fom
Posts:
1,717
Registered:
12/4/12
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Re: Matheology S 224
Posted:
Apr 17, 2013 10:48 AM
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On 4/17/2013 9:36 AM, Frederick Williams wrote:
> Nam Nguyen wrote:
>
>> "x is in a non-empty subset of S" could be _expressed_ as a FOL language
>> expression: x e S' /\ Ay[ y e S' -> y e S].
>>
>> On the other hand, in "x is proven to be in a non-empty subset of S",
>> the _meta phrase_ "is proven" can not be expressed by a FOL language:
>> "is proven" pertains to a meta truth, which in turns can't be equated
>> to a language expression: truth and semantics aren't the same.
>
> "x is in a non-empty subset of S" can be expressed in the language of a
> first order theory with a binary predicate e. The intended meaning of e
> is given by the non-logical axioms of that theory.
>
> What reason is there to suppose that "x is proven" cannot be expressed
> in the language of a first order theory with a unary predicate p (say)?
> The intended meaning of p would then be given by the non-logical axioms
> of that theory.
>
> Note that set theory can express its own provability predicate.
>
Really? Are you referring to, say, Kunen's discussion of
Tarski's undefinability of truth by representing formulas
with their Goedel numbers?
I tried to warn Nam that your question would come up.
news://news.giganews.com:119/aoSdnaXPlOO2gfPMnZ2dnUVZ_sKdnZ2d@giganews.com
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