An uncountable countable set
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From: Virgil on 24 Jul 2006 16:03 In article <1153764073.036121.173100(a)s13g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Mathematics does not exist other than in reality. Therefore it *is* > reality (like everything which really *is*). There are only some > persons who think that they need no reality for thinking. But that is > thoughtless thinking. Mathematics is all thinking, it is all in the mind. Though we have learned to record our thoughts in physical ways, it is in those thoughts themselves that mathematics exists, and nowhere else.
From: Randy Poe on 24 Jul 2006 16:06 mueckenh(a)rz.fh-augsburg.de wrote: > Randy Poe schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Of course. Real space is Euclidean or non-Euclidian irrespective of the > > > axioms which mathematicians prefer. Aithmetics is finite or infinite > > > irrespective of the axioms which mathematicians prefer. > > > > You mean there's something called "arithmetic" which > > exists outside mathematics? > > It is the core of mathematics. So the core of mathematics is outside mathematics? Strange property of a "core". - Randy
From: Franziska Neugebauer on 24 Jul 2006 17:45 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > >> Dear Wolfgang! >> >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > Franziska Neugebauer schrieb: >> > >> >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> >> >> > >> >> > Dik T. Winter schrieb: >> >> > >> >> >> With the axiom of infinity there are infinitely many finite >> >> >> numbers. And I still do not know the exact difference between >> >> >> actual infinite >> >> >> and potential infinite. But that is (in my opinion) something >> >> >> that is best left to philosphy. >> >> > >> >> > In set theory "infinity" means "actual infinity". Potential >> >> > infinity can neither be counted nor be surpassed. >> >> >> >> _In_ set theory "actual" or "potential" have no meaning. You >> >> produce meaningless verbiage. >> > >> > Obviously you have not the foggiest idea about set theory. >> >> I'm not keen on ancient set theory, that's true. >> >> > Here are few quots from many: Fraenkel, Abraham A., Levy, Azriel: >> > "Abstract Set Theory" (1976) >> > p. 6 the set of all integers is infinite (infinitely comprehensive) >> > in a sense which is "actual" (proper) and not "potential". >> >> 1. Page 6 (six)! What's the name of the Chapter? "Preface"? >> 2. "in a sence". You know what that means? An eidetic elucidation. >> 3. What this sentence states is: omega exists. >> 4. Where do Fraenkel/Levy use (!) the aforementioned terms >> _mathematically_? > > In set theory. It is full of the actual infinite. When I ask for "usage of terms" I want you to name contemporary papers/books on set thery that mention/use these terms. > Set theory is theory of the transfinite. Transfinite is actually > beyond the finite. > > Potentially infinite means always finite but without border. I don't ask for an elucidation of the terms but of accepted and widespread use in current literature. >> > One may doubt whether this example really illustrates the abyss >> > between finiteness and actual infinity. >> >> One may also doubt that you understood my objection. _In_ (not in the >> elucidation of) set theory there is no meaning of "actual" or >> "potential". > > So you have no clue of set theory. Not on /ancient/ set theory, that's true. > I suspected that. Consider at the > begining for instance the axiom of infinity or the cardinal number of > the reals which is larger than the infinite cardinal numbe of the > naturals. You did not understand my objection. In contemporary literature on set theory /I/ cannot see any widespread use of these terms. >> > p. 240 Thus the conquest of actual infinity may be considered an >> > expansion of our scientific horizon no less revolutionary than the >> > Copernican system or than the theory of relativity, or even of >> > quantum and nuclear physics. >> > >> > Potential infinity existed already long before set theory. >> >> Where is the mathematical use of the aforementioned terms? > > In set theory! You will certainly find means in the world wide web which enable you to substanciate your claim by giving us statistical evidence. F. N. -- xyz
From: Dik T. Winter on 24 Jul 2006 18:50 In article <1153750179.424604.45850(a)75g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > With the axiom of infinity there are infinitely many finite numbers. > > And I still do not know the exact difference between actual infinite > > and potential infinite. But that is (in my opinion) something that is > > best left to philosphy. > > In set theory "infinity" means "actual infinity". Potential infinity > can neither be counted nor be surpassed. Read Cantor where he wrote that w is not actual infinity but potential infinity, because you could not count to it. About the surpassing I am not so sure. > > > The proof is given by the list: > > > > > > 1 0.1 > > > 2 0.11 > > > 3 0.111 > > > ... ... > > > w 0.111... > > > > That is neither a proof, nor a list. > > The list is given in the upper part. Call the whole an extended list. > It shows that w is not the number of the naturals, because w appears in > the sequence of ordinal numbers only once. That is only a statement, *not* a proof. > Here we see, that it appears > only after all naturals and measures the cardinality of the extended > list, not of the list of the naturals alone. The cardinality of the "list" above (including w) is indeed aleph-0 (as would be the cardinality of the "list" without w. The ordinality of the "list" (with w) = w + 1. But that is completely irrelevant, and not a proof. > > > The set of unary numbers cannot result in the *+sum 0.111... . (I > > > defined: the *+ sum is 1 if there is at least one 1 in it.) With > > > respect to the left column this means the cardinality of 1,2,3,... > > > cannot be w without w in it. > > > > You again state such without a proof, prove it, assuming that there is > > no last element. Let's start with the list: > > A1 = 0.1 > > A2 = 0.11 > > A3 = 0.111 > > assuming decimal notation (because in that way we can define everything > > we need in the easiest possible way). Now it is easy to define *+ on > > two operands, Ap *+ Aq = Ap when p > q else it is Aq. And now it is > > easy to do a finite set of operations, and we can proof that: > > (*+){i = 1 .. n} Ai = An > > no problem so far. We get a problem when we try to define the operation > > for *all* Ai. As there are not finitely many Ai, we can not use the > > finite case to extend to all elements of