details re Z-R |- PA consistent
in [Math]
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From: Nam Nguyen on 5 Jul 2010 16:14 MoeBlee wrote: > On Jul 5, 11:52 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> Well, then, PA is consistent can be proven in T = {(x=x) /\ ~(x=x)}, > > Sure, but with the one technical quibble that the language for T > provides a formulation of "PA is not consistent". But, yes, on the > basic point, we agree. > >> which I did mention already. That should settle the issue of formal >> proof of PA's consistency! Why bother with any thing as complicated >> as Z-R or ZFC, or what have we? > > Exactly! I mean that NOT sarcastically. This is part of what we've > been saying ALL ALONG (in other threads, in various books, especially > as well explained in Franzen's incompleteness book). So OK then. Do you now agree with me that: [...] there's no formal proof that PA is consistent in a consistent theory (formal system)? A clear answer and that would settle the discussion here.
From: MoeBlee on 5 Jul 2010 16:20 On Jul 5, 1:14 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > On Jul 5, 11:52 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > >> Well, then, PA is consistent can be proven in T = {(x=x) /\ ~(x=x)}, > > > Sure, but with the one technical quibble that the language for T > > provides a formulation of "PA is not consistent". But, yes, on the > > basic point, we agree. > > >> which I did mention already. That should settle the issue of formal > >> proof of PA's consistency! Why bother with any thing as complicated > >> as Z-R or ZFC, or what have we? > > > Exactly! I mean that NOT sarcastically. This is part of what we've > > been saying ALL ALONG (in other threads, in various books, especially > > as well explained in Franzen's incompleteness book). > > So OK then. Do you now agree with me that: > > [...] there's no formal proof that PA is > consistent in a consistent theory (formal system)? > > A clear answer and that would settle the discussion here. I GAVE YOU a clear answer to that question in some detail in another thread. My answer does not take the form yes/no, for the reasons that can be seen in my fuller answer. Also, I just remarked in that thread that I'm not interested in responding to you in the manner of a deposition. I'm not going to give you bare "Yes"/"No" answers that require elaboration as to the actual sense and framework of the question. MoeBlee
From: Nam Nguyen on 5 Jul 2010 16:46 MoeBlee wrote: > On Jul 5, 1:14 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> MoeBlee wrote: >>> On Jul 5, 11:52 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> Well, then, PA is consistent can be proven in T = {(x=x) /\ ~(x=x)}, >>> Sure, but with the one technical quibble that the language for T >>> provides a formulation of "PA is not consistent". But, yes, on the >>> basic point, we agree. >>>> which I did mention already. That should settle the issue of formal >>>> proof of PA's consistency! Why bother with any thing as complicated >>>> as Z-R or ZFC, or what have we? >>> Exactly! I mean that NOT sarcastically. This is part of what we've >>> been saying ALL ALONG (in other threads, in various books, especially >>> as well explained in Franzen's incompleteness book). >> So OK then. Do you now agree with me that: >> >> [...] there's no formal proof that PA is >> consistent in a consistent theory (formal system)? >> >> A clear answer and that would settle the discussion here. > > I GAVE YOU a clear answer to that question in some detail in another > thread. My answer does not take the form yes/no, for the reasons that > can be seen in my fuller answer. Also, I just remarked in that thread > that I'm not interested in responding to you in the manner of a > deposition. I'm not going to give you bare "Yes"/"No" answers that > require elaboration as to the actual sense and framework of the > question. That's why our discussion in this thread (at least) goes around and around, despite _your and mine_ wishes to the contrary: you simply refused to answer direct to the point important but very easy to understand question. (Btw, I already mentioned "I don't know" is a valid question too!) For example, I'd answer "NO" (and already did) to the question. And I'd provide to you a clear cut explanation if you ask. Basically since _syntactically_ none of us can prove in meta level a disproof of a formula in _any_ T, which is required as part of syntactical definition of a T's consistency, so the answer to the question must be a NO. See, I don't have to invoke "another thread" or use the whole page to summarize or explain my answer. Again, MoeBlee, can you simply answer the question, as I've done so?
From: MoeBlee on 5 Jul 2010 16:48 On Jul 5, 1:46 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > On Jul 5, 1:14 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> MoeBlee wrote: > >>> On Jul 5, 11:52 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>> Well, then, PA is consistent can be proven in T = {(x=x) /\ ~(x=x)}, > >>> Sure, but with the one technical quibble that the language for T > >>> provides a formulation of "PA is not consistent". But, yes, on the > >>> basic point, we agree. > >>>> which I did mention already. That should settle the issue of formal > >>>> proof of PA's consistency! Why bother with any thing as complicated > >>>> as Z-R or ZFC, or what have we? > >>> Exactly! I mean that NOT sarcastically. This is part of what we've > >>> been saying ALL ALONG (in other threads, in various books, especially > >>> as well explained in Franzen's incompleteness book). > >> So OK then. Do you now agree with me that: > > >> [...] there's no formal proof that PA is > >> consistent in a consistent theory (formal system)? > > >> A clear answer and that would settle the discussion here. > > > I GAVE YOU a clear answer to that question in some detail in another > > thread. My answer does not take the form yes/no, for the reasons that > > can be seen in my fuller answer. Also, I just remarked in that thread > > that I'm not interested in responding to you in the manner of a > > deposition. I'm not going to give you bare "Yes"/"No" answers that > > require elaboration as to the actual sense and framework of the > > question. > > That's why our discussion in this thread (at least) goes around > and around, despite _your and mine_ wishes to the contrary: you simply > refused to answer direct to the point important but very easy to > understand question. (Btw, I already mentioned "I don't know" is > a valid question too!) We're done, Nam. MoeBlee
From: Nam Nguyen on 5 Jul 2010 16:53
MoeBlee wrote: > On Jul 5, 1:46 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> MoeBlee wrote: >>> On Jul 5, 1:14 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>> MoeBlee wrote: >>>>> On Jul 5, 11:52 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >>>>>> Well, then, PA is consistent can be proven in T = {(x=x) /\ ~(x=x)}, >>>>> Sure, but with the one technical quibble that the language for T >>>>> provides a formulation of "PA is not consistent". But, yes, on the >>>>> basic point, we agree. >>>>>> which I did mention already. That should settle the issue of formal >>>>>> proof of PA's consistency! Why bother with any thing as complicated >>>>>> as Z-R or ZFC, or what have we? >>>>> Exactly! I mean that NOT sarcastically. This is part of what we've >>>>> been saying ALL ALONG (in other threads, in various books, especially >>>>> as well explained in Franzen's incompleteness book). >>>> So OK then. Do you now agree with me that: >>>> [...] there's no formal proof that PA is >>>> consistent in a consistent theory (formal system)? >>>> A clear answer and that would settle the discussion here. >>> I GAVE YOU a clear answer to that question in some detail in another >>> thread. My answer does not take the form yes/no, for the reasons that >>> can be seen in my fuller answer. Also, I just remarked in that thread >>> that I'm not interested in responding to you in the manner of a >>> deposition. I'm not going to give you bare "Yes"/"No" answers that >>> require elaboration as to the actual sense and framework of the >>> question. >> That's why our discussion in this thread (at least) goes around >> and around, despite _your and mine_ wishes to the contrary: you simply >> refused to answer direct to the point important but very easy to >> understand question. (Btw, I already mentioned "I don't know" is >> a valid question too!) > > We're done, Nam. If you say so (without giving an clear cut answer, even the framework, you had sought for clarification, was confirmed to you)! I've done my best to encourage a spirited discussion on foundational problems! |