Re: Nonfirstorderizability

"Michael De" <mikejde@xxxxxxxxx> writes:

> I was attempting, at all costs, to formalize your nonfirstorderizable
> sentence. Actually I didn't really mean to mention infinitary formulas.
> It would have sufficed to say that Eyy=x is true for at least one
> substitution instance given 0, f(0),... as substituends. I just thought
> that version was less popular.

While it is informative that the condition is formalizable as x=0 v
x=f(0) v x=f(f(0)) v ..., that is, in the much-studied infinitary
logic L(w_1,w), the substitutional quantification version is
essentially vacuous. Anyway, it's unclear why you should seek a logic
in which the condition is formalizable, since the point of the example
was that the unformalizability in first order logic of
"x is one of 0, f(0), f(f(0)) and so on" is precisely the reason why
the standard model of arithmetic cannot be characterized in first
order logic, and thus points the way to a direct argument using
compactness for the unformalizability of the GK-sentence.


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