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From: worldsofsolution on 13 Apr 2005 11:53 There is something I've been wondering about cantor's proof: the decimal number generated to prove the contradiction, was taken to be a real number. There is a tacit assumption that all decimal numbers represent a real number. Does that not require a proof?
From: Torkel Franzen on 13 Apr 2005 12:00 worldsofsolution(a)yahoo.com writes: > There is a tacit assumption that all decimal numbers > represent a real number. Does that not require a proof? No, since not every real number has a successor. This makes mincemeat of the whole thing.
From: Chris Menzel on 13 Apr 2005 11:52 On Wed, 13 Apr 2005 13:56:09 +0200, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> said: > On 4/12/2005 6:34 PM, Chris Menzel wrote: > >>> Whether or not an infinite set is countable depends on its structure. >> >> Sets don't have structure. What you seem to have in mind is that a set >> is to be considered countable or not depending on how it is *ordered*. > > You are correct. To me the real numbers only make sense together with > their ascending order. It is good that you are acknowledging the centrality of your own cognitive intuitions here. > I am avoiding the word well-ordered because it is burdened with > expectations that cannot be fulfilled with the reals. But it is you who are bringing in the idea of a well-ordering. I only pointed out that the SET of reals is UNordered. This is independent of the question of whether or not they can be well-ordered, and that question (as you seem not to realize) is independent of the question of whether or not the set of reals is *larger* than the set of natural numbers. >> So do you think the set of rational numbers is uncountable when >> ordered by the less-than relation and countable when well-ordered in >> some familiar fashion? > > No. The set of rational numbers is countable because it consists of > genuine numbers in the sense, each of them is approachable with a > finite amount of numerals. >> >>> The reals are obviously not countable because one cannot even >>> numerically approach/identify a single real number. >> >> It's not clear what it is to "numerically approach" a number -- >> certainly what you say is false if we understand "approach" in terms of >> limits -- but could you demonstrate your thesis with regard to, say, the >> real number 2? Haven't I just identified it? > > The 2 is a natural number. After embedding into the reals it reads in > decimal representation either 2.000000000... or 1.9999999999999... > There is no chance to completely address its successor. Er, well, reals don't have successors in their natural ordering, so it shouldn't be possible to "address" a real's successor no matter what you mean by "address". But, more generally, until you rigorously define what "M.NNNN..." means, or what it means to "numerically approach" a number, or to "completely address" a number's successor, your assertions are meaningless gibberish. Cantor's work is compelling in part because his definitions of "smaller than" and "same size as" are simple and clear. Your own views might be compelling as well, or at least meaningful, if you were to aspire to similar clarity. Chris Menzel
From: Ross A. Finlayson on 13 Apr 2005 12:12 Hi, If your real numbers have basically a least positive real or iota-value, then they are naturally well-ordered by their normal ordering for positive reals or the widening spiral for positive and negative real numbers. If they don't, then nobody has an example for you, of a well-ordering of the reals. Many agree that there is a well-ordering of the real numbers or for that matter any set. They do. That's pretty simple. Where they do, the proof of nested intervals doesn't apply, and sets of numbers are measurable, with at worst non-standard measure. The antidiagonal result, that's kind of a different thing, and that basically resolves to ultrafinitism, or a dually minimal and maximal ur-element, that corresponds well to the Russell set, Burali-Fortian Ord, the Liar, the empty set, and the Ding-an-Sich and Being and Nothing, towards a theory that can be consistent, and Goedelianly complete, Quineanly. Using these proper names is concision, conciseness. If you get to looking at Turing, a variety of statements about Turing machines, that would seem to collide with, for example, what I say, in the infinite may be true because they're about the finite. I have a lot to learn about complexity and Kolmogorov. A lot of people base their proofs upon statements that ZF is consistent. Many of those theorems are still correct when ZF is determined to be inconsistent. Where the physical universe is all physical objects, and a physical object, that's similar, in a way, to a set being a set of all sets, about concreteness. That means it's empirical evidence of that kind of thing. Infinite sets are equivalent. Ross -- "This style is hard."
