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Re: Matheology S 224
Posted:
Apr 21, 2013 12:13 PM


On 21/04/2013 10:03 AM, Frederick Williams wrote: > 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.
I didn't say "every x and every set S". In the underlying context I was talking about, x is _an_ individual and S is _a_ set (however general each might be).
> >> 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.
  There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI 



