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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: 2,164
Registered: 10/4/10
Re: Matheology S 224
Posted: Apr 20, 2013 4:40 PM
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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.

> Fwiw, I had claimed I'd "borrow", say, 'e' and others like '='
> as much as I could to express meta level objects (unformalized sets,
> individuals, and what not).


--
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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