ONT Re: Model Theory
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Note 38
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| 1. Introduction
|
| 1.4. Theories and Examples of Theories (cont.)
|
| 1.4.12. Example. [Theories of Sets].
|
| We shall now discuss some examples of set theories.
|
| There are two quite different reasons to include a
| discussion of set theories in a book on model theory.
| The first reason is that, if we wish to be completely
| precise, we should formulate our whole treatment of
| model theory within an appropriate system of axiomatic
| set theory. Actually, we are taking the more practical
| approach of formulating things in an informal set theory,
| but it is still important that, 'in principle', we could
| do it all in an axiomatic set theory. We have left for the
| Appendix an outline of the informal set theory that we are
| using. The other reason for discussing set theories is that
| they are among the most interesting and important examples of
| theories. The second reason is the one which concerns us at
| this time. The theory of models is particularly well suited
| to the study of models of set theory. In the Appendix we have
| listed the axioms for four of the most familiar set theories:
| Zermelo, Zermelo-Fraenkel, Bernays, and Bernays-Morse. The first
| two of them are formulated in the language $L$ = {in}, while the
| other two are formulated in the language $L$’ = {in, V}, where
|'in' is a binary relation symbol and V is a unary relation symbol.
| Zermelo set theory is a subtheory of Zermelo-Fraenkel, and Bernays
| set theory is a subtheory of Bernays-Morse.
|
| The deepest results in set theory use constructions of models.
| However, these constructions are often of a special nature,
| for models of set theory only, and are therefore outside
| the scope of this book. For instance, the notion of
| constructible sets was used by Gödel (1939) to show
| that if Bernays set theory is consistent, then it
| remains consistent if we add to it the axiom of
| choice and the generalized continuum hypothesis;
| in other words, if Bernays set theory has a model,
| then it has a model in which the axiom of choice
| and the generalized continuum hypothesis are true.
| The same proofs and results are also well known to
| hold for Zermelo-Fraenkel set theory. Cohen's forcing
| construction has been used by Cohen and others to obtain
| a remarkable series of additional consistency results (see
| Cohen, 1963). For example, if Bernays (or Zermelo-Fraekel)
| set theory has a model, then it has a model in which the
| axiom of choice is false, and another model in which
| the axiom of choice is true but the generalized
| continuum hypothesis is false.
|
| In the rest of our discussion we use the abbreviation ZF
| for "Zermelo-Fraenkel set theory". Whether or not we can
| prove that ZF is consistent depends on just how much we are
| assuming in our intuitive set theory. If our intuitive set
| theory is just a replica of ZF, then we cannot prove the
