Cantor Confusion
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From: Virgil on 3 Nov 2006 20:20 In article <1162562035.655584.297570(a)h54g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > David Marcus schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > The diagonal is an infinite sequence. So the diagonal is longer than > > > > > any of the finite sequences. But the diagonal consists of elements of > > > > > the finite sequences. So it cannot be longer than the maximum of the > > > > > finite sequences. If this maximum does not exist, you cannot take the > > > > > supremum omega for it, because the supremum is not a member of the > > > > > sequences and does not supply elements of the diagonal. > > > > > > > > Let's try a simpler problem. Consider the following list. > > > > > > I would hesitate to call it a list, because injectivity is lacking. > > > > I've never seen injectivity required for a list. Usually, people use > > "list" to mean sequence. If your lists require injectivity, you should > > explicitly state that. People can't read your mind. > > "List" is no a technical term but taken from civil life. There it is > used to list different items, not the same item twice or more. Then that is a very peculiar use of lists, which does not allow anything to be listed more than once. Can Wm point out anywhere in the following any requirement that no repetitions are alloWed? <begin quote> Websters Concise Electronic Dictionary 7 sense(s) for lists 1. list (noun) [lists] series of names or items (verb) [listed; listed; listing; lists] make a list of put on a list lister (noun) [listers] listing (noun) [listings] 2. list (homograph) (verb) [listed; listed; listing; lists] tilt or lean over (noun) [lists] slant Proximity/Merriam-Webster U.S. English Thesaurus 6 meaning(s) for lists 1. (noun) [plural of list] a series of items (as names) written down or printed especially as a memorandum or record (synonym) catalogs, registers, rolls, roll calls, rosters, schedules (related) checklist, handlist, index, inventory 2. (verb) [present 3rd person sing. of list] to enter in a list (synonym) books, catalogs, enrolls, inscribes (related) file, index, note, post, schedule, tabulate, record, register, roster 3. (verb) [present 3rd person sing. of list] to take in (as a person) by entering identification in a list, catalog, or roll (synonym) enrolls, registers (related) enter, insert, catalog, inscribe, record, enlist, line up, recruit, sign up, join, matriculate (contrast) discard, omit, reject 4. (verb) [present 3rd person sing. of list] to specify one after the other (synonym) enumerates, numerates, ticks off (related) run over, tell off, identify, mention, recite, recount, relate, specify 5. (verb) [present 3rd person sing. of list] to set down in detail or by particulars (synonym) itemizes, enumerates, inventories, particularizes, specializes, specifies (see also) specify (related) circumstantiate, document, count, number, cite, instance, mention, spell out (antonym) summarize 6. (verb) [present 3rd person sing. of list] to set or be set at an angle (synonym) slants, cants, heels, inclines, leans, reclines, slopes, tilts, tips (related) bank, decline, descend, bend, deviate, diverge, splay, swerve, veer Proximity/Franklin U.S. English Thesaurus 3 meaning(s) for lists 1. (noun) [plural of list] a sequential record of items (synonym) listings, catalogs, rolls, rosters, schedules 2. (verb) [present 3rd person sing. of list] to set at an angle (synonym) angles, biases, skews, slants, cants, inclines, leans, reclines, slopes, tilts, tips, pitches 3. (verb) [present 3rd person sing. of list] to note down items one by one (synonym) catalogs, enrolls, itemizes, enumerates, inventories, particularizes, registers, tallies, checks off, ticks off (related) note down, itemise, specify <end quote>
From: Virgil on 3 Nov 2006 21:22 In article <1162562518.437541.264190(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > If real numbers are to be represented by sequences then any two > > sequences which are "arbitrarily close" represent the same number. > > Therefore Cantor's proof is invalid. Non-sequitur. "arbitrarily close" means that the distance between them is smaller that any positive real number, which makes them equal, even in Robinson's non-standard analysis. > With increasing length of the > list, the difference introduced by exchanging the diagonal becomes > smaller and smaller. For an infinite list it vanishes at all. For which term of the sequence has it vanished entirely? > > > > > > In Cantor's list there are those unique representations required. > > > > Not so. Even in decimal, Cantor's diagonal rule allows for certain > > rationals having dual representation. > > Which two numbers could that be? WM misses the point, as usual. It is not two numbers at all, but one number with two representations. Like 1.000... and 0.999... in decimal arithmetic. And this will be the case for every rational q in a base b representation for which there exists any n in N with q*b^n in N. Which is a set of rationals dense in the set of all reals. > > > > > > > > > Therefore I do not understand why you say "Numbers are fixed entities". > > > They are merely defined by sequences. > > > > The sequence 1 + 1/2 + 1/4 + 1/8 + ...+ 1/2^n + ... "defines" a fixed > > number. That that number has other representations does not make that > > number into a variable quantity. > > The sequence 1 - 1/3 + 1/5 - 1/7 +-... does not define a fixed number. What sort of motion does WM ascribe to that number?
