Surreal numbers and omnific integers
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From: Rupert on 9 Aug 2010 19:18 At the end of Chapter 4 of "On Numbers and Games" Conway writes "The reader mightbe tempted to suppose that the subRing of omnific integers described in the next chapter was in a similar way a non- standard Model for the ordinary integers. But of course this is not so, since for instance x^2=2y^2 has many non-zero omnific integer solutions. In fact deep logical theorems tell us that we could not hope to find a non-standard model for Z in so simple a way." I wonder which logical theorems Conway has in mind here.
From: Herman Jurjus on 10 Aug 2010 02:07 On 8/10/2010 1:18 AM, Rupert wrote: > At the end of Chapter 4 of "On Numbers and Games" Conway writes > > "The reader mightbe tempted to suppose that the subRing of omnific > integers described in the next chapter was in a similar way a non- > standard Model for the ordinary integers. But of course this is not > so, since for instance x^2=2y^2 has many non-zero omnific integer > solutions. In fact deep logical theorems tell us that we could not > hope to find a non-standard model for Z in so simple a way." > > I wonder which logical theorems Conway has in mind here. Tennenbaum's theorem? -- Cheers, Herman Jurjus
From: Rupert on 10 Aug 2010 02:58 On Aug 10, 4:07 pm, Herman Jurjus <hjm...(a)hetnet.nl> wrote: > On 8/10/2010 1:18 AM, Rupert wrote: > > > At the end of Chapter 4 of "On Numbers and Games" Conway writes > > > "The reader mightbe tempted to suppose that the subRing of omnific > > integers described in the next chapter was in a similar way a non- > > standard Model for the ordinary integers. But of course this is not > > so, since for instance x^2=2y^2 has many non-zero omnific integer > > solutions. In fact deep logical theorems tell us that we could not > > hope to find a non-standard model for Z in so simple a way." > > > I wonder which logical theorems Conway has in mind here. > > Tennenbaum's theorem? > > -- > Cheers, > Herman Jurjus It's not really clear to me how that would apply. The omnific integers are a proper class and certainly do not constitute a recursive model. I think I have figured it out. The field of quotients of the omnific integers are a real-closed field. But the field of quotients of a nonstandard model of the integers cannot be a real-closed field.
From: George Greene on 10 Aug 2010 20:56 On Aug 10, 2:58 am, Rupert <rupertmccal...(a)yahoo.com> wrote: > I think I have figured it out. The field of quotients of the omnific > integers are a real-closed field. But the field of quotients of a > nonstandard model of the integers cannot be a real-closed field. What exactly is it about a non-standard model of the integers that would make it non-standard? Equivalently, what is the defining feature of the standard model? What makes THAT model standard? The other two canonical cases (PA and ZFC) are "extremal" in some sense; the standard model of PA is a submodel of the others and the standard/ intended model of ZFC is the "biggest" (the one with "full" powersets). I am guessing that a standard model of the integers would be standard in something more akin to the PA sense, but I really have no idea.
From: Aatu Koskensilta on 10 Aug 2010 21:05
George Greene <greeneg(a)email.unc.edu> writes: > What exactly is it about a non-standard model of the integers that > would make it non-standard? Not being isomorphic to the integers. As with the naturals we can characterize the integers (up to isomorphism as usual) as the smallest model of a certain formal theory. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |