013117369989006.post-88216178948783625652011-03-08T01:52:06.116+02:002011-03-08T01:52:06.116+02:00I think it's fine to have multiple algorithms;...I think it's fine to have multiple algorithms; the problem is putting them all in a textbook (not a reference book) as though they're all crucially important for someone learning the subject. Consider the various different proofs of the infinitude of primes, the uncountability of the reals, the fundamental theorem of algebra. Of course, it's rare to find more than one of any of these in the same book!Xamuelhttp://www.xamuel.comnoreply@blogger.com