An uncountable countable set
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From: Virgil on 23 Aug 2006 15:00 In article <echqkh$ikm$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > David R Tribble wrote: > > In contrast, your Big'un does not have any such well-defined > > properties. You define it as "the number of reals in [0,1]", which > > of course is c, but then you tack on "and it operates arithmetically > > just like a finite value" without proof or elucidation. Which is just > > silly. > > > > It's not silly, because it works. If it didn't work, I wouldn't push it. > What does it break? It is not our responsibility to show what it breaks. If TO claims it works, it is entirely his responsibility to show it works. To argue that something works because no one has shown it doesn't work is the fallacy of argumentum ad ignorantiam, and is unacceptable in sound mathematics. See http://www.infidels.org/library/modern/mathew/logic.html http://philosophy.lander.edu/logic/ignorance.html http://skepdic.com/ignorance.html http://en.wikipedia.org/wiki/Argument_from_ignorance or any other of about 57000 hits in a Google search for argumentum ad ignorantiam. Of course, TO eschews all sound mathematics in favor of his unsound versions. > >> There is no least infinity, any more than there's a greatest finite, > >> where addition or subtraction changes the value. It always should. > > > > In standard arithmetic and standard set theory? Then prove it. > > No, obviously standard tranfinitology holds that omega is the first > limit ordinal and aleph_0 the first transfinite cardinality, but that > follows from the definition of the von Neumann ordinals and their > acceptance as a model of the naturals. Like I said, I don't consider > that model to be correct. Then prove it wrong. While we do not absolutely claim ZF or NBG to be consistent, thousands have tried to prove it otherwise and every single one of them has failed. While absence of proof of inconsistency is not absolute proof of consistency, sufficient absence of and proof of inconsistency can be, and is, persuasive evidence of consistency. > > > > > If you mean in your theory, you still have to prove it. But first you > > have to define your system consistently. > > > > It follows directly form infinite case induction. But TO's version of infinite case induction is false in ZF, ZFC and NBC. What is your system, TO? > > Yes, but omega is considered the smallest infinity, which notion itself > leads to inconsistencies Not in ZF,ZFC of NBG. That adding any of TO's assumptions to one of ZF,ZFC or NBG causes inconsistencies does not count. , such as the ability to classify the number of > bits required to list the naturals. It's neither finite nor countably > infinite. It's a direct result of omega's status as "littlest giant". > > > > > David R Tribble wrote: > >>> What do you call the "size" of an uncountable set? You don't > >>> have a name for it, do you? > > > > Tony Orlow wrote: > >> Uh, yeah. Infinite. Infinite sets can be finely ordered formulaically. > > > > Do you have different names for those different formulaically > > ordered infinite sets? Or do you agree with Ross that all infinite > > sets are the same, so one name is enough? > > > > Different names? I specify infinite set sizes using formulas on Big'un, > which can be ordered by applying infinite case induction, of course. I > like to call the big one Moe, though. > > > > > David R Tribble wrote: > >>> If you are going to keep taking this political approach to > >>> mathematics, condemning what's already established and functional > >>> but not providing any workable alternatives, you might want to > >>> consider running for office instead. > > > > Tony Orlow wrote: > >> It's not functional, and I have provided functional alternatives. > > > > But not consistently defined alternatives. > > > > (sigh) Where are the inconsistencies? (Sigh) Where is the TO system? Absent any TO system , we must rely on such systems as ZF, and then TO's stuff, being inconsistent with what we must use, goes down the tubes.
