what to do about the Successor axiom once it reaches 999....9999#318; Correcting Math
in [Logic]
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From: Nam Nguyen on 31 Jan 2010 11:26 Nam Nguyen wrote: > Here is my attempt: basically summation of x and y would be defined as the > minimal _common_ upper bound of both x and y. > > Definition1: > ============ > > Sum(x,y,z) <-> l.u.b(x,y,z) > l.u.b(x,y,z) <-> cLim(x,y,z) /\ Az'[cLim(x,y,z') -> (z=z')] > cLim(x,y,z) <-> lim(x,z) /\ lim(y,z) > limt(x,y) <-> (x <= y) > > It looks less complicated than I feared, but would this seem a _reasonable_ > definition? Thanks. There's a typo that needs correction and a simplification that's desired. So here's a slightly different rendition: Sum(x,y,z) <-> cLim(x,y,z) /\ Az'[cLim(x,y,z') -> (z=z')] cLim(x,y,z) <-> lim(x,z) /\ lim(y,z) lim(x,y) <-> (x <= y)
From: Nam Nguyen on 31 Jan 2010 11:32 Nam Nguyen wrote: > > Here is my attempt: basically summation of x and y would be defined as the > minimal _common_ upper bound of both x and y. > > Definition1: > ============ > > Sum(x,y,z) <-> l.u.b(x,y,z) > l.u.b(x,y,z) <-> cLim(x,y,z) /\ Az'[cLim(x,y,z') -> (z=z')] > cLim(x,y,z) <-> lim(x,z) /\ lim(y,z) > limt(x,y) <-> (x <= y) > > It looks less complicated than I feared, but would this seem a _reasonable_ > definition? Thanks. There are a typo and a desired simplification in my above. So the following is a slightly different rendition of the same idea: Sum(x,y,z) <-> cLim(x,y,z) /\ Az'[cLim(x,y,z') -> (z=z')] cLim(x,y,z) <-> lim(x,z) /\ lim(y,z) lim(x,y) <-> (x <= y)
From: Nam Nguyen on 6 Feb 2010 01:40 Nam Nguyen wrote: > Aatu Koskensilta wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> OK. It's good to hear that. What would you think this might mean? >> >> It means whether we take 0, +, * and < or 0, S, +, * as primitives >> doesn't make much any indifference. >> >>> Say, don't you think it doesn't make sense now to hold something up as >>> "the" standard language of arithmetic, together with its "the" >>> standard model? >> >> No, I don't think that. What is meant by "the standard language of >> arithmetic" is dictated by context. In some cases we throw in >> exponentiation for good measure, in other contexts a more frugal supply >> of primitives is appropriate, and so on. Whatever mathematical substance >> there is to the logical study of number theory does not lie in such >> technical details, nor is there any philosophical insight to be gleaned >> from essentially notational bits of logical arcana. >> > > No it's not about philosophy or "logical arcana" at all: it's about > Induction, or not! OK. Let me start with some intuitions and then progressively present the technical aspect of the issue of induction-or-not-induction in arithmetic formal systems. Canadian 6/49 lottery is a sequence of 6 distinct numbers, each is between 1 and 49. Suppose you've bought this following ticket for next week lottery: (1) 1 2 3 30 34 47 Suppose further that next week has come and the winning number has been generated _randomly_ and is bein announced in TV, but there's a power outage in your area in which nobody would have a slightest clue what the winning number is, other than it would be like: (2) x1 x2 x3 x4 x5 x6 where the x's are _complete unknown_. Not just about money prize or any legality of it, but _even logically speaking_ no one would dare to say that you don't have a winning ticket but neither you could claim you've been a millionaire. The essence of what I'm going to say in this thread (or a next one) is that you got to accept this, say, _UT_ (_Unknown-ability Thesis_) that: it's _impossible_ for one to know if (1) and (2) be the same or different. If you don't accept this UT, there's not much we could discuss. But if you do accept, I'll progressively demonstrate that there's a same-essence UT regarding to the body of knowledge known as the standard model of any FOL formal system sufficiently portraying basic arithmetics of the natural numbers. *** Assuming you've accepted the original UT I'll technical present steps to accept the counterpart (FOL) UT regarding to the naturals. First let's assume the following languages: - L1 = L2(0,S,+) - L2 = L1(0,<,*) - L = L(0,S,+,*,<) Now the steps: Step1: Demonstrate that given a same infinite universe U of individuals, there are an infinite list List1 described solely in L1 and an infinite List2 in L2 that it's impossible to determine/formulate their _exact_ alignment against each other. (This is in essence the FOL counterpart UT). Step2: Demonstrate that since L is the combined language of L1 and L2, the undefinability of the exactness mentioned in Step 1 will transform into a an arithmetic truth value that if being "true" would be impossible to know. Step3: Demonstrate that, as alluded in Step2, if there's an an arithmetic truth value that would be impossible to know as true, if being "true", there are infinite arithmetic truth values whose truth values can't be assigned! These are effectively truth-unassigned- able formulas, reflecting concepts independent of the concept of the natural numbers. Step4: Given the infinitely many truth-unassigned-able formulas mentioned in Step3, we could arrive at the meta statement (say, G2IT [Godel- Goldbach Incompleteness Theorem]) which would say: For _any_ overall concept A of the natural numbers, there's at least a concept say G(A) that's independent from A, in the sense that it's impossible to assign a truth value - in the sense of A - to that concept.
