
Re: Matheology S 224
Posted:
Apr 19, 2013 11:18 PM


On 19/04/2013 5:55 AM, Frederick Williams wrote: > Nam Nguyen wrote: >> >> On 18/04/2013 7:19 AM, Frederick Williams wrote: >>> Nam Nguyen wrote: >>>> >>>> On 17/04/2013 8:48 AM, fom wrote: >>>>> On 4/17/2013 9:36 AM, Frederick Williams wrote: >>>>>> Nam Nguyen wrote: >>>>>> >>>>>>> "x is in a nonempty subset of S" could be _expressed_ as a FOL language >>>>>>> expression: x e S' /\ Ay[ y e S' > y e S]. >>>>>>> >>>>>>> On the other hand, in "x is proven to be in a nonempty subset of S", >>>>>>> the _meta phrase_ "is proven" can not be expressed by a FOL language: >>>>>>> "is proven" pertains to a meta truth, which in turns can't be equated >>>>>>> to a language expression: truth and semantics aren't the same. >>>>>> >>>>>> "x is in a nonempty subset of S" can be expressed in the language of a >>>>>> first order theory with a binary predicate e. The intended meaning of e >>>>>> is given by the nonlogical axioms of that theory. >>>> >>>> Frederick seemed to be confused: what I'm doing here has nothing to >>>> do with formal systems, theories, axioms of formal systems. >>> >>> You wrote: >>> >>> '"x is in a nonempty subset of S" could be _expressed_ as a FOL >>> language expression: x e S' /\ Ay[ y e S' > y e S].' >>> >>> How does the FOL expression express "x is in a nonempty subset of S"? >>> It can only do so if "e" has a particular meaning. How is that meaning >>> established? >> >> By establishing the interpretation that "e" would mean "is a member of". > > How is that done? Note that "is a member of" does not have one > meaning.
So how many meanings it would have? 10? 14? Can you list out all the meanings it would have?
> >>> 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?
  There is no remainder in the mathematics of infinity.
NYOGEN SENZAKI 

