what to do about the Successor axiom once it reaches 999....9999#318; Correcting Math
in [Logic]
|
From: Nam Nguyen on 1 Feb 2010 20:26 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Nam Nguyen wrote: >> >>> The both '<' and ZF's 'e' are 2-nary symbols so we'd translate >>> L-formulas into L(ZF), and then transform them further to L' by >>> replacing 'e' by '<' >>> >>> It'd still be "horrible" formulas to look at in the end, but we know >>> for sure, thanks to ZF, the translation/transformation is >>> syntactically possible. As for semantics, being a member of a set x >>> could be interpreted as being "less-than" - at least in part-hood >>> (mereology) sense. >>> >>> Would this "solution" work? Thanks again. >> [This seems to expose some element of circularity in Godel's work.] > > How is that? > Can't we encode the naturals using ZF?
From: Nam Nguyen on 1 Feb 2010 21:01 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Can't we encode the naturals using ZF? > > Sure. How does this "expose some element of circularity in G�del's > work"? > Two given formal systems _equally_ able to express arithmetic and _equally_ able to encode each other and you don't see any circularity?
From: Nam Nguyen on 1 Feb 2010 21:05 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Reviewing my purposes, however, I think I also chose the wrong >> _lesser_ language to express some arithmetic concepts (e.g. GC). I >> should have chosen L'' = L''(0,+,*,<). I'll into L'' more. > > I don't know about your purposes but it's a trivial exercise to > translate any statement in the usual language of arithmetic into L'' > (and vice versa). > OK. It's good to hear that. What would you think this might mean? 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?
From: Nam Nguyen on 1 Feb 2010 21:19 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Two given formal systems _equally_ able to express arithmetic and >> _equally_ able to encode each other and you don't see any circularity? > > You've mentioned ZF -- what's the other formal system? In any case, you > have provided no indication of any "element of circularity in G�del's > work". > One other system would be some written in L''. (Sorry I have to run now but I'll come back about circularity later).
From: Nam Nguyen on 1 Feb 2010 21:20
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! |