From: Nam Nguyen on 6 Feb 2010 01:49 Nam Nguyen wrote: > Nam Nguyen wrote: >> Aatu Koskensilta wrote: >>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>> >>>> OK. It's good to hear that. What would you think this might mean? >>> >>> It means whether we take 0, +, * and < or 0, S, +, * as primitives >>> doesn't make much any indifference. >>> >>>> Say, don't you think it doesn't make sense now to hold something up as >>>> "the" standard language of arithmetic, together with its "the" >>>> standard model? >>> >>> No, I don't think that. What is meant by "the standard language of >>> arithmetic" is dictated by context. In some cases we throw in >>> exponentiation for good measure, in other contexts a more frugal supply >>> of primitives is appropriate, and so on. Whatever mathematical substance >>> there is to the logical study of number theory does not lie in such >>> technical details, nor is there any philosophical insight to be gleaned >>> from essentially notational bits of logical arcana. >>> >> >> No it's not about philosophy or "logical arcana" at all: it's about >> Induction, or not! > > OK. Let me start with some intuitions and then progressively present the > technical aspect of the issue of induction-or-not-induction in arithmetic > formal systems. > > Canadian 6/49 lottery is a sequence of 6 distinct numbers, each is > between 1 and 49. Suppose you've bought this following ticket for next > week lottery: > > (1) 1 2 3 30 34 47 > > Suppose further that next week has come and the winning number has been > generated _randomly_ and is bein announced in TV, but there's a power > outage in your area in which nobody would have a slightest clue what the > winning number is, other than it would be like: > > (2) x1 x2 x3 x4 x5 x6 > > where the x's are _complete unknown_. Not just about money prize or any > legality of it, but _even logically speaking_ no one would dare to say > that you don't have a winning ticket but neither you could claim you've > been > a millionaire. > > The essence of what I'm going to say in this thread (or a next one) is that > you got to accept this, say, _UT_ (_Unknown-ability Thesis_) that: > > it's _impossible_ for one to know if (1) and (2) be the same or > different. > > If you don't accept this UT, there's not much we could discuss. But if you > do accept, I'll progressively demonstrate that there's a same-essence UT > regarding to the body of knowledge known as the standard model of any FOL > formal system sufficiently portraying basic arithmetics of the natural > numbers. > > *** > > Assuming you've accepted the original UT I'll technical present steps to > accept the counterpart (FOL) UT regarding to the naturals. > > First let's assume the following languages: > > - L1 = L2(0,S,+) > - L2 = L1(0,<,*) > - L = L(0,S,+,*,<) > > Now the steps: > > Step1: Demonstrate that given a same infinite universe U of individuals, > there are an infinite list List1 described solely in L1 and an > infinite List2 in L2 that it's impossible to determine/formulate > their _exact_ alignment against each other. > > (This is in essence the FOL counterpart UT). > > Step2: Demonstrate that since L is the combined language of L1 and L2, the > undefinability of the exactness mentioned in Step 1 will transform > into a an arithmetic truth value that if being "true" would be > impossible to know. > > Step3: Demonstrate that, as alluded in Step2, if there's an an arithmetic > truth value that would be impossible to know as true, if being > "true", there are infinite arithmetic truth values whose truth > values can't be assigned! These are effectively truth-unassigned- > able formulas, reflecting concepts independent of the concept of > the natural numbers. > > Step4: Given the infinitely many truth-unassigned-able formulas mentioned > in Step3, we could arrive at the meta statement (say, G2IT [Godel- > Goldbach Incompleteness Theorem]) which would say: > > For _any_ overall concept A of the natural numbers, there's at least > a concept say G(A) that's independent from A, in the sense that it's > impossible to assign a truth value - in the sense of A - to that > concept. Please make that "...a concept say G2(A)...", in tandem with G2IT.
From: Nam Nguyen on 6 Feb 2010 11:04
Nam Nguyen wrote: > First let's assume the following languages: > > - L1 = L2(0,S,+) > - L2 = L1(0,<,*) > - L = L(0,S,+,*,<) > What a horrible typo! (Sorry). It should have been: > - L1 = L1(0,S,+) > - L2 = L2(0,<,*) > - L = L(0,S,+,*,<) |