Re: Finest partition - exercise in Suppes's book

On Sat, 19 Nov 2005, MoeBlee wrote:

> On page 84 of Suppes's 'Axiomatic Set Theory' (Dover), he mentions an
> "intriguing" exercise (and he says, "the problem is to prove it"). But
> the exercise seems trivial to me while there are exercises in the book
> that are much harder but with no mention that they are difficult. So I
> am wondering whether I've missed something in my proof.
>
> Suppes's definition:
>
> Df. (P is a partition of S & Q is a partition of S) ->
> (P is finer than Q <-> (P not= Q & Ax(xeP -> Eb(beQ & x subsetof b)))
>
> For a conditional definition of a predicate symbol, I like to make the
> antecedent part of the definens, and I think that 'finer than' is a
> 3-place predicate symbol, so for precision, but without any important
> effect on the problem, I modiy to:
>
> Df. P is finer per S than Q <->
> (P is a partition of S & Q is a partition of S & P not= Q & Ax(xeP ->
> Eb(beQ & x subsetof b)))
>
Little to no difference.

> Df. P is a finest partition of S <->
> (P is a partition of S & AQ((Q not= P & Q is a partition of S) -> P is
> finer per S than Q))
>
The finest partition of S is { {s} | s in S }

> Proposition. AS EP P is a finest partion of S. The proof seems trivial
> to me, so I want to make sure I haven't missed what Suppes claims to be
> "intriguing" about it:
>
> {{y}| yeS} exists since it is a subset of the power set of S.
>
> {{y}| yeS} is a partition of S, as follows:
>
Oh, yes yes, I suppose.
Better ascii notation is y in S.

.

Relevant Pages

  • Re: Finest partition - exercise in Suppess book
    ... > the exercise seems trivial to me while there are exercises in the book ... > finer per S than Q)) ... So x subsetof b. ... > non-empty x that's a member of Q but x is not a singleton. ...
    (sci.logic)
  • Finest partition - exercise in Suppess book
    ... the exercise seems trivial to me while there are exercises in the book ... So x subsetof b. ... Further, (| yeS} is unique as a finest partition of S, as follows: ... intriguing and a problem to prove? ...
    (sci.logic)
  • Re: Finest partition - exercise in Suppess book
    ... > On Sat, 19 Nov 2005, MoeBlee wrote: ... >> the exercise seems trivial to me while there are exercises in the book ... >> For a conditional definition of a predicate symbol, ... leave dangling what happens if the antecedent is false, ...
    (sci.logic)
  • Re: Characterization, I dentification and Definition
    ... Px and Am(mex -> m subsetof Pm)) but x not well ordered by the ... Maybe you can do that as an exercise? ... My ordinals defines a well ordered set. ...
    (sci.math)