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Topic: Matheology § 224
Replies: 2   Last Post: Apr 12, 2013 10:14 AM

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Posts: 2,777
Registered: 12/13/04
Re: Matheology § 224
Posted: Apr 12, 2013 10:14 AM
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On 12/04/2013 7:50 AM, Alan Smaill wrote:
> Nam Nguyen <> writes:

>> On 12/04/2013 3:29 AM, Alan Smaill wrote:
>>> Nam Nguyen <> writes:

>>>> But what is "meta-logic of the meta-language", in the context of FOL
>>>> structure? Or is that at best just intuition and at worst just a
>>>> buzzword?

>>> You tell us that it is possible to reason about language structures.
>>> What logic are you using to do that -- or is that at best just intuition?

>> I've used FOL ( _First Order Logic_ ) definitions that one should be
>> familiar with.
>> If anything, notation like "this" is defined in term of FOL
>> terminologies.

> ie, you use a first-order meta-logic.

OK. Why don't we just call it First Order Logic as everyone is supposed
to understand already?
>> So I don't see all that historical context of "meta-logic" would
>> have anything to do with the issue of, say, whether or not it's
>> impossible to construct the naturals as a language model.

> The strength of the meta-logic does have a bearing.
> Do you allow proof by induction over the syntax of formulas,
> for example?

So you're talking the strength (and weakness) of First Order Logic.
>> If you could construct it, as I did construct Mg, M1, ..., then present
>> the construction, otherwise at least for the time being admit you
>> couldn't do it.
>> Why would that be such a difficult task for one to do?

> That's not what's at issue here.
> Suppose you have a language structure for the language of Peano
> Arithmetic where all the axioms of PA are true, and suppose
> that the underlying set is X, and "0" is the constant used
> for the number zero.

"Suppose" is hypothetical but sure, I could hypothesize such.
> Is there a way of characterising the element of Z correponding to "0"?
> Or is that a matter of opinion?

But, FOL definition of language structure already provides that, e.g.:

('0', {}).


There is no remainder in the mathematics of infinity.


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