Chris, (01)
All I was trying to say is what you have just affirmed: (02)
CM> If you simply mean a *list*, then obviously no infinite set can
> be specified extensionally; but that's not very interesting. (03)
I agree. But Pat seemed to object. (04)
CM> For instance, the following specifies the set ω of finite
> von Neumann ordinals in ZF set theory:
>
> ω = the smallest set x containing ∅ and such that,
> if y∈x, then y∪{y}∈x. (05)
That is what I was calling a specification by a finite statement
of some rule or axiom for determining the elements of the set.
But Pat seemed to object. (06)
JFS>> Grammar rules:
>>
>> N -> 0
>> N -> S N
>>
>> This is a finite specification that generates a denoting expression
>> for all and only the natural numbers. You can add a few Peano-style
>> axioms to get the standard model without the weird number-like things
>> of the non-standard models. (07)
CM> If one were to formalize the grammar above as a first-order theory,
> even an infinite, undecidable theory (which of course wouldn't be much
> of a theory), there would be no way to rule out nonstandard
> interpretations of the constant "N". (08)
I agree. But I did *not* formalize it in FOL. What I did was to
*formalize* it as a grammar. A formal grammar is just as formal as
any statement in any version of logic. But nobody talks about
nonstandard models of a grammar. They talk about "all and only
those sentences generated by the grammar." That last clause
goes beyond ordinary FOL. (09)
CM> Seems to me clear that we "know" the standard model or we wouldn't
> recognize the distinction between it and the nonstandard ones. (010)
Yes indeed. We recognize that difference very well because we know
excellent formalizations of the standard model: (011)
The model of Peano's axioms that is limited to those numbers that
are denoted by the numerals generated by the above grammar. (012)
The fact that something cannot be formalized in FOL, by itself,
does not mean that it cannot be formally specified. (013)
John (014)
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