From: Virgil on 3 Nov 2006 21:38 In article <1162563346.217156.157990(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > > > > > > > > > > > 0 > > > > > > 1 2 > > > > > > 3 4 5 > > > > > > 6 7 8 9 > > > > > > > > > > > > The entries surpass every finite entry. Nevertheless you call all of > > > > > them finite. > > > > > > > > I don't know what you're trying to say. > > > > > > Because you did not read what I wrote. I defined it above: "better say > > > finite sequences or numbers or entries" > > > > No, I read it over a few times. When I say I don't understand > > something, you can take me at my word that I mean just that - I read it > > a few times, thought about it, and don't understand it. Thus, you can > > save yourself the wasted typing of saying false things such as that I > > didn't read what you wrote. > > Entries are 1 2 and 3 4 5 and so on. The numbers written in a line. > > > > I don't know what you mean by entries SURPASSING every finite entries. > > What entries surpass which other entries? What does 'surpass' mean? If > > you give me ordinary discourse, then I'll have a better chance of > > understanding you, just as I defined each of my terms, 'sequence', > > 'entry', etc. in my own remarks. > > The digits of the numbers written down in your list (above) are not > bounded by a finite number. The digits certainly ARE bounded, unless WM suggests using an alphabet of infinitely many different characters. If WM is talking about the numerals, which is quite a different thing, he should use the correct term. > > > > > > > Maybe, if you say so. But omega is not the maximum of all finite > > > sequences. > > Yes, since omega is not a sequence at all, let alone being a finite > > sequence, let alone being the maximum of all finite sequences. > > > > > Therefore the width of the list is less than omega. > > > > My argument does not mention 'width of the list'. If YOU want to refer > > to 'width of the list', then YOU need to define it. And that means > > first proving that there exists a unique object that meets the > > description. > > The width of the list is the number of digits of the number with most > digits. That presumes a condition often contrary to fact, that in an infinite list there is any maximal object. That certainly is not true for the list f(n) = n, expressed in decimal notation. Wm's continual reliance on falsehoods to support his arguments does not inspire any confidence in his thought processes, nor in the validity of his claims. > > You claim it is a simple truth without proving it. And your claim is > > not even compatible with the simple intuitive picture that uses > > ellipses. So not only do you not have a mathematical proof of your > > claim, you don't have an intuitive explanation, except an argument by > > ANALOGY in which you analogize between the finite and infinite, only > > assuming, as a form of question begging, that what holds for a finite > > sequence must hold for an infinite sequence. > > I did not introduce a number omega which is larger than every natural > number. You have not produces any logical arguments at all about anything, as far as I can tell. > But IF such a number is introduced, THEN we should be allowed to use > the inequality omega > n for every natural number n, i.e. for the n > digits of the n-th list entry. What makes WM think that the nth entry in a list is limited to no more than n digits? What about the list f(n) = 13^n, for n in N? > So you do not man that omega > n holds fo every n e N? As WM has not explained what order relation he is invoking to claim omega > n, we have no idea what he is asking. > > > > Therefore, there exists a denumerable sequence S of finite sequences > > such that the diagonal of S is denumerable. > > Now map it on a line. Why. This is not geometry. The only "lines" so far have been finite sequences of characters, and the whole point has been to show that there is a diagonal that does not "map" to any of them.
From: Virgil on 3 Nov 2006 21:42 In article <1162563567.735020.246810(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1162470874.593282.36250(a)b28g2000cwb.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > WM merely repeated his automatic error several more times here. > > > > WM claims that a list in which the nth listed element is a string of > > length at least n characters cannot produce a diagonal of length > > greater that any finite number of characters. > > > > The diagonal needs an element from every line, the n-th element from > the n-th line. Therefore it cannot be longer than every line. The "diagonal"must be longer than every finite "line". The only way it will ever fail to be longer than some "line" is if that line is itself infiitely long.
From: Virgil on 3 Nov 2006 21:50
In article <1162564016.193720.60290(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Do you agree that A n: n < omega is incorrect? > > > > It is ambiguous in ZF, thus effectively meaningless in ZF until a > > context for the universal quantifier is made. > > A n eps N. In ZF, N = omega as a set, and sets are not allowed to be members of themselves. > > > > If it cannot be a fraction because ZF does > > > not yet know how to divide elements, then it can only be a whole > > > number, I would guess.# What are "whole numbers" in ZF? I know of no definition explaining what a "whole number" is in ZF. > > > > Except that ZF does not know what whole numbers are. There is no > > definition within ZF for "whole number". > > ZF doesn't seem to know much, not even the main property of the only > species of numbers which it deals with. ZF knows about ordinal numbers and natural numbers, and in its extensions, even about rational numbers and integral numbers and real numbers, and a few others, but nowhere in ZF is there any definition of "whole numbers". So when WM goes on and on about "whole numbers" he is talking garbage, at least until he tells us what he means by it. |