Re: Elementary questions, need some clarification (model theory)

On Jun 21, 9:20 pm, Augustine <xinyan...@xxxxxxxxx> wrote:
|another related question, in this book, Page 38 line 14
|it is said that the cardinality of a language is the number of
|relation, function and constant symbols in it, which is quite
|confusing, does that mean
||L|=max(|relations|,|function|,|constant symbol|) ?

I would take that to mean |relations| + |functions|
+ |constant symbols|. E.g. if there was one relation,
two functions, and a constant symbol, the cardinality
would be 4 by this definition.

This is closely related to the number of well-formed
formulas of the language. Assuming the axiom of
choice, for infinite sets S and T, the cardinality of
the union of S and T is equal to the cardinality of
whichever is larger, and the cardinality of S is equal
to the cardinality of finite sequences of elements
taken from S.

Without assuming the axiom of choice, it's possible
to show that countability of the language by their
definition is equivalent to the countability of the set
of well-formed formulas in it. Here countability is used
in the sense which includes finite sets.

|seems in this book, many thms presume the language is countable, but
|this assumption is always omitted?

I'm not familiar with the book, and authors sometimes
do choose at some point to consider only a special case
like countable languages. Many results however do
extend to the general case. Do you have an example in
mind of a theorem stated in the book that you think might
be false for some uncountable languages?

Keith Ramsay

.

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