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Re: Matheology S 224
Posted:
Apr 21, 2013 12:03 PM


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 nonempty 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 nonempty > >>>>>> 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 nonempty you > >>>>> have to say that something is in x, not that x is in something. > >>>> > >>>> but the claim was that x *is in* a nonempty set  > >>>> in this case S', which is nonempty, 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 (definitioninuse) > >>> > >>> 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 secondorder 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 nonempty 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 nonempty 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



