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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 20, 2013 5:59 PM
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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 <> 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?

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

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.

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