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fom
Posts:
1,968
Registered:
12/4/12


Re: Matheology S 224
Posted:
Apr 20, 2013 5:59 PM


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?
Or are we reverting to the distinction between real and apparent variables from the first "Principia Mathematica"?
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.