From: Will Twentyman on 13 Apr 2005 12:13
Eckard Blumschein wrote: > On 4/12/2005 11:21 PM, David Kastrup wrote: > >>And the core of >>Cantor's argument is sound, > > > It is fallacious. He introduces non existing infinite numbers in order > to misinterpret the unquestionably different qualities of rational und > real numbers. Just the latter are uncountable. Cantor's thinking is at > best fallatious. He ignores that both the rational and the real numbers > do likewise have the quality to be infinite, and infinite cannot be > enlarged. Consequently, the missing possibility to represent the reals > in a list cannot be attributed to a larger size (Maechtigkkeit, > cardinality) of the reals but it relates to something else. I think the problem may be with the concept of ordinals. These are an extension of the idea of numbers. Ordinals are the formalized version of numbers that includes the "infinite numbers" that you seem to have an issue with. Cantor's original exposition on this may not have been clear (I haven't read it), but it can be formalized in a meaningful way. One of your objections is you believe oo+a = oo. Formally, what is being said is that card(w+a) = card(w) where a is a finite successor of infinite ordinal w (usually countable). >>>So he tried to show that there are more reals as compared to the >>>"size" of the set of the rationals. >> >>Nope. He showed that no bijection can be established. > > His second diagonal argument was an evidence by contradiction. > Cantor misinterpreted it by claiming that there are more real numbers > than his list contained. He overlooked the correct possiblity that his > assumed list simply did not represent the reals. That was the conclusion of his proof. He assumed he had a complete list and then showed it had to be incomplete. Translation, his assumed list did not represent the reals. It seems clear you didn't understand something when you were reading the proof. >>And that means >>that there is an order of cardinalities, where cardinalities are >>considered as an indicator of surjectability of sets. > > Please indicate the pertaining pages where he claimed and proved that. I do not have his text, nor can I read it in the original German. It is easy to show by observing that given two sets A and B, if they have different cardinalities C1 and C2, then they have bijections f:A->C1, g:B->C2 between the representative sets of those cardinalities. Suppose A and B have a bijection h:A->B, then C1 and C2 have a bijection f^-1 h g:C1 -> C2, which is a contradiction, since they were assumed to be different cardinalities. If C1 is a lower cardinality than C2, there is a surjection i:C2->C1, and g i f^-1 : B->A is also a surjection. Otherwise, C1 has higher cardinality than C2, which implies there is a surjection i:C1->C2 and f i g^-1 : A->B is also a surjection. Note: the argument does not distinguish between finite and infinite cardinalities. >>The existence and >>non-existence of surjections is a _hard_ fact that has nothing to do >>with any philosophy of "infinity". And it also is a hard fact that >>being surjectable is a transitive and reflexive property. > > What do you think about surjection between IR and IR+? Note: I'm assuming 0 in R+. If you prefer otherwise the following can be easily modified. f:R -> R+ by f(x)=abs(x) is a surjection. g:R+ -> R by g(x)=floor(x)/2+x-floor(x) if floor(x) is even, g(x)=-(floor(x)-1)/2 -(x-floor(x)) if floor(x) is odd is a surjection. f implies Card(R)>= Card(R+) and g implies Card(R+) >= Card(R). Therefor they have the same cardinality, which is consistent with surjections in both directions implying the existence of a bijection. >>If the reals obeyed the laws of ordinary numbers, >>they would be structural equivalent to them and could be put into a >>one-on-one correspondence with them. > > Yes. I agree with you that the reals do differ from ordinary numbers. What are "ordinary numbers"? >>This is what his second diagonal argument is about. Exactly that, and >>nothing else. > > So far I do not object. > And who added the whole cardinality story? Infinite numbers are insane. > Cantor got famous by means of a incredible misinterpretation. I am > suggesting a less spectacular explanation that does not contradict to > very fundamental rules. I suggest you read up on ordinals, and don't think of them as numbers, though there are some analogous concepts. -- Will Twentyman email: wtwentyman at copper dot net |