re-reading some of the Shoenfield's book, "Mathematical logic", I noticed a statement incomprehensible. The offending statement is on page 89, where it says:
With the result mentioned in (iv), this give a new proof of completness of ACF.
If, kindly, someone in possession of the text could take a look (pages 86-89), should detect the strangeness that I mentioned. For this I am curious to hear independent opinions.
(Note: just look at the pages I mentioned taking the definitions and results presented, i.e. there is no need to go into details of the demonstrations.)