Cantor Confusion
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From: Virgil on 30 Nov 2006 16:30 In article <1164877402.469222.169650(a)n67g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1164818051.564389.180950(a)14g2000cws.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > MoeBlee schrieb: > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > And in particular because the expression "contradiction" is not pat of > > > > > ZFC. > > > > > > > > In Z set theory, we can formulate definitions of 'S is a contradiction' > > > > and 'T is inconsistent'. > > > > > > One could, but one wouldn't. Never! > > > > One has. But defining something does not instanciate it. > > > > One can define square circles and 4 sided triangles, too. > > Wrong definitions alltogether, the four-sided triangle as well as the > possibility of set theory being proven inconsistent by one of its > advocates. Definitions in mathematics, being merely abbreviations, cannot be "right" or "wrong", they can only be more or less useful. The more useful ones tend to persist. Useless ones get discarded.
From: Virgil on 30 Nov 2006 16:35 In article <1164879948.009723.297190(a)16g2000cwy.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > The set of transcendentals is not a connected set in R, so violates the > > hypotheses of the proof. > > So the set of transcendentals cannot be proved uncountable by this > proof. Nor can it be proved uncountable by the diagonal argument. Do > you have an idea why this is so? Could it be that Cantor's proof have > not at all to do with countability? Since it CAN be shown that the set of transcendentals differs by a countable set from the set of all reals and the first proof applies to the set of reals, one can deduce that the set of trannscendentals is equally uncountable. > > In my paper I stress the fact that his proof is equally vaild or > invalid, when applied to the rational numbers alone as well as when > applied to the transcenental numbers alone. What is wrong with this > statement? Nothing. But see above for how to show the uncountability of the transcendentals.
From: Dik T. Winter on 30 Nov 2006 19:34 In article <1164878462.838895.276420(a)n67g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > How can you judge about that without the slightest idea of what they > > > wrote? Only for the lurkers: Fraenkel et al. write: "Platonistic point > > > of view is to look at the universe of all sets not as a fixed entity > > > but as an entity capable of "growing", i.e., we are able to "produce" > > > bigger and bigger sets." So a set (like the set of all sets) is not a > > > fixed entity. > > > > There is nothing in that that shows that a set is not a fixed entity. > > The universe of all sets can grow. Define: "The universe of all sets is > called the set of all sets", and you see it. And that is the confusion. If it is a set it cannot grow, but as Fraenkel et al. do not define it as a set it is allowed to grow. They do not state that a set can grow because they do not state that the universe is a set. > > You are able to produce bigger and bigger sets, but they are all > > different. > > That is a matter of definition. If you consider a fixed set then it is > fixed. Small wonder. If you consider a variable set then it is > variable and perhaps changes its cardinal number. An easy example which > should not escape you: The set of states of the EC has been growing and > probably will continue to grow. That is not "the set of states". You can talk about "the current set of states" or about "the set of states in 1957" or whatever. At least mathematically. In mathematics, by definition, a set can not grow. You are, of course, entitled to use another definition, but that will not clarify the discussion at all (and you are not using standard set theory). > > When the universe has grown it allows bigger sets than > > where originally allowed, but the sets originally allowed are still > > sets and still the same and did not grow. How you conclude from the > > above statement that sets themselves are growing escapes me. > > It is simply a matter of definition. Yes, with your definition a set can grow, but you put yourself outside set theory, and you must at first consider all results from set theory unproven theorems in your theorem, and you need to prove them (if possible). -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 30 Nov 2006 19:37 In article <456EEA86.20001(a)et.uni-magdeburg.de> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> writes: > On 11/30/2006 3:19 AM, Dik T. Winter wrote: .... > > There is nothing in that that shows that a set is not a fixed entity. > > You are able to produce bigger and bigger sets, but they are all > > different. > > So far I recall "the set IN of naturals is countable". > I understood set theory means a hypothetical (alias fictitious) set off > all natural numbers. Oh, for once, try to talk mathematics. By the axiom of infinity the set of all naturals is neither hypothetical nor fictitious. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Dik T. Winter on 30 Nov 2006 20:14
In article <1164879523.091494.212240(a)l12g2000cwl.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > > > > > > In my tree there are no finite (terminating) paths! > > > > Sorry, as far as I see your tree has also finite paths. I see paths > > from the root to the first nodes below the roots. > > That is an edge. But no path does stop there. Every path is continued, > infinitely. I see paths from the root to the nodes below the nodes below the tree. > > What is the tree? What are the nodes? You again switch to something > > different. > > I only want to make clear that decimal representations of real numbers > and binary representations of real numbers (like the paths of the tree) > are not different. Indeed, they are not. Base 'pi' or base 'sqrt(2)' are also not inherently different. > > But in a decimal tree, where each node is connected by > > edges to ten subnodes and where each of those subnodes is assigned a > > decimal digit, it can be shown that also each node represents a > > finite decimal number. And in that tree 1/3 is not represented by > > any node. But 0.333... is not a path in that tree, just as I wrote > > earlier. And, indeed, 0.333... does not represent a number until > > that notation has been defined. > > And I use exactly the same definition, adapted for the case of binary > representations. Instead of writing a number like Ok. > And that is all the difference! I cannot understand how someone > familiar with decimal representations of real numbers should have any > difficulties with the binary tree. I have difficulty with the tree because your explanations are confused and sometimes contradictionary. > > The difference is that infinite decimal representations are not defined > > as given, there is a concept behind it: limit. > > Take simply the same concept for my tree: The limit. But whether there > is a concept behind or not: Your mentioning of an infinite distance > from the root to a node fails completely. In any case there is no > digit of a real infinitely far from the decimal point. Why? Its > position could not be indexed by a natural number. Its position, > therefore, would be undefined. The same is true for the nodes of the > binary tree. Its paths are infinitely long but no node has an infinite > distance from the root. Do you see your error? What has this to do with everything else? > > > As there is no digit "3" at an infinite distance from the "0." in > > > 0.333..., there is no infinite decimal expansion. > > > What is the difference to a representations by paths? > > > > Limits. > > No. Paths are only another notation for the reals in usual > representation and usual definition. You state that you are using limits with your infinite paths? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |