Date: Apr 21, 2013 12:03 PM Author: Frederick Williams Subject: Re: Matheology S 224 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.

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