the list. > > The finite case applies as long a only finite numbers are concerned. > How should a rule which is proven for any finite number become invalid > for any other finite number? No, the finite case applies as long as only finitely many finite numbers are concerned. As you have not given a definition of (*+){i = 1 .. oo} Ai I have no idea how to calculate it. Nevertheless you want to do that. > > In my opinion the > > only sane way to define the *+ sum of all elements is to define it as > > lim{n -> oo} An > > but you apparently disagree. > > That is only a convention. We do not know if it holds. In order to find > out, my list was designed. Therefore we cannot use it as input. What do you mean here? We do not know if it holds when it is not defined. And as long as it is not defined we have no idea what it means. Above I suggest one definition. With that definition it certainly holds. Without that definition I have no idea. > > Whatever, with this definition we get that > > (*+){i = 1 .. oo} Ai = 1/9 > > and 1/9 is not equal to any of the Ai. > > We cannot arbitrarily forget that for all finite numbers (*+){i = 1 .. > n} Ai = An. Because this is not an arbitrary definition but a result of > the precise defintion of the *+sum of a column, namely to be 1 if at > least one 1 is present. Therefore your arbitrary definition(*+){i = 1 > .. oo} Ai = 1/9 cannot be correct, because 1/9 is not equal to any of > the Ai. I do not see why it cannot be correct. If you have a finite list, the *+ sum is the last element of the list. If you have an infinite list, there is no last element, so the *+ sum can not be the last element. And indeed, even the assumption that the *+ sum of the list (say K) is one of the elements (say An) leads to a contradiction. Because K *+ A_(n+1) is not equal to K but equal to A_(n+1), so the assumption is false. You are continuously ignoring that (*+){i = 1 .. oo} Ai is an ongoing process that will never be complete, if we do the individual *+ sums. You *can not* count to w. > > Except that there is no digit position with ordinal number w in the > > standard decimal notation. > > Yes, just that is the problem. All numbers of the true list have a > finite number of digits. If it is claimed that all digits of 0.111... > can be indexed, then 0.111... has omega 1's and then it belongs to the > position omega and then omega is necessary to index it. The last conclusion requires proof (and it is false). What is the case is that for every Ai the indexing of digits will terminate at some point, and the indexing of digits of 0.111... will never terminate when we start at the first digit and go on. Because there will always be a further digit. (As there will always be a further An.) > > > > In both cases, when you insert the *+ sum early in the list you get either > > > > (case 1) a non-natural number in the list, so it makes no sense, or > > > > (case 2) a different rational in the list, and in this case the *+ sum > > > > is that rational. But in the latter case I see no problem. > > > > > > In the list there are only terminating rationals. 0.111... has omega > > > digits. Therefore we see perfect parallels. > > > > Again, you make no sense here. I think you are arguing (case 2). If > > you add the *+ sum in the list the list no longer contains only terminating > > rationals. > > Forall terminating rationals we have a terminating *+sum. For all finite lists of terminating rationals we have a terminating *+ sum. For infinite lists this is not necessarily the case (as it is in general not the case with other operations). E.g. 1/4 + 1/16 + 1/64 + ... is (using the common definitions of limits etc.) 1/3. > > Apparently you are thinking that 0.111... is a terminating > > rational. > > It has two different definitions, w
From: Dik T. Winter on 24 Jul 2006 19:03
In article <1153750477.050796.204330(a)s13g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1153674914.649461.71180(a)m73g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > In article <1153478871.303029.5360(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > > But it is obvious for *all* *finite* *unary* number. And only > > > > > finite unary numbers are involved in the second list below: > > > > > > > > > > The *- sum of > > > > > > > > > > 0.1 > > > > > 0.11 > > > > > 0.111 > > > > > ... > > > > > 0.111... > > > > > > > > 0.111... is not a finite unary number. Rather, I would say it is not a > > > > unary number at all. > > > > > > 0.111.. is not a terminating rational number. > > > > Eh, can we please keep to the subject. You state you have a list that > > only finite numbers are involved in the "list" above. I state that > > 0.111... is not a finite unary number. Pray show that 0.111... is a > > finite unary number. (I neither have stated that it is a terminating > > rational number, because it is not.) > > The pure list stretches only from top to the three ... The last element > does not beong to it. Sorry, you stated "the second list below". As the first list you gave terminated with "...", I can only conclude that your second "list" includes 0.111... . > My reply concerned your statement that 0.111... was not a unary number > at all and the difference between decimal and unary interpretation you > could perhaps try to derive from your observation. Indeed, it is not a unary natural number. Perhaps it is the unary representation of something else, but not of a natural number. And seen as a rational, it is not the representation of a terminating rational number, but it is the representation of a rational number. The difference is that natural numbers always have a terminating unary representation (by definition), while rational numbers not always have a terminating (decimal) representation. However, let us consider further. Seeing the list of unary numbers and the operation *+, we can state that given any finite subset of natural numbers S, the *+ sum of them is nothing more, nor less, than max{n in S} n. As infinite subsets of N have no maximum, it should be immediately clear that the *+ sum on such sets is not defined. Let's now look at them as decimal representations. We find that Ak = (1 - 10^(-k))/9. Given any finite subset S of N we find that (*+){k in S} Ak = (1 - 10^(-(max{k in S} k))). Again, not defined for infinite subsets, for exactly the same reason. You use both cases above, switching between the two at will, but in neither case there is a proper definition of what it should mean with infinite lists. See my earlier response (where I look at them as strings of symols) how you *could* define them. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |