Date: Apr 20, 2013 4:40 PM
Author: Frederick Williams
Subject: Re: Matheology S 224
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