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Topic: Matheology § 224
Replies: 84   Last Post: Apr 20, 2013 4:43 PM

 Messages: [ Previous | Next ]
 fom Posts: 1,968 Registered: 12/4/12
Re: Matheology S 224
Posted: Apr 17, 2013 1:03 PM

On 4/17/2013 11:29 AM, Frederick Williams wrote:
> fom wrote:
>>
>> 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?

>
> Yes, that is what I had in mind. Have I misrepresented Kunen?
>

Ran across this yesterday. It has a mention about set theory
that is relevant to your statement.

http://plato.stanford.edu/entries/truth-axiomatic/#2.1

It is not much. Just that truth definitions do not really
help since one needs something stronger than set theory
upon which to base the construction. I guess the arithmetic
of natural numbers relies on the Gentzen consistency proof
as a basis for the sensibility of truth predicates. But,

I do not really know much about arithmetic and its
metamathematics.

Date Subject Author
4/12/13 Alan Smaill
4/12/13 namducnguyen
4/12/13 Frederick Williams
4/12/13 fom
4/13/13 namducnguyen
4/13/13 fom
4/13/13 namducnguyen
4/13/13 fom
4/13/13 namducnguyen
4/13/13 Peter Percival
4/13/13 namducnguyen
4/13/13 Peter Percival
4/13/13 namducnguyen
4/13/13 Peter Percival
4/13/13 namducnguyen
4/13/13 Jesse F. Hughes
4/14/13 namducnguyen
4/14/13 namducnguyen
4/14/13 namducnguyen
4/14/13 Peter Percival
4/14/13 fom
4/14/13 namducnguyen
4/14/13 fom
4/14/13 namducnguyen
4/14/13 namducnguyen
4/14/13 fom
4/14/13 namducnguyen
4/14/13 namducnguyen
4/14/13 namducnguyen
4/14/13 Jesse F. Hughes
4/14/13 namducnguyen
4/14/13 Jesse F. Hughes
4/14/13 namducnguyen
4/16/13 namducnguyen
4/16/13 namducnguyen
4/16/13 Jesse F. Hughes
4/16/13 namducnguyen
4/16/13 fom
4/17/13 namducnguyen
4/17/13 fom
4/17/13 namducnguyen
4/17/13 Jesse F. Hughes
4/17/13 Jesse F. Hughes
4/17/13 namducnguyen
4/20/13 namducnguyen
4/17/13 Frederick Williams
4/17/13 Frederick Williams
4/17/13 fom
4/17/13 Frederick Williams
4/17/13 fom
4/17/13 fom
4/18/13 namducnguyen
4/18/13 Frederick Williams
4/18/13 namducnguyen
4/19/13 Frederick Williams
4/19/13 namducnguyen
4/20/13 Frederick Williams
4/19/13 Frederick Williams
4/19/13 namducnguyen
4/20/13 Frederick Williams
4/14/13 Jesse F. Hughes
4/14/13 namducnguyen
4/14/13 namducnguyen
4/14/13 Jesse F. Hughes
4/14/13 namducnguyen
4/14/13 Peter Percival
4/15/13 Peter Percival
4/14/13 namducnguyen
4/14/13 namducnguyen
4/13/13 Frederick Williams
4/13/13 Peter Percival
4/13/13 Peter Percival
4/13/13 namducnguyen
4/15/13 Peter Percival
4/13/13 fom
4/13/13 namducnguyen
4/13/13 Peter Percival
4/13/13 namducnguyen
4/13/13 Frederick Williams
4/14/13 Frederick Williams
4/14/13 namducnguyen
4/13/13 Peter Percival
4/13/13 namducnguyen
4/13/13 namducnguyen