From: Virgil on 23 Aug 2006 15:06 In article <echr52$jgb$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > > Virgil wrote: > >>> Anyone wishing to extend an operation only defined on one domain to a > >>> larger domain must give a clear definition of what that extended > >>> definition is to mean. TO has not done this. > > > > Tony Orlow wrote: > >>> The domain is the real line. > > > > Virgil wrote: > >> Which specifically does not include any points at infinity, and at best > >> only defines arithmetic on the finite numerical values associated with > >> geometrical points, and on nothing else. > > > > And that's only because we can equate real numbers to points on > > a geometric "line" (which actually involves other concepts such as > > "distance" and metric spaces). > > > > Show us where Big'un exists as a point on the real line. > > Is it somewhere near 1/0? Or maybe near 3/0? > > > > Big'un is best viewed on one or the other version of the number circle, Since TO declared that all his numbers were on a number line, his answer avoids and evades the issue of where on that line his "big'un" is alleged to reside. > opposite 0. It's 1/Lil'un. And where on the line is "lil'un"? > 3*Big'un can be considered to be the size of > three infinite lines So now we have to find a position of the line for 3 infinite lines. > perhaps the first three of an infinite number > comprising a 2D space. GIGO!
From: mueckenh on 23 Aug 2006 16:02 Tony Orlow schrieb: > >> However, I wonder whether you would ever consider the existence of > >> infinite natural numbers. > > > > Then these numbers would not deserve the name "natural number". > > Why is that? If they are whole numbers, each with successor, does that > not fit the bill? What would you call them? I suppose they should be > called hypernaturals, to distinguish from the more limited standard > naturals. Would you consider the existence of any infinite values? > No. > > > >> After all, your argument says that either you > >> have a finite set OR you have infinite values in the set. Is the second > >> option objectionable for you? > > > > What would infinite values be good for? Distinguishable infinite values > > are provably inconsistent (see my binary tree) and to have one infinity > > the approved potential infinity is sufficient. > > Sufficient for what? Sufficient to do mathematics with limits. Sufficient to address all points. You cannot address more than a countable number of them, because there are only a finite number of names (including finite strings of digits). And the points which cannot be addressed do not exist. In what form should they exist? As carbon atoms from a pencil or of chalk particles? No. In real mathematics (as opposed to matheology) we have the valid foundation: What cannot be addressed, that does not exist. Don't call me a finitist. There is no largest number but only a largest set of less than 10^100 elements. Regards, WM
From: mueckenh on 23 Aug 2006 16:05 Tony Orlow schrieb: > > Ordinals and cardinals are necessities if we want to talk about > > set "order" and "size" in any kind of logical, well-defined way. That is wrong. We can talk about finite sets and about infinite sets which all have the same magnitude by the measure of intercession instead of bijection. > > > What do you call the "size" of a countable set with no end? > > You don't have a name for it, do you? > > No, I find focused concentration on the Twilight Zone, the "boundary" > between finite and infinite, to be a rather fruitless exercise. There is > no such distinct boundary. > > Because there is no boundary at all. All sets are finite, but some are without end. Those are called potentially infinite. All of them have the same magnitude if measured by intercession. Definition: "A and B intercede": An order can be defined such that: if b_1 and b_2 are elements of set B, then a at least one element a of set A lies between them in this order, and if a_1 and a_2 are elements of set A, then at least one element b of set B lies between them in this order. Example: Rational and irrational numbers intercede in the natural order by magnitude. Regards, WM
From: mueckenh on 23 Aug 2006 16:07
Dik T. Winter schrieb: > In article <ec8if0$rtd$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > Indeed, I agree with WM's logic concerning the identity relationship > > between element count and value in the naturals. He's quite correct in > > that regard. > > Well, you and he are not. The logic is flawed. For 1, 2, and 3, order and value are identical, even according to your logic, I suppose. Where does the first deviation happen? Which one does deviate first, i.e., which one does first be larger than the other? And why? The cardinal number aleph_0 is infinite while the ordinal number remains finite in order to have infinitely many finite numbers. > > And his proof is not a proof. O course not, because every proof which has an unpleasant result is not a proof. > > > For my part, I agree that the set of finite > > naturals is finite, though unbounded, > > In that case you are not using standard mathematical terminology. I > have no idea what a finite but unbounded set is. That's why you cannot understand mathematics. You fall back behind Cantor. He knew it. Regards, WM |