Cantor Confusion
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From: stephen on 1 Dec 2006 09:52 Virgil <virgil(a)comcast.net> wrote: > In article <eko0cd$ail$1(a)news.msu.edu>, stephen(a)nomail.com wrote: >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > 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. >> >> So you do not know what a path is, or a tree apparently. >> In a tree, there exists exactly one path between any two nodes. >> >> Stephen > There are those for whom what connects *adjacent* nodes is called an > edge, and others call it a branch. > Most of those whom I am aware, other than possibly WM, reserve "path" > for the chain of connections starting at the root node and ending, if at > all, in a leaf node. In a graph theory context, and a tree is just a special case of a graph, a path is just a set of adjacent edges (with some additional caveats). A path can consist of just a single edge. Stephen
From: William Hughes on 1 Dec 2006 09:59 mueck...(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > O.K. we now know that there is no concept of potential infinity > > in ZF (this comes under the heading "the pacific ocean is wet"). > > However, you have not actually given a defintion of potential > > infinity. Do you have one? > > Take the Peano axioms, in particular the axiom of induction. They show > a precise definition of what a potentially infinite set is. O.K. We assume that there exists an entity to which the Peano axioms apply. Call it B. B is then a potentially infinite set. Define elements inductively. 0 is an element of B. x is an element of B if there exists y an element of B such that x is the successor of y. (This is a bit more restrictive than my 'Generalized Set" defintion, but does not confict with it in any way) > > > > > > > > > >als etwas Werdendem, Entstehendem, > > > Erzeugtem, d.h., wie man sagt, mit dem potentiell Unendlichen zu tun. [David Hilbert: Über > > > das Unendliche, Math. Ann. 95 (1925) p. 167] > > > > In short: A potentially infinite entity is a variable, always finite > > > but surpassing every border. The atually infinite is fixed, i.e., it is > > > a quantity. > > > > > > > Nothing in the above, in either language, precludes in > > any way giving a meaning to > > "x is an element of the potentially infinite entity A". > > That is correct. 7 is an element of the set of natural numbers. But > this set is not a set in the sense of set theory. I.e., it has no > cardinal number which is a quantity. It has at most the "cardinal > number" oo. But that is a property, not a quantity. A cardinal number is an equivalence class of sets under the equivalence relation bijection (under some definitions to be a cardinal, the equivalence class must contain an ordinal). We extend this definition to include potentially infinite sets. A function from the set A to the set B is a set of ordered pairs (a,b) such that a is an element of A and b is an element of B. We extend this to potentially infinite sets: A function from the set potentially infinite set A to the potentially infinite set B is a potentially infinite set of ordered pairs (a,b) such that a is an element of A and b is an element of B. We can now define bijections on potentially infinite sets and extend the bijection equivalence relation to include potentially infinite sets. Thus we can define equivalence classes under bijection of potentially infinite sets. Thus we can define "cardinal numbers" of potentially infinite sets. - William Hughes
From: Eckard Blumschein on 1 Dec 2006 12:12 On 12/1/2006 1:37 AM, Dik T. Winter wrote: > 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. This axiom combines flawless Archimedean reasoning with an at least questionable replacement of the notion number by the notion set. Numbers are breaks of a stepwise process without limitation. Consequently, infinite numbers are impossible or more strictly speaking, they are fictitious. The elements of a set can seemingly be imagined to be set altogether at a time. Therefore, it is tempting to anticipate a set of all natural numbers. So the purpose of axiomized set theory is to veil the categorical difference between rational and real numbers. People are in position to continue the formerly naive believe that reals are not fictitious but genuine numbers.
