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

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Frederick Williams

Posts: 2,166
Registered: 10/4/10
Re: Matheology S 224
Posted: Apr 21, 2013 12:03 PM
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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.

> 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.



--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting



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