Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Matheology § 224
Replies: 2   Last Post: Apr 12, 2013 10:25 PM

 Messages: [ Previous | Next ]
 namducnguyen Posts: 2,777 Registered: 12/13/04
Re: Matheology § 224
Posted: Apr 12, 2013 10:25 PM

On 12/04/2013 8:23 AM, Alan Smaill wrote:
> 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

>
> 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.

I don't know what (or why) you're talking about _two_ logic. Over the
years and in many threads what I've been presenting is about FOL
definition of language structure. And there's only _one_ of such a thing
referred to as _the_ concept FOL language structure.
>
>>>> 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?

Your use of "proof" is too loose to make any sense: _exactly what_ is
being proven by induction "over the syntax of formulas"? Iow, proof
of _what_ "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?

Listen. I merely repeated (to emphasize) what you've stated "Suppose".
>
>>> 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.

What an odd observation. It's not the issue of "happiness": it's the
issue of my presentation follows FOL definition of language structure!
>
> Now can you characterise the element of X that corresponds to "S(0)",
> with "S" the symbol for successor?

What does that have anything to do with my presenting the issue under
debate, using FOL definition of language structure?

--
----------------------------------------------------
There is no remainder in the mathematics of infinity.

NYOGEN SENZAKI
----------------------------------------------------

Date Subject Author
4/12/13 Alan Smaill
4/12/13 namducnguyen