From: MoeBlee on 1 Dec 2006 13:26 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > > > > 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. > > > > As to what Fraenkel, Bar-Hillel, and Levy wrote, they are underscoring > > the fact that different axioms yield different universes of sets. That > > is what they mean by the universe of sets "growing" (scare quotes in > > original text). > > You should try to distinguish between "to differ" and "to grow". You > may scream as loud as you can. These verbs are different and denote > different processes. You need to re-read that section of the book - toward the aim of understanding rather than that of finiding out-of-context quotes for ill-premised polemics. Then you'd see what they mean by "grow" (in scare quotes in the original). > > > An easy example which > > > should not escape you: The set of states of the EC has been growing and > > > probably will continue to grow. > > > > Which you'll have a hard time proving to be a set in Z set theory. > > Of course in set theory there are variables denoting sets. In any book > on ZF set theory you can find sentences like: "The letters X and Y in > these expressions are variables; they stand for (denote) unspecified, > arbitrary sets." By such tools it is very easy to deal with a set like > the set of states of the EC in ZF. It turns out again and again that > you have very little knowledge about set theory and its philosophy. I said that in Z set theory, you'd have a hard time PROVING the existence of the set of states of the EC. In Z set theory, as the axiom of extensionality is formulated to preclude urelements you will not be able to prove the existence of the set of states of the EC. And even in Z set theory with the axiom of extensionaliy formulated to allow urelements, you will not be able to prove the existence of set of states of the EC, since no PARTICULAR urelements (the states, or whatever the states themselves might be regarded as sets of) are provided for by a formulation of the axiom of extensionality that allows urelements. To prove the existence of the set of states of the EC in set theory, you'd have to add an axiom that states the existence of certain particular European entitites or something like that. > I am sorry but as your behaviour parallels your expertise I will no > longer discuss with you. A last hint may help you to become a decent > person and socialize with your surrounding: You should know that it is > not appropriate to speak out everything one thinks. In general it is > not useful to injure persons. What would it help if I publicly uttered > what I think of you? Oh boo hoo hoo, have you been "injured"? And you already have said enough of what you think about me. Your sermon in Internet etiquette is sanctimonious and hypocritical. MoeBlee
From: MoeBlee on 1 Dec 2006 13:57
mueckenh(a)rz.fh-augsburg.de wrote: > Here is a book on ZF set theory: Karel Hrbacek and Thomas Jech: > "Introduction to Set Theory" > Marcel Dekker Inc., New York, 1984, 2nd edition. they write: The > letters X and Y in these expressions are variables; they stand for > (denote) unspecified, arbitrary sets. > > Therefore we can denote a set by X and we can say that the set X grows. I've been mentioning the axiom of extensionality. But it occurs to me that this is even more basic. Identity theory itself precludes an object from having different members at different times, points, or stages. If x has different members from y, then x and y are different sets - just by plain identity theory. > That is nothing else than to say that the number of states in the EC > grows. The number of states in the EC grows. But the states in the EC is not an object of Z set theory. > Of course the number 6 has not gown to 25. But it is simply a > matter of definition, how one interprets "to grow" and "number". And Z set theory has no definition of 'grow' or 'number' that allows a set to have different members at different times, stages, points, or whatever. > > 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. > > Wrong. Read my explanation above. You're wrong. Your "explanations" are about everyday senses of the notion of set, which are looser than mathematics that uses identity theory. Your inability to see the difference between common, informal notions about things and precise and rigid mathematical treatments is blocking you from understanding the most basic things about mathematics and set theory in particular. > > 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). > > Hrbacek and Jech teach standard set theory including the fact that in > ZF everything is a set. And they don't allow anything that violates identity theory. If x has has different members from y, then x and y are different. If a theory includes identity theory (as ordinary set theories do), then no set has different members at different times, points, stages, or whatever. > > > > 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). > > Hrbacek and Jech teach standard set theory including the fact that in > ZF everything is a set. Yes, and just as Fraenkel, Bar-Hillel, and Levy mention the universe of sets as "growing" as different AXIOMATIZATIONS entail differently what will BE the universe of sets. But no set itself, and not even a class that is the universe of sets, grows or has different members at different times, points, stages, or whatever. Rather what we REGARD or CONCLUDE TO BE the universe of sets is different depending on what axioms we adopt. THAT is the sense of "grow" (scare quotes in ORIGINAL text) that Fraenkel, Bar-Hillel, and Levy were conveying. But those authors did not do a good job of counting on the fact that over-opinionated, polemical ignoramuses like you would exploit their very loose couching of the idea to twist it into something actually antithetical to even plain identity theory. MoeBlee |