PROOF INFINITY DOES NOT EXIST! Not even ONE type
in [Math]
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From: Transfer Principle on 13 Jul 2010 03:29 On Jul 10, 3:36 am, ah...(a)FreeNet.Carleton.CA (David Libert) wrote: > Transfer Principle (lwal...(a)lausd.net) writes: > > I recommend that Herc read some of the other sci.math > > finitists to see what they have to say. (Of course, he > > already knows about WM.) > You are writing above about the 2 axioms ~Infinity > and D=0. Thanks for the information. And so now we know that Srinivasan's theory NBG-Infinity+D=0 will work, even if we drop Replacement Schema and Regularity, as described by Libert in this post. > I posted in ZFC - Infinity transitive closures might not exist as a set: > [4] David Libert "Axiom of infinity and the set of all hereditary finite sets" > sci.logic Oct 3, 2007 > http://groups.google.com/group/sci.logic/msg/7593d4adf17732b7 > So if just expand the usual definition above as Russell is problems as for omega above. > In > [5] David Libert "Recursive cardinals" > sci.logic, sci.math Jan 3, 2010 > http://groups.google.com/group/sci.logic/msg/02248254025cb4c8 > I posted how to defince TC as a class in ZF - Infinity. > The same definition would work in Z - Infinity - R . Ah, I remember that discussion about zuhair and his "singleton towers" that work in ZF-Infinity. I mentioned zuhair's theory in other threads, hoping that maybe it might satisfy the desiderata of other posters, but to no avail. If Herc were merely a finitist, then he could choose Srinivasan's theory (or possibly even zuhair's), but in this thread he implies that he might either be a ultrafinitist or have WM-like gaps in his set I of natural numbers. Still, Libert's post was interesting and helpful.
From: FredJeffries on 13 Jul 2010 09:29 On Jul 12, 2:24 pm, Dan Christensen <Dan_Christen...(a)sympatico.ca> wrote: > Correction > > On Jul 10, 7:22 pm, FredJeffries <fredjeffr...(a)gmail.com> wrote: > > > > > On Jul 9, 10:17 am, Dan Christensen <Dan_Christen...(a)sympatico.ca> > > wrote: > > > > I can't imagine that you would be able to do very much using > > > "finitist" methods. How do they handle such basic concepts as the > > > square root of 2? > > > Terence Tao in "A computational perspective on set theory"http://terrytao.wordpress.com/2010/03/19/a-computational-perspective-... > > > in which he explores the question "what is the finitary analogue of > > statements such as Cantors theorem or the Banach-Tarski paradox?" > > With your "countably infinite loops" (see link), it seems you are > sneaking infinite sets in through the back door. You posit an > algorithm that can complete an infinite, countable number of > iterations (ranging over ALL the natural numbers) and arrive at some > conclusion. Have such notions ever been successfully formalized > without referring to the set of natural numbers as a whole? > I am not qualified to answer your question (if I even understand it) and as I work in the real world as opposed to some theory I don't have time to dig out the references now, but I believe that Ed Nelson and Alexander Yessenin-Volpin (both regularly mentioned in these kinds of threads) have tried to address that issue. See Nelson's Predicative Arithmetic at http://www.math.princeton.edu/~nelson/books/pa.pdf My impression is that the answer to the "successfully" part of your question is open.
