Pat replied to John:
> On Jan 16, 2010, at 9:49 AM, John F. Sowa wrote:
>
>> Pat,
>>
>> I have nothing against using sets, finite or infinite. I was just making the
>observation that you can't give an extensional specification for infinite sets. (01)
John, I'm a bit unclear on what one mean by an extensional specification of a
set. If you simply mean a *list*, then obviously no infinite set can be
specified extensionally; but that's not very interesting. However, I'm having
trouble formulating a clear idea of anything more substantial. For instance,
the following specifies the set ω of finite von Neumann ordinals in ZF set
theory: (02)
ω = the smallest set x containing ∅ and such that, if y∈x, then
y∪{y}∈x. (03)
(This can of course be written out formally in the language of ZF.) I don't
see any sense in which this specification is intensional. By my lights, an
intensional specification would be one that makes essential use of intensional
concepts, e.g., modal or epistemic notions. But the specification above
(ultimately) uses only the apparatus of first-order logic and the notion of
membership, which, in ZF at least, seems to be the very paradigm of
extensionality. (04)
>> JFS>> ... the integers themselves must be specified by some rules or axioms.
>>
>> PH> Most mathematicians would disagree. The natural numbers cannot be fully
>specified by any finite number of axioms, yet we all feel that we know what
>they are, and are quite happy to refer to them.
>>
>> I agree. But you can state a finite set of metalevel rules for generating a
>denoting expression for each natural number.
>
> Of course, they are called the numerals. Now you have to somehow say that the
>things denoted by these are precisely the natural numbers, and to do that you
>have to be able to refer to the natural numbers, and thats what you can't do
>in any finite axiomatization.
>
>> [JFS] For example, I can say that any string of terminal symbols generated
>by the following grammar denotes a natural number:
>>
>> Terminal symbols: {S, 0}
>>
>> Nonterminal symbols: {N}
>>
>> 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.
>
> [PH] No you can't. There is no finite axiomatization, in any form, that has
>just the standard model as its only satisfying interpretation. (05)
Right. 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". (06)
> (Goedel.) (07)
The existence of nonstandard models of Peano Arithmetic certainly follows from
Gödel's Theorem, but we get the simple existence of nonstandard models of PA
just from the Löwenheim-Skolem theorem. What we get from Gödel is the even
stronger result that there are nonstandard models of Peano Arithmetic (or ZF)
in which some true sentences of arithmetic (i.e., sentences that are true in
the standard model) are false. (08)
> And yet, we still feel that we 'know' the standard model. And I'm quite happy
>to take that for granted and move on. (09)
Right. Seems to me clear that we "know" the standard model or we wouldn't
recognize the distinction between it and the nonstandard ones. (010)
Chris Menzel (011)
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