Re: Fwd: SUO Quo Vadis
John, Merry Christmas!
> AS> Yes, but then you would be writing about yourself, not
> > about _how_ you write about yourself. And if you write about
> > how you write about yourself, you don't write about how you
> > write about how you write about yourself. It is plain that
> > self reference does create an infinite regress.
>
> It's not infinite. You continue that as far as you like.
> And when you get tired, you stop. At every step, it's finite.
>
> That's an example of Tarski's multiple metalevels, which are
> always finite, but could be extended if desired. For further
> discussion and references to Tarski's levels, see my article
> on Laws, Facts, and Contexts:
>
> http://www.jfsowa.com/pubs/laws.htm
Sure, if you put it that way. Having category A, if A is a
subcategory of itself, A>A, then there is the infinite
regress A>A>A>..... In set theory there is the axiom of
foundation/regularity to avoid infinite chains.
> 3. If you have a theory T about theories, it would be true of
> models whose elements are theories (and T could very well
> be an element in one of those models).
So, there is a theory T about theories t1,t2,... If T is not
a tautology, it must give some account of the theories; some
non a priori information about how the theories truly are.
Every permutation of the theories can be taken as a set of
elements of some model, like m1={t1}, m2={t2}, ..., m1,2={t1,t2},
and so forth. There are altogether 2^(n-1) models if you disclude
the model(s) that have no elements. If T itself is an element
of one of these models, then T must also give account of itself,
and this again leads to the infinite regress, or then T does not
thoroughly describe itself. Of course, the other option is that
T is an element of a model which is satisfied by every possible
element a priori. But, as you said, you can continue as far
as you like. And when you get tired, you stop. But nothing can
never ever describe itself thoroughly.
> > If there are things outside of the scope of the present
> > languages, these things must be excluded from the ontology
> > because it is impossible to describe them.
>
> What do you mean "present languages"? If it's not describable
> in one language, you can move to another language. There are
> infinitely many possible languages -- for example, Tarski's
> hierarchy of metalanguages, all of which have exactly the same
> grammar, but language N+1 includes the domain D of everything
> that N can describe plus all the syntactic features of the
> language N.
>
> However, I agree that there are many things that cannot be
> described in any finite sentence in any language that can be
> written in a finite alphabet. Only a countable subset of
> the real numbers, for example, are describable.
You are absolutely right.
> > A person realizes a category...
>
> In mathematics, it's common practice to avoid introducing the notion
> of the mind of any particular agent. Many mathematicians adopt a
> Platonic stance: all possible mathematical structures that can be
> consistently axiomatized exist in a Platonic heaven.
>
> But I believe that it's possible to do good mathematics with any
> of these conceptions. If you like, you can talk about what some
> person thinks or what God thinks. Or you can take a formalist
> stance: if a theory T is consistent and T contains an existential
> quantifier (Ex), then it's grammatically OK to say that x exists.
>
> To me, this seems like a discussion of metalevel questions that
> had been answered many years ago. I don't see what is left to be
> finished "before Christmas".
All is now clear to me too, thanks, and merry Christmas to all!
http://www.helsinki.fi/~astyrman/ChristmasLights.jpg
Arvil