An uncountable countable set
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From: mueckenh on 25 Sep 2006 06:45 Mike Kelly schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Mike Kelly schrieb: > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > Mike Kelly schrieb: > > > > > > > > > It is meaningful to say that a natural drawn uniformly at random from a > > > > > set of consecutive naturals 1 thru 3n has a 1/3 probabaility of being > > > > > divisible by 3. Nobody disputes this. But talking about the probability > > > > > of "a natural" being divisible by 3 implies a uniform distribution over > > > > > the naturals. Such a thing does not exist. > > > > > > > > Talking about sinx / x for x --> 0 does not imply the existence of sin0 > > > > / 0. Neither does the result 1/3 imply the distribution for a realy > > > > infinite set f naturals. There is no real (actual, finished) infinity, > > > > neither in physics > > > > > > Ok. > > > > > > >nor in mathematics. > > > > > > Why? > > > > Because all we are and all we think with (brain, neurons, currents, > > loads, and ideas, letters, words, pictures, i.e., hardware and > > software) and all we think and all we think we are thinking: all that > > is physics. > > So it is impossible to think of something that does not physically > exist? Of course. Whatever yo think does exist physically as a thought in your head. Be it the famous unicorn or an infinite set. Only the numbers completing this set do not exist - not all of them, because you are unable to think them. Regards, WM
From: mueckenh on 25 Sep 2006 07:01 Mike Kelly schrieb: > > > > So lim [n-->oo] n/aleph_0 < 1 > > > > > > Division is not defined for infinite cardinal numbers. > > > > Is that your only escape? If you dare to say that aleph_0 > n for any > > n e N, then we can conclude the above inequality. > > No, because division is not defined on infinite cardinal numbers. The > above inequality is meaningless. For cardinals we have well defined n*aleph_0 = aleph_0 for any natural n (see any book on set theory). Multiply with 1/n for any natural n, then you get well defined A n e N: aleph_0 / n = aleph_0 > 1. Hence, my followin statement is correct. > > >But remedy is easy. > > Take lim [n-->oo] aleph_0 / n > 1 and reverse the following fractions > > analogously. > > > [...] > Your position seems very inconsistent. You claim that numbers have no > existence outside their representation. And now you are claiming there > exists a "true" arithmetic. It is obvious. You can verify it by experiment: II + III = IIIII. [...] > It is not a proof. Division is not defined where either operand is an > infinite cardinal number. But you can conclude n / aleph_0 < 1 by inserting aleph_0 > n which is definied *if aleph_0 is a number in trichotomy with natural numbers*. You cannot have both, assert that aleph_0 is a number larger than any n but on the other hand prohibit that the inequality n < aleph_0 be utilized. Regards, WM
From: MoeBlee on 25 Sep 2006 07:16 Han.deBruijn(a)DTO.TUDelft.NL wrote: > MoeBlee wrote: > > > Han de Bruijn wrote: > > > to mainstream mathematics and its illusionary "rigour". > > > > The rigor is in a recursive axiomatization with recursive rules of > > inference. You haven't shown that this is an illusion. > > That rigor turns out to be incompatible with useful infinitesimals. You think the rigor itself is incompatible? Why don't your read a book on mathematical logic? MoeBlee
From: mueckenh on 25 Sep 2006 07:27 Dik T. Winter schrieb: > > > No. There are "numbers" (exactly one) that can be indexed by that list > > > but that is not in that list. > > > > That is a false belief. And why should there be only one of those > > numbers? Why couldn't it be two at least or even ten? If you answer > > this question, you have a good chance to recognize that not even one > > exists that is not in that list but can be indexed by the numbers of > > that list. > > Because there is only one set that contains *all* natural numbers. Why? Your assertion is without proof. "There is a set ..." does not mean that there is only *one* set. Why should there be only one number 0.111... ? By what property is this 0.111... different from all the numbers in the list? And why can't there be more than one number with infinitely many digits? You cannot answer these questions because already one infinite set is a contradiction. > > > > But there is a single "number" that can be indexed by your list but not > > > covered by your list. 0.111... is one (as I defined it above). But it > > > is *not* a natural number. > > > > Therefore it cannot be indexed by natural numbers. Try to find out, why > > you insist on only one of those strange numbers. There could be > > infinitely many. Could they all be indexed by natural numbers?. > > There is only *one* set of natural numbers, so there is only *one* > sequence 0.111... that can be indexed using *all* natural numbers. Who said so? If, in fact, one set of all natural numbers could exist, then also many of them could exist. Think of the axiom of comprehension. Which index distinguishes 0.111... from all the numbers of the list? You cannot answer? So we cannot answer which index distinguishes the many different infinite digit sequences 0.111... from each other. > > > > Do you agree that the non-terminating list can index non-terminating > > > numbers? If the answer is no, why? > > > > Because it should index at least two different of those infinite > > numbers if it could index one of them. > > Why? Either: There is an index which distinguishes 0.111... from any number of the list. Or: There is a number which cannot be distinguished by indexes. But if there was one such number admitted, how could the existence of many of them be excluded? Regards, WM
From: mueckenh on 25 Sep 2006 07:34
Dik T. Winter schrieb: > In modern mathematics there are no self-evident truths. There are the > axioms that are the basic material to work with and the theorems that > follow from them. In that sense, Cantor's Grundsatz is completely > equivalent with an axiom as used in modern mathematics. And so, > translating it with "axiom" is neither wrong, nor misleading. It is wrong and misleading because Cantor was not a modern mathematician and did not accept "axiom" in the modern meaning of this notion. His "Grundsatz" is not subject to arbitrary choice. > > > > > > Although I disagree that > > > > > changing them leads to rubbish. > > > > > > > > because you have not yet understood what Cantor's truths are. > > > > > > Educate me. What are they? Pray provide sources. But your rubbish > > > is not my rubbish... > > > > Cantor's truths are self-evident truths which cannot be changed > > arbitrarily in contrast to the axioms of modern set theory. For instance: I + I = II. > > Yeah, whatever. Is this a reply to my question? That was my intention, yes. If I failed, you should improve the precision of your question. Regards, WM |