An uncountable countable set
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From: MoeBlee on 18 Sep 2006 15:37 Han de Bruijn wrote: > MoeBlee wrote: > > > Tony Orlow wrote: > > > >>Unbounded but finite may > >>be considered potentially, but not actually, infinite. > > > > That will be jiffy, once you give axioms and/or our definitions for > > 'potentially infinite' and 'actual infinite'. Until then, it's pure > > handwaving. > > Come on, Moeblee, don't be silly! Let Google be your friend! I'm well aware of the notions of 'potential infinity' and 'actual infinity' that go back through at least a couple thousand years of philosophy and philosophy of mathematics. But to use those notions in an axiomatic mathematics requires either defining the terms 'potentially infinite' and 'actually infinite' in the axiomatic theory or taking them as primitive and giving axioms for them in the axiomatic theory. So far, that has not been done in this thread or in any other thread I've happened to read. Meanwhile, rather than direct you to an Internet search, I recommend 'The Philosophy Of Set Theory' by Mary Tiles for more about 'potential infinity' and 'actual infinity' and debates through history about infinity. MoeBlee
From: MoeBlee on 18 Sep 2006 15:43 Han.deBruijn(a)DTO.TUDelft.NL wrote: > Mike Kelly wrote: > > > Infinite natural numbers. Tish and tosh. Good luck explaining that idea > > to schoolkids. > > Look who is talking. Good luck explaining alpha_0 to schoolkids. Not that what is understandable to children should be determinative, but your analogy is irrelevent on the terms of the original point, which is that as far as explaining counting numbers, infinite natural numbers are not as understandable as finite natural numbers. Going on to ANOTHER matter, viz. infinite cardinals, is not relevant to the original comparison between the notions of finite natural numbers vs. infinite natural numbers. MoeBlee
From: Randy Poe on 18 Sep 2006 16:09 Han.deBruijn(a)DTO.TUDelft.NL wrote: > Mike Kelly wrote: > > > Infinite natural numbers. Tish and tosh. Good luck explaining that idea > > to schoolkids. > > Look who is talking. Good luck explaining alpha_0 to schoolkids. I think I was 10 when I saw the proof that the rationals are countable, and first saw the notation "aleph_0". I don't remember having a problem with it. - Randy
From: Mike Kelly on 18 Sep 2006 16:23 Han.deBruijn(a)DTO.TUDelft.NL wrote: > Mike Kelly wrote: > > > Infinite natural numbers. Tish and tosh. Good luck explaining that idea > > to schoolkids. > > Look who is talking. Good luck explaining alpha_0 to schoolkids. > > Han de Bruijn Sure, the theory of infinite cardinals is beyond (most)schoolkids. But this is a bad analogy, because school kids don't need to know about cardinals but they do need to know how to work with natural numbers. My point, if you really missed it, was that Tony's ideas of "infinite natural numbers" don't match up to our "naive" or "intuitive" idea of what numbers should be - how we were taught to do arithmetic in school. I for one don't understand what the hell an "infinite natural number" is. And yet supposedly the advantage of his ideas are that they're more intuitive than a standard formal treatment. -- mike.
From: Mike Kelly on 18 Sep 2006 16:38
Han de Bruijn wrote: > Mike Kelly wrote: > > > Han de Bruijn wrote: > > > >>Mike Kelly wrote: > >> > >>>Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>> > >>>>What I meant to express is that you are about to be parrotting > >>>>mainstream arguments, without adding to it much thoughts of > >>>>yourself. And that is quite senseless because we have gone > >>>>through all this already. > >>> > >>>Yes, and your position was utterly ripped apart in the very first > >>>response to you by David C. Ullrich, and then by several others. You > >>>were unable to defend your claim. So, why repeat it as though it were > >>>in any way valid? > >> > >>That's because I did a _honest attempt_ to fit my position into the > >>framework of nonstandard analysis (Robinson's theory), which failed. > > > > Your claim was that *standard* set theory + calculus contradicts > > *standard* probability theory. This is untrue. Do you admit it? > > Of course not. Of course you don't admit it? Even though you're *wrong*? Let me try to paraphrase the contradiction you claim you percieve between *standard* calculus and *standard* probability theory both built upon *standard* set theory. In probability theory, we say that there is no uniform distribution over the naturals because a countably infinite sum of a constant cannot sum to 1, so it is not a probability measure. If the constant is 0 it sums to 0 and otherwise it doesn't sum to a number at all. You claim that this contradicts with calculus because in calculus infinitely many 0s can sum to a positive number. You point to the Reimann integral as an example. Is this an accurate paraphrase? Anything you would reword? If not : your argument is bullshit. Integration isn't the same thing as summation. Infinitely many 0s sum to 0 in calculus, too. > >>Of course it failed. Mainstream mathematicians have never understood > >>how infinitesimals work. > > > > NSA is part of mainstream mathematics. What mathematics have never > > understood is why some people are so enarmoured of vigorous handwaving > > as a form of mathematical argument. > > Huh, huh. How cute. Take a look at yourself and your aleph_0s. Looking. I see no handwaving. Maybe you could point it out? -- mike. |