PROOF INFINITY DOES NOT EXIST! Not even ONE type
in [Math]
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From: Don Stockbauer on 12 Jul 2010 12:51 On Jul 12, 11:31 am, "Milton J. Smuthworthy, I" <tonworthyCLOT...(a)gmail.com> wrote: > Then Don Stockbauer says: > > > > >>> "|-|ercules" <radgray...(a)yahoo.com> wrote > >>>>> =A0To me, it is self-evident that the axiom of infinity is true. > > >I have an affinity for infinity. > > I'm finicky about these infinity foibles. Coy bulls make poor breeders.
From: Dan Christensen on 12 Jul 2010 15:11 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?" > summarizes thus: > > <quote> > The above discussion suggests that it is possible to retain much of > the essential mathematical content of set theory without the need for > explicitly dealing with large sets (such as uncountable sets), but > there is a significant price to pay in doing so, namely that one has > to deal with sets on a "virtual" or "incomplete" basis, rather than > with the "completed infinities" that one is accustomed to in the > standard modern framework of mathematics. Conceptually, this marks > quite a different approach to mathematical objects, and assertions > about such objects; such assertions are not simply true or false, but > instead require a certain computational cost to be paid before their > truth can be ascertained. This approach makes the mathematical > reasoning process look rather strange compared to how it is usually > presented, but I believe it is still a worthwhile exercise to try to > translate mathematical arguments into this computational framework, as > it illustrates how some parts of mathematics are in some sense "more > infinitary" than others, in that they require a more infinite amount > of computational power in order to model in this fashion. It also > illustrates why we adopt the conveniences of infinite set theory in > the first place; while it is technically possible to do mathematics > without infinite sets, it can be significantly more tedious and > painful to do so. > </quote> With your "countably infinite loops" (see link), it seems to 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? Dan Download my DC Proof software at http://www.dcproof.com
From: Dan Christensen on 12 Jul 2010 17:24 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?" > summarizes thus: > > <quote> > The above discussion suggests that it is possible to retain much of > the essential mathematical content of set theory without the need for > explicitly dealing with large sets (such as uncountable sets), but > there is a significant price to pay in doing so, namely that one has > to deal with sets on a "virtual" or "incomplete" basis, rather than > with the "completed infinities" that one is accustomed to in the > standard modern framework of mathematics. Conceptually, this marks > quite a different approach to mathematical objects, and assertions > about such objects; such assertions are not simply true or false, but > instead require a certain computational cost to be paid before their > truth can be ascertained. This approach makes the mathematical > reasoning process look rather strange compared to how it is usually > presented, but I believe it is still a worthwhile exercise to try to > translate mathematical arguments into this computational framework, as > it illustrates how some parts of mathematics are in some sense "more > infinitary" than others, in that they require a more infinite amount > of computational power in order to model in this fashion. It also > illustrates why we adopt the conveniences of infinite set theory in > the first place; while it is technically possible to do mathematics > without infinite sets, it can be significantly more tedious and > painful to do so. > </quote> 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? Dan Download my DC Proof software at http://www.dcproof.com
From: K_h on 12 Jul 2010 18:21 "Nam Nguyen" <namducnguyen(a)shaw.ca> wrote in message news:uNc_n.6705$KT3.5193(a)newsfe13.iad... > K_h wrote: >> "Nam Nguyen" <namducnguyen(a)shaw.ca> wrote in message >> news:MTSZn.2663$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. >>>> The equation 10+20=30 is an absolute truth and that truth does exist. >>> Again, in your mind perhaps. Others working in modulo arithmetic >>> may state 10+20=0 is absolutely true, just as you stated "10+20=30 >>> is an absolute truth". What's the difference anyway? >> >> If you don't believe that 10+20=30 is true in regular arithmetic then there's >> not much point in arguing it. Obviously I was not referring to modulo >> arithmetic. > > I didn't say I don't believe such in regular arithmetic. But if you have to > refer to regular arithmetic then that isn't "an absolute truth" as you had > incorrectly claimed! An absolute mathematical truth is a statement which is > just true _independent of any context_ that you're referring to. And there > isn't such an absolute truth. 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. >>>> So you have existential doubts about the truth of 4+5=9? >>> People have no doubt that 4+5=9 is false in some modulo arithmetic. >> >> So we agree that there are absolute truths in both regular and modulo >> arithmetic. > > No, I didn't agree to that. A truth that requires a context for it to be true > isn't an absolute truth. And in any rate it's not "outside of the human mind" > as you incorrectly stated above. Those truths exist and are perceived by the human mind so I stated nothing incorrectly. _
From: Marshall on 12 Jul 2010 20:28
On Jul 12, 12:50 pm, 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. Our understand of the universe, and these laws > of nature we created to explain it are all predictions about the future > derived from past experience. Such predictions NEVER become absolute > truths. NO 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 |