[FOM] To Vladimir Sazonov and others doubting the unambiguity of N

Aatu Koskensilta aatu.koskensilta at xortec.fi
Thu Jun 12 01:39:36 EDT 2003

Vladimir Sazonov wrote:
> What (as you say) "disinformed" me? Some deeper that in the school 
> things like Goedel's theorems, especially on incompleteness and  
> Goedel/Cohen proof on independence of CH, what demonstrated (to me)  
> that both N and continuum are vague concepts, [ --- ]

Like [] I can understand your position with regards to the continuum, 
but as to N, I'm still baffled. Surely the notion of *non*-standard 
model of arithmetic is much more illusive, as any such model must 
necessarily be non-recursive?

There seems to be no such clear distinction with standard and 
non-standard models of set theory (let alone the notion of "the" 
standard model of set theory), and thus I can appreciate the idea that 
there is something inherently vague to the continuum or the even more 
substantially infinitistic set theoretic objects. But N? There seems to 
be a genuine *mathematical* distinction here; the standard model is the 
recursive model, and the non-standard ones are the non-recursive ones.

 From your postings I gather this won't satisfy you, but I'd be 
interested to know whether Gödel's theorems merely motivated you to 
question the platonistic picture of mathematics or do you believe they 
server as arguments against such a position?

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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