Date: Apr 12, 2013 10:23 AM
Author: Alan Smaill
Subject: Re: Matheology § 224

Nam Nguyen <namducnguyen@shaw.ca> writes:

> On 12/04/2013 7:50 AM, Alan Smaill wrote:
>> Nam Nguyen <namducnguyen@shaw.ca> writes:
>>

>>> On 12/04/2013 3:29 AM, Alan Smaill wrote:
>>>> Nam Nguyen <namducnguyen@shaw.ca> 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?


Because you now have two logics --
good old statements like cGC are not statements about syntax,
they are statements about numbers. The language structures for
your theory of syntax is different from language structures for
natural numbers.

>>> 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.


Not only -- see above.
But do you allow proof by induction over the syntax of formulas?

>>> 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.


How could it not be the case?

>> 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', {}).
>
> Right?


If you're happy with that, fine.

Now can you characterise the element of X that corresponds to "S(0)",
with "S" the symbol for successor?


--
Alan Smaill