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

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Posts: 1,968
Registered: 12/4/12
Re: Matheology S 224
Posted: Apr 21, 2013 1:21 PM
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On 4/21/2013 11:13 AM, Nam Nguyen wrote:
> 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).

the mistake is mine.

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