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Re: Eliminating Quantifiers For Dummies! A(x) E(y) ALL(t)EXIST(u)EaAb ..
Posted:
Jan 26, 2012 5:46 PM
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On Jan 26, 1:10 pm, Frederick Williams <freddywilli...@btinternet.com>
wrote:
> Tony Orlow wrote:
>
> > On Jan 26, 9:43 am, Frederick Williams <freddywilli...@btinternet.com>
> > wrote:
>
> > > Can all sets be got by starting with the null set and singletons; adding
> > > single elements; and taking unions of two sets?
>
> > Actually, many are gotten by defining xeS <-> P(x), but that makes the
> > whole discussion rather trivial, wouldn't you say? The set union
> > statements are meant to cover combinations of sets over which P(x) is
> > true. What's your point, exactly?
>
> My point was that starting with the null set and singletons, adding
> single elements, and taking unions of two sets gets you very little. It
> will not allow you to show that
>
> {0, 1, 2, 3, ...}
>
> is a set, for example.
>
> --
> 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
So? You throw in the axiom of infinity, if that's the set you want to
talk about. It's defined piecemeal anyway, element-wise and
recursively. What I am talking about is quantification over a base
set.
Peace,
Tony
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