Mike, The stronger system adds a stronger axiom of mathematical induction, which goes up to the transfinite.
http://en.wikipedia.org/wiki/Peano_axioms#Consistency >>> The number of possible descriptions is countable > > I disagree. I cite my source-- the book "Meta Math!". The argument is reproduced here: http://jyte.com/cl/there-are-uncountably-many-real-numbers-that-no-one-can-ever-describe-in-any-manner Sorry that that is not a particularly reliable source, but there isn't anything wrong with the argument. It's about what appears in Chaitin's book (and, I assume, technical papers). --Abram On Mon, Oct 27, 2008 at 3:33 PM, Mark Waser <[EMAIL PROTECTED]> wrote: >>> The number of possible descriptions is countable > > I disagree. > >>> if we were able to randomly pick a real number between 1 and 0, it would >>> be indescribable with probability 1. > > If we were able to randomly pick a real number between 1 and 0, it would be > indescribable with probability *approaching* 1. > >>> Which side do you fall on? > > I still say that the sides are parts of the same coin. > >>> In other words, we're proving arithmetic consistent only by adding to its >>> definition, which hardly counts. The classical viewpoint, of course, is that >>> the stronger system is actually correct. Its additional axioms are not >>> arbitrary. So, the proof reflects the truth. > > What is the stronger system other than an addition? And the viewpoint that > the stronger system is actually correct -- is that an assumption? a truth? > what? (And how do you know?) > > Also, you seem to be ascribing arbitrariness to constructivism which is > emphatically not the case. > > > ----- Original Message ----- From: "Abram Demski" <[EMAIL PROTECTED]> > To: <[email protected]> > Sent: Monday, October 27, 2008 2:53 PM > Subject: Re: [agi] constructivist issues > > > Mark, > > The number of possible descriptions is countable, while the number of > possible real numbers is uncountable. So, there are infinitely many > more real numbers that are individually indescribable, then > describable; so much so that if we were able to randomly pick a real > number between 1 and 0, it would be indescribable with probability 1. > I am getting this from Chaitin's book "Meta Math!". > > "I believe that arithmetic is a formal and complete system. I'm not a > constructivist where formal and complete systems are concerned (since > there is nothing more to construct)." > > Oh, I believe there is some confusion here because of my use of the > word "arithmetic". I don't mean grade-school > addition/subtraction/multiplication/division. What I mean is the > axiomatic theory of numbers, which Godel showed to be incomplete if it > is consistent. Godel also proved that one of the incompletenesses in > arithmetic was that it could not prove its own consistency. Stronger > logical systems can and have proven its consistency, but any > particular logical system cannot prove its own consistency. It seems > to me that the constructivist viewpoint says, "The so-called stronger > system merely defines truth in more cases; but, we could just as > easily take the opposite definitions." In other words, we're proving > arithmetic consistent only by adding to its definition, which hardly > counts. The classical viewpoint, of course, is that the stronger > system is actually correct. Its additional axioms are not arbitrary. > So, the proof reflects the truth. > > Which side do you fall on? > > --Abram > > On Mon, Oct 27, 2008 at 1:03 PM, Mark Waser <[EMAIL PROTECTED]> wrote: >>>> >>>> I, being of the classical persuasion, believe that arithmetic is either >>>> consistent or inconsistent. You, to the extent that you are a >>>> constructivist, should say that the matter is undecidable and therefore >>>> undefined. >> >> I believe that arithmetic is a formal and complete system. I'm not a >> constructivist where formal and complete systems are concerned (since >> there >> is nothing more to construct). >> >> On the other hand, if you want to try to get into the "meaning" of >> arithmetic . . . . >> >> = = = = = = = >> >>>> since the infinity of real numbers is larger than the infinity of >>>> possible names/descriptions. >> >> Huh? The constructivist in me points out that via compound constructions >> the infinity of possible names/descriptions is exponentially larger than >> the >> infinity of real numbers. You can reference *any* real number to the >> extent >> that you can define it. And yes, that is both a trick statement AND also >> the crux of the matter at the same time -- you can't name pi as a sequence >> of numbers but you certainly can define it by a description of what it is >> and what it does and any description can also be said to be a name (or a >> "true name" if you will :-). >> >>>> If the Gödelian truths are unreachable because they are undefined, then >>>> there is something *wrong* with the classical insistence that they are >>>> true >>>> or false but we just don't know which. >> >> They are undefined unless they are part of a formal and complete system. >> If >> they are part of a formal and complete system, then they are defined but >> may >> be indeterminable. There is nothing *wrong* with the classical insistence >> as long as it is applied to a limited domain (i.e. that of closed systems) >> which is what you are doing. >> >> >> ----- Original Message ----- From: "Abram Demski" <[EMAIL PROTECTED]> >> To: <[email protected]> >> Sent: Monday, October 27, 2008 12:29 PM >> Subject: Re: [agi] constructivist issues >> >> >> Mark, >> >> An example of people who would argue with the meaningfulness of >> classical mathematics: there are some people who contest the concept >> of real numbers. The cite things like that the vast majority of real >> numbers cannot even be named or referenced in any way as individuals, >> since the infinity of real numbers is larger than the infinity of >> possible names/descriptions. >> >> "OK. But I'm not sure where this is going . . . . I agree with all >> that you're saying but can't see where/how it's supposed to address/go >> back into my domain model ;-)" >> >> Well, you already agreed that classical mathematics is meaningful. >> But, you also asserted that you are a constructivist where meaning is >> concerned, and therefore collapse Godel's and Tarski's theorems. I do >> not think you can consistently assert both! If the Godelian truths are >> unreachable because they are undefined, then there is something >> *wrong* with the classical insistence that they are true or false but >> we just don't know which. >> >> To take a concrete example: One of these truths that suffers from >> Godelian incompleteness is the consistency of arithmetic. I, being of >> the classical persuasion, believe that arithmetic is either consistent >> or inconsistent. You, to the extent that you are a constructivist, >> should say that the matter is undecidable and therefore undefined. >> >> --Abram >> > > > ------------------------------------------- > agi > Archives: https://www.listbox.com/member/archive/303/=now > RSS Feed: https://www.listbox.com/member/archive/rss/303/ > Modify Your Subscription: https://www.listbox.com/member/?&; > Powered by Listbox: http://www.listbox.com > > > > > ------------------------------------------- > agi > Archives: https://www.listbox.com/member/archive/303/=now > RSS Feed: https://www.listbox.com/member/archive/rss/303/ > Modify Your Subscription: > https://www.listbox.com/member/?&; > Powered by Listbox: http://www.listbox.com > ------------------------------------------- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244&id_secret=117534816-b15a34 Powered by Listbox: http://www.listbox.com