From: |-|ercules on 13 Jul 2010 10:03 "Transfer Principle" <lwalke3(a)lausd.net> wrote > On Jul 10, 6:42 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: >> Herc's Axiom Of Pseudo Infinity (based on above equation AOF) >> There is a set, I, that includes all the natural numbers that could physically be computed >> (before the end of the computer sustainable Universe) > > Out of curiosity, what does Herc consider to the the largest > number in I? > > If Herc is like AP, then his upper bound might be somewhere > around 10^500 or so. But I suspect that Herc might be more > like WM (who influenced him greatly), and perhaps the set > I contains gaps -- so that googolplex is in I, but there > exist natural numbers less than googolplex but not in I. I could redefine pseudo infinity to be any finite length = any computable length with a program that terminates, if you can compute n then you can (theoretically) compute n-1, so pseudo infinity might be the initial sequence of N up to the largest physically computable number. Let me clarify the argument. let x = the length of some incremental sequence of natural numbers, starting at 1 let y = the last value of such sequence As x->oo, y->oo y = x The limit of y does not exist Therefore the limit of x does not exist Quite Easy Done! Herc
From: Marshall on 13 Jul 2010 10:51 On Jul 12, 7:19 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > K_h wrote: > > "Nam Nguyen" <namducngu...(a)shaw.ca> wrote in message > >news:uNc_n.6705$KT3.5193(a)newsfe13.iad... > >> K_h wrote: > >>> "Nam Nguyen" <namducngu...(a)shaw.ca> wrote in message > >>>news:MTSZn.2663$Bh2.125(a)newsfe04.iad... > >>>> K_h wrote: > >>>>> Mathematical truth exists. > >>>> Sure. In your mind for example! > >>> And also outside of the human mind. > >> Did you mean _physically outside of human mind_ ? That's very bizarre to say > >> of mathematical abstractions that human thinks of. No? > > > The truth underlying the abstractions does exist. > > Sure. In your mind for example! > > > > > No, I correctly claimed that the truth underlying regular arithmetic does exist > > and that truth is independent of context. 1+1=2 is true for any two objects: two > > cars, two houses, two people, etc. > > But 1+1=2 is true in many ... many _modulo arithmetics_ "for any two objects: two > cars, two houses, two people, etc.", right? That's a different plus. (Or alternatively a different equals.) They aren't the same sentences, even if we use the same strings to write them. So your whole argument about "relativity" reduces to pointing out that sometimes words mean different things, like when "blue" can mean a particular color, or an emotion, or "marked by blasphemy." Your only *technical* point about "no absolute truth" is that there is no absolute authoritative definition. Whoop de do! What an amazing thing you've discovered! Marshall
From: Curt Welch on 13 Jul 2010 13:59
Marshall <marshall.spight(a)gmail.com> wrote: > On Jul 12, 12:50=A0pm, c...(a)kcwc.com (Curt Welch) wrote: > > "K_h" <KHol...(a)SX729.com> wrote: > > > > > No, there are absolute truths of the universe, for example > > > conservation of electric charge. > > > > Not absolute in ANY sense. =A0Our understand of the universe, and these > > l= > aws > > of nature we created to explain it are all predictions about the future > > derived from past experience. Such predictions NEVER become absolute > > truths. =A0NO matter how many times we flip the coin and see it comes > > up heads, do we _ever_ get to make the claim that the next time we flip > > it, = > we > > will get heads again, with ABSOLUTE certainty. > > I see you are absolutely certain that there is no absolute certainty. > > Marshall Your attempt at being clever fails. I already explained the error in such an assumption but maybe you didn't see my post when you wrote that? There are no absolutes, so how would I be absolutely certain there are no absolutes? I'm not of course and and I of course am not absolutely certain so what you think I believe is clearly wrong. The answer is quite simple and if you are good at figuring out puzzles, you would have known the answer before you wasted your time posting. We can't be absolute about any position, including the position that there are no absolutes. All we can do is assign a rough probability of certainty to any belief. By how I talk about there being "no absolutes" you can assume this means I've assigned a very high (but NOT ABSOLUTE) probability to the belief. And as such, I have decided it's valid to ACT AS IF it were absolute, even though I know it's not. I believe it to the true beyond a reasonable doubt, but I don't believe it's an absolute truth. There are no contradictions in the statement because the statement is not a statement of absolutes, it's a statement of estimated probabilities (as all statements must be). -- Curt Welch http://CurtWelch.Com/ curt(a)kcwc.com http://NewsReader.Com/ |