The End of an Era - That of The Natural Numbers
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From: Nam Nguyen on 21 May 2010 02:05 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Why don't you make some reflections of your own and tell us if F >> is true or false in the naturals, > > The universal closure of (x = 0 \/ 0 < x) is certainly true in the > naturals. What does this have to do with anything? So then F is provable in Shoenfield's version of Robinson Arithmetic? Care to lay of the basic steps of the proof? > >> since you seem to believe the knowledge of the naturals is not of >> intuitive nature. > > On your peculiar definition of "intuitive" our knowledge of pretty much > anything is indeed intuitive. >
From: Nam Nguyen on 21 May 2010 02:08 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> William Hughes wrote: >> >>> Does an inconsistent system have a model? >> Yes it does: a false model. > > Nonsense. An inconsistent theory proves everything, including the > sentence (Ex)(x=x), which is not true in a model with an empty universe. Where did I say any formula is true in a false model?
From: William Hughes on 21 May 2010 02:12 On May 21, 3:01 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: > > On May 21, 2:41 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> William Hughes wrote: > >>> On May 21, 1:49 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >>>> William Hughes wrote: > > >>>>> ... an inconsistent system T does not have a model ... > >>>> Right. > >>> So can you answer yes or no: Does an inconsistent system > >>> have a model? > >> Yes it does > > > You are being inconsistent about inconsistent systems. > > If you keep chopping away my (technical) explanations, then > sure you could say anything you're pleased. No. I only comment on stuff you actually say. You have made two claims an inconsistent system T does not have a model an inconsistent system T does have a model You can make as many "explanations" as you like and use any definition of "model" you want, even introducing the heretofore unknown "false model". The two claims are inconsistent. - William Hughes
From: Aatu Koskensilta on 21 May 2010 02:16 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Aatu Koskensilta wrote: > >> The universal closure of (x = 0 \/ 0 < x) is certainly true in the >> naturals. What does this have to do with anything? > > So then F is provable in Shoenfield's version of Robinson Arithmetic? Why should that follow? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 21 May 2010 02:22
William Hughes wrote: > On May 21, 3:01 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> William Hughes wrote: >>> You are being inconsistent about inconsistent systems. >> If you keep chopping away my (technical) explanations, then >> sure you could say anything you're pleased. > > No. I only comment on stuff you actually say. > > You have made two claims > > an inconsistent system T does not have a model > an inconsistent system T does have a model > > You can make as many "explanations" as you like > and use any definition of "model" you want, even > introducing the heretofore unknown "false model". > The two claims are inconsistent. In your response on May 20, 11:31 PM, you wrote: "yes" and almost immediately thereafter you also had: "no". See, I'm not the only one around here who is inconsistent in making statements, in _THAT_ way! |