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Topic: Matheology S 224
Replies: 16   Last Post: Apr 21, 2013 6:53 PM

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namducnguyen

Posts: 2,699
Registered: 12/13/04
Re: Matheology S 224
Posted: Apr 21, 2013 12:13 PM
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On 21/04/2013 10:03 AM, Frederick Williams wrote:
> fom wrote:
>>
>> On 4/20/2013 3:40 PM, Frederick Williams wrote:

>>> Nam Nguyen wrote:
>>>>
>>>> On 20/04/2013 8:59 AM, fom wrote:

>>>>> On 4/20/2013 5:25 AM, Alan Smaill wrote:
>>>>>> Frederick Williams <freddywilliams@btinternet.com> writes:
>>>>>>

>>>>>>> Nam Nguyen wrote:
>>>>>>>>
>>>>>>>> On 19/04/2013 5:55 AM, Frederick Williams wrote:

>>>>>>>>> Nam Nguyen wrote:
>>>>>>>>>>
>>>>>>>>>> On 18/04/2013 7:19 AM, Frederick Williams wrote:

>>>>>>>
>>>>>>>>
>>>>>>>>>
>>>>>>>>>>> Also, as I remarked elsewhere, "x e S' /\ Ay[ y e S' -> y e S]"
>>>>>>>>>>> doesn't
>>>>>>>>>>> express "x is in a non-empty subset of S".

>>>>>>>>>>
>>>>>>>>>> Why?

>>>>>>>>>
>>>>>>>>> It says that x is in S' and S' is a subset of S.

>>>>>>>>
>>>>>>>> How does that contradict that it would express "x is in a non-empty
>>>>>>>> subset of S", in this context where we'd borrow the expressibility
>>>>>>>> of L(ZF) as much as we could, as I had alluded before?

>>>>>>>
>>>>>>> You really are plumbing the depths. To express that x is non-empty you
>>>>>>> have to say that something is in x, not that x is in something.

>>>>>>
>>>>>> but the claim was that x *is in* a non-empty set --
>>>>>> in this case S', which is non-empty, since x is an element of S',
>>>>>> and S' is a subset of S.
>>>>>>
>>>>>> (Much though it would be good for Nam to realise that
>>>>>> some background set theory axioms would be kind of useful here)
>>>>>>

>>>>>
>>>>> Yes. I thought about posting some links indicating
>>>>> that primitive symbols are undefined outside of a
>>>>> system of axioms (definition-in-use)
>>>>>
>>>>> The other aspect, though, is that Nam appears to be using an
>>>>> implicit existence assumption. So,
>>>>>
>>>>> AxASES'(xeS' /\ Ay(yeS' -> yeS))
>>>>>
>>>>> clarifies the statement and exhibits its second-order nature.
>>>>> This is fine since he claims that his work is not in the
>>>>> object language.

>>>>
>>>> Right.

>>>
>>> If fom's formula is to express "x is in a non-empty subset of S" then it
>>> needs to have both x and S free, so delete the first two quantifiers.
>>>

>>
>> Do you have a particular x and S in mind?

>
> I probably misunderstood. If Nam saying that, for every x and every set
> S, x is in a non-empty subset of S, then your formula expresses that.
> But clearly it is false.


I didn't say "every x and every set S". In the underlying context I was
talking about, x is _an_ individual and S is _a_ set (however general
each might be).

>
>> Or are we reverting to the distinction between real
>> and apparent variables from the first "Principia
>> Mathematica"?

>
> I call them free and bound respectively.
>

>> Or are we interpreting a statement in relation to a
>> general usage over an unspecified domain? My quantifiers
>> are in place to make clear the meaning for general usage.
>>
>> Within any context involving proof, the leading quantifiers
>> obey rules:
>>
>> |AxASES'(xeS' /\ Ay(yeS' -> yeS))
>> |ASES'(teS' /\ Ay(yeS' -> yeS))
>> |ES'(teS' /\ Ay(yeS' -> yeP))
>> ||(xeP' /\ Ay(yeP' -> yeP))
>>
>> The original statement is assumed (hence, is stroked)
>>
>> The existential statement is assumed (hence, a second stroke)
>>
>> Now the presuppostions of use are clear.
>>
>> That was my only purpose.



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

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



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