An uncountable countable set
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From: David R Tribble on 6 Sep 2006 18:46 Virgil wrote: > Actually many fire engines are not red. White and yellow are at least as > popular at least for newer engines, and are a god deal safer. > http://www.psychologymatters.org/solomon.html Stripes of highly contrasting colors (such as red and yellow, or black and orange) would actually be much more visible. Which is probably why those color combinations are used to mark hard-to-see stairsteps and high-voltage boxes.
From: David R Tribble on 6 Sep 2006 19:09 Tony Orlow wrote: > So, what is it you think I DON'T get about inductive proof, sets, and > recursion? For one thing, you seem to think that successively adding 1 to the naturals, starting with 0, leads to an infinite value. That shows that you don't understand induction over the naturals. Another is the total absence of stated axioms for your alleged system of "infinite arithmetic". Yet another is your confusion between "countable", "uncountable", and "infinite". > As far as I can tell, very few dare to question the > predefined rules as set forth, Actually quite a few people question them. It would be fair to say that most of us questioned the theorems of set theory when we first encountered them. However, most folks eventually understand the logical consistency of set theory. It's only the minority who continue to misunderstand it. > but I have yet to see any valid > counterexample to my rules regarding inductive proof in the infinite > case. Since you still have not provided a meaningful definition of your "infinite case", that should not be surprising. I suggested that you show a proof in your system (using your new axioms) that the number of reals in [0,2] is twice the number of reals in [0,1], and that both intervals are dense in the reals. That should be easy for you, right? Assuming that you could prove such a thing, you then have to realize that such a system is incompatible with standard arithmetic and set theory. You see why, don't you? > Where an equality between expressions is proven for all n, it is > valid for infinite n. Again, this is something you have to prove instead of just claiming as true. A good start is to define exactly what you mean by "infinite", since you reject the accepted definitions everyone else uses. > Where an inequality is proven for all n greater > than some finite m, and the difference upon which the inequality is > based does not have a limit of 0 as n->oo, it holds also in the infinite > case. What does "in the infinite case" mean? > The staircase in the limit vs. the diagonal line was a valiant > effort at a counterexample, but obviously flawed in its reliance on the > limitations of point set topology, and the only other attempt was based > on a half hidden limit of 0 for the nested inequality as n->oo, based on > a function discontinuity, so was dismissible without any fanfare. You're starting to sound like Ross. Slow down, you're using too many loaded words in too small a space. > So, if > you have what you think is a valid counterexample which shows how little > I know about inductively defined structures and how they suddenly change > character when n=oo, well, bring it on. :) It would help if you could define what it means for n=oo in the first place. But be warned that since that has no meaning in standard arithmetic, whatever system you come up with in which it does have some kind of meaning is a system that cannot be compatible with standard arithmetic.
From: David R Tribble on 6 Sep 2006 19:19 Aatu Koskensilta wrote: > As a general observation, it seems that when arguing with cranks people > often have a habit of preferring overtly formal expositions and > arguments, stressing somewhat irrelevant technicalities. This is > certainly understandable - and some people probably just enjoy going > through the formal details, possibly working out some of the details for > themselves for the first time - but seems, to me, ultimately > counterproductive; if the objective is to actually establish > intellectual contact with the crank, and have him see the error of his > ways, surely most simple, informal, conceptual arguments and > explanations are to be preferred, setting aside all technicalities that > really are irrelevant - cutting through the crud, focusing solely on the > actual issue or confusion? The problem is that using too simple a language can lead to further confusion, or at least maintain the misunderstandings. I've been chastised for talking about "set size" instead of using the more specific term "cardinality", for instance. So there may be times when a more technically precise meaning is called for in order to cut through the confusion of multiple meanings.
From: Dik T. Winter on 6 Sep 2006 19:10 In article <1157574332.235885.113030(a)d34g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: Talking about 0.111... defined as: for all natural p digit p is 1, there are no other digtits. And the list is the list of natural numbers. > > > Your "each" means in symbols of logic: "A = (for) all". > > > The number is nothing than all of its digit positions. > > > Therefore your statement is a self contradiction. > > > > Where? In logical terms (A meaning "for all" and E meaning "there is"): > > (1) A{p = digit position} E{q = list item} {such that q indexes p} > > That is your definition. But what we can safely say is only: > (1') A{p = digit position of list item} E{q = list item} {such that > q indexes p} Why can we only say that? The definition of 0.111... is such that (1) holds. If it does not hold you should be able to give an index position such that it is false. > > (2) A{p = digit position} E{q = list item} {such that q covers p} > > If your definition could be satisfied, the construction of the list > would imply this, yes. What we can safely say, however, is only: > (2') A{p = digit position of list item} E{q = list item} {such that > q covers p} The same here. But apparently you think my definition of 0.111... can not be satisfied. Why not? > Now we may ask: Is it possible that a list item indexes or covers other > digits than those which are indexed or covered by list items? The > answer is: no. You are repeating yourself again, and I have already agreed to that. Why then repeat again and again? > And we may further ask: How can 0.111... be distinguished from all list > items? The answer is the only one possible: By digit positions occupied > by 1's which are neither indexed nor covered by list items. Why? This is false. 0.111... is distinguished from all list items in that it does not terminate. Consider them as decimal fractions. The list consists of the numbers (1-10^(-n))/9, 0.111... is 1/9. > because 0.111... does not have any other component. It does not have any other component, but also it does not terminate, in contrast to all numbers on the list. > So we may finally ask: Is it possible that all digit positions of > 0.111... are indexed or covered by list items? > The answer is obvious: no. Every digit position can be indexed and covered. Otherwise state which digit position can not be indexed, use my definition: Talking about 0.111... defined as: for every natural p digit p is 1, there are no other digtits. but there is no natural that covers all digits. And if you think that that definition is wrong, please *prove* that. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Tony Orlow on 7 Sep 2006 09:01
Virgil wrote: > In article <44fe1466(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> A better way to view the universe of sets is as a relation between the >> objects within it and 1-place predicates concerning objects, where each >> object has a truth value associated with each predicate, between 0 and 1 >> inclusive. Thus, these predicates define subsets of the universe, eh? > > TO appears to be suggesting that there are truth values strictly between > 0 and 1, as in "fuzzy logic". I am actually referring to probabilistic logic, which doesn't have the kludgy aspects of "fuzzy" logic. > > That doesn't work well in mathematics, TO. Probability doesn't work well in mathematics? > > And allowing unrestricted predicates leads to all sorts of anomalies, > like sets which are not members of themselves. Yes, predicates like "is a set" cause self-referencing problems. > > Besides which mathematics has no need to include the entire universe, > and those why try to make it do so get themselves in over their heads. In which case they must swim. It's really rather nice out here in the middle of the pond. Come wallow with us, and see if the deep end isn't to your liking. The universe is worth considering. :) Tony |