Re: Separation,Power and Countability.


On Jun 14, 9:36 am, zuhair <zaljo...@xxxxxxxxx> wrote:
|The word uncountable set is not acceptable to me,because the language
|of any set theory T should be countable, otherwise it will affect the
|recurssive ability of that theory and by then it will lead to the
|emergence of the shadowy sets i.e the indefinable sets, which is un
|acceptable.

Declaring uncountable or undefinable sets "unacceptable"
has no substance to it.

People often find the relationship between constructivism,
definability and countability confusing. Part of the problem
in this context is that part of the time "definable" is
being used to mean "first-order definable". The sets
definable by the axiom schema are all first-order definable
(with parameters). It's entirely arbitrary to imagine that
those are all that there are. If to each formula there
exists a set, by separation, then there is a function from
the formulas to the sets, but this function isn't itself
first-order definable. But that's ok! Don't allow yourself
to suffer from chronic first-order-itis, folks.

It may be the case that constructivists typically don't
accept the general power set axiom, and probably you can
find various who don't accept impredicative definitions.
But there's not an inherent conflict here. It's consistent
to believe that there are uncountably many subsets of the
integers and that for one to exist means that it can be
defined (somehow). It's just that you can't also believe
that there is some one fixed language in which they're all
definable. But believing that all definitions of sets of
integers can be given in a fixed language, let alone a
fixed first-order language, is a very strange belief.

Keith Ramsay

.

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