FOL and AST

Hi,

I am reading Shoenfield, Joseph R. Mathematical Logic. ASL. A K
Peters, Ltd. 1967 and have some questions to ask:

It is often suggested that to escape from the ‘bootstrapping’ or
‘chicken or egg’ astonishment when developing FOL and AST, one should
view sets or classes as merely collections of objects (in a naïve, pre-
theoretic, informal way). In fact, Shoenfield p. 9 asks the reader to
do exactly that by introducing set theoretic wording just as a matter
of ‘terminology and notation’ in order to ‘explain things more
succinctly’.

Models of first-order theories (in fact, structures for first-order
languages) become, then, collections of objects together with
collections of n-tuples of objects. These collections of objects and
of n-tuples of objects are to be thought of purely extensionally, that
is, regardless of the means by which the objects or the n-tuples of
them have been collected. Note also that the nature of the objects is
totally irrelevant.

This immediately suggests:

1. To which extent is it justified to identify such abstract
collections of objects with set theoretic sets or classes?

and

2. How should this identification or replacement be performed?

We know from set theory that not every collection of sets is a set
(proper classes). So sets are not good candidates to collections of
objects. Classes still could do, however they are usually defined
intensionally (see, for example, Kunen’s Set Theory. An introduction
to independence proofs or Levy’s Basic Set Theory). I suppose one
should ignore this and suppose that a class is any collection of sets,
independently that they can be referred to via a first-order formula
in the language of set theory. This way it looks like collections of
objects should be replaced by classes of sets.

Another intriguing fact for me is:

3. How much set theory is necessary to prove completeness of FOL?

particularly

4. Is it possible to prove completeness of FOL without making use of
the axiom of choice (in Shoenfield’s proof, of Zorn’s Lemma which
implies Teichmüller-Tukey Lemma accounting for Lindenbaum’s theorem)?

Thank you in advance for your help.
.

Relevant Pages

  • Re: Scott on CH in 2nd order set theory
    ... first order variables range over the whole of the cumulative hierarchy ... it is not at all obvious what the second order variables are supposed to ... So if the first-order variables are ranging over the whole of the ... observation can be made w.r.t second order Zermelo set theory and much ...
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  • Re: Kuratowski Ordered Pair
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  • Re: Kuratowski Ordered Pair
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    (sci.math)
  • Re: Question: Given |X|>0 and |Y|>0, can X x Y be empty?
    ... needed context, but seriously, this was so silly that it cannot ... In any first-order logical treatment of set theory, ... if you wanted to prove existence of cross-product, ...
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  • Re: Godel misuses ZF
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