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From: Virgil on 31 Oct 2005 18:08 In article <MPG.1dd03a3077c987098a5b4(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Oh geeze. Here we go again. "No largest finite!!!" No kidding. There > is no single step where that happens, as you well know. Let me ask > you this. At what point does the count of naturals become infinite? > That's the point at which the element values become infinite. (sigh). And that never happens outside of TOmatics.
From: David R Tribble on 31 Oct 2005 18:15 David R Tribble wrote: >> Consider the set of reals in the interval [0,1], that is, the set >> S = {x in R : 0 <= x <= 1}. The elements of this set cannot be >> enumerated by the naturals (which is why it is called an "uncountably >> infinite" set). But all sets have a size, so this set must have a >> size that is not a natural number. It is meaningless (and just >> plain false) to say this set "has no size" or "is not a set". > Albrecht Storz wrote: > I'm not shure if the reals build a set in spite of you and Cantor and > others are shure. > A set is defined by consisting of discrete, distinguishable, individual > elements. Now tell me: what separates a point on a line from the very > next point on the line to be discrete? What separates sqrt(2) from the > very next real number to be discrete? > If you look only on individual points, you may have a set. But if you > look on all of them? > > So, your above argumentation has no relevance to me. Proof the reals to > be a set, then let's talk again. Well, for a start, the set of naturals (N) is a subset of the set of reals (R), i.e., every member in N is also a member of R. If N is a set, then it would appear that R is also a set. Or are there entities that can have subsets taken from them but themselves are not sets? What do you call those entities then? I can also form the set S = {pi, 2 pi, 3 pi, 4 pi, ...} which is obviously an infinite set containing real numbers. S does not contain all the real numbers, of course, but it is a subset of all the real numbers. This, too, implies that the set of all real numbers is a set.
From: Virgil on 31 Oct 2005 18:15 In article <MPG.1dd03d44bdc121af98a5b6(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Randy Poe said: > > > > Tony Orlow wrote: > > > David R Tribble said: > > > All this proves is that there is no size of the set, as Albrecht has been > > > saying, since the set size is equal to its largest element, which doesn't > > > exist. > > > > Hmmm. Adding the TO-axiom that "set size is equal to its largest > > element" leads to the conclusion that "sets without maximal elements > > have no size". Since without this axiom we have a perfectly > > self-consistent notion of cardinality (informally, "size"), > > there seems no point in adding this axiom to our system. > Except to make sense. > > > > I know, let's not make the assumptiont that "set size is > > equal to its largest element" and see where that goes. > It's not an assumption for the naturals. It's a derived theorem, given true > logic and graphical fact. For any bounded and therefore Dedekind finite set of one-origin naturals the cardinality is less than or equal to that of the initial set of naturals ending with the same natural, but this is false for any unbounded and therefore Dedekind infinite set of naturals, of which there are infinitely many more than of bounded sets of naturals. So TO must have derived it from excessive ingestion of some controlled substance, as it does not come from any legitimate logic or mathematcs.
From: Virgil on 31 Oct 2005 18:17 In article <MPG.1dd03db61dd6992598a5b7(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > Daryl McCullough said: > > Tony Orlow says... > > > > >Look again at the bijection I offered. > > > > It's not a bijection. As you say, no element > > is in the set mapped to it. It follows that > > no element is mapped to *N. From that, it follows > > that the mapping is not a bijection. By definition. > > > > -- > > Daryl McCullough > > Ithaca, NY > > > > > No element is ever identified, in any infinite bijection, as mapping to the > last element of an endless set. To expect this of the power set bijection is > consistently inconsistent, and proof of nothing but the bogusness of the > assumptions of transfinite set theory. To require a bijection to biject is considered bogus in TOmatics? Then TOmatics is totally bogus in any and every standard version of set theory.
From: Virgil on 31 Oct 2005 18:23
In article <MPG.1dd040925340036c98a5b8(a)newsstand.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > > If there exists y such there is no x with f(x) = y, then you > > provably do not have a bijection. By definition. > That is true if y is well defined, but the entire set, with no > identified end, is not a well defined set in the sense that you could > ever map anything to it besides an element that also has no end. How about f:*N -> P(*N): 0 -> *N , otherwise n -> {n}. Assuming that TO's *N is zero origined, otherwise do 1 -> *N So it appears that TO is WRONG! AGAIN! And a supposed bijectin that is not well efines is not a bijectin outside of TO's delusional world. |