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 29 Jan 2010 13:23 Jesse F. Hughes wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > > There is no minimal element in the integers and thus, according to > your definition, the integers are all infinite. That's right. But I'm 100% certain I used the word "tweak" before with you in this thread-topic. (Anyone can tweak the definition to make it work for integers, if one's really interest to do so).
From: Nam Nguyen on 29 Jan 2010 13:50 Nam Nguyen wrote: > Nam Nguyen wrote: > > Another example, meant to be about certain numbers, would be: > > Interesting(x) <-> (x=S0)-> (x+x=0)] > NotInteresting(x) <-> ~Interesting(x) > > How would *you* decide the truth of Interesting(S0) in N and arithmetic > modulo 2? Explain *why* you'd make such decisions? > > Do you now see the issue of defining concept based _solely on semantics_ > of symbols? And if somehow the word "Interesting" isn't mathematical then: Modulo(x) <-> (x=S0)-> (x+x=0)]
From: Nam Nguyen on 29 Jan 2010 14:05 OK. One final thrust from me here on the topic before creating a new thread that would involve a lot of semantics-definitions and translations. We know L(PA) = L(0,S,+,<). Suppose then L' = L'(0,<,*), would we be able to express GC strictly in L'? At least what would be a road map? Any constructive idea would be appreciated.
From: Nam Nguyen on 29 Jan 2010 14:09 Nam Nguyen wrote: > OK. One final thrust from me here on the topic before creating > a new thread that would involve a lot of semantics-definitions > and translations. > > We know L(PA) = L(0,S,+,<). Suppose then L' = L'(0,<,*), would > we be able to express GC strictly in L'? At least what would be a > road map? Any constructive idea would be appreciated. Sorry for the typo: of course L(PA) = L(0,S,+,*,<).
From: Nam Nguyen on 29 Jan 2010 14:27
Nam Nguyen wrote: > Nam Nguyen wrote: >> OK. One final thrust from me here on the topic before creating >> a new thread that would involve a lot of semantics-definitions >> and translations. >> >> We know L(PA) = L(0,S,+,<). Suppose then L' = L'(0,<,*), would >> we be able to express GC strictly in L'? At least what would be a >> road map? Any constructive idea would be appreciated. > > Sorry for the typo: of course L(PA) = L(0,S,+,*,<). OK, let me start. Basically the set of the 2-tuples for 'S' is a subset of that for '<', meaning this subset could be defined strictly in term of '0', '=', '<' and other logical symbols. Therefore in formulas involving S, S could be eliminated. In addition, '+' can be defined in term of 'S' and hence would also be eliminated-able. Would you think this road map is logically sound? |