Godels theorem ends in paradox-simply proof
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From: Nam Nguyen on 8 Jun 2010 23:05 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Meantime, are you then saying the naturals is collectively a model >> of PA? > > No -- I don't understand what is meant by "the naturals is collectively > a model of PA". Collectively as opposed to what? It's actually simple: is the set of the naturals the U of a model of PA?
From: Aatu Koskensilta on 8 Jun 2010 23:07 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > It's actually simple: is the set of the naturals the U of a model of > PA? Sure, there are many models of PA with the naturals as the domain. -- 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 8 Jun 2010 23:21 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> It's actually simple: is the set of the naturals the U of a model of >> PA? > > Sure, there are many models of PA with the naturals as the domain. > How do you prove PA is syntactically consistent in the first place to even have a model, let a lone one with something called "the naturals" as its U? Can I also claim, say, PA + ~(1) to have "the naturals" as the U of its many models way you claim them be for PA's models? What about PA + (1)? [Btw, out of the naturals as the U, which model would you think should be called as "the standard model" of PA?]
From: Daryl McCullough on 9 Jun 2010 00:01 Nam Nguyen says... > >Aatu Koskensilta wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> It's actually simple: is the set of the naturals the U of a model of >>> PA? >> >> Sure, there are many models of PA with the naturals as the domain. >> > >How do you prove PA is syntactically consistent in the first place >to even have a model, let a lone one with something called "the naturals" >as its U? You've got the order wrong. People understood the natural numbers a long before they were formalized in PA. The axioms of PA were specifically chosen *because* they were clearly true of the naturals, as people already understood them. A collection of true statements cannot be inconsistent. Now, of course you can speculate that we are deluding ourselves by believing that we understand what the naturals are. It's not a very *interesting* speculation, but you are free to speculate. >Can I also claim, say, PA + ~(1) to have "the naturals" as the U of its >many models way you claim them be for PA's models? What about PA + (1)? I don't remember what your statement (1) is, so I can't answer. But in general, either a statement is true of the naturals, or its negation is. Adding one or the other will force nonstandard models. >[Btw, out of the naturals as the U, which model would you think should >be called as "the standard model" of PA?] If M is any model of PA, with an associated domain U, then M is standard if it has no proper submodels. That is, if you replace the domain U by a proper subset U', then the result is not a model of PA. -- Daryl McCullough Ithaca, NY
From: Aatu Koskensilta on 9 Jun 2010 00:05
stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: > I don't remember what your statement (1) is, so I can't answer. But in > general, either a statement is true of the naturals, or its negation > is. Adding one or the other will force nonstandard models. Sure, but a non-standard model may well have the naturals as its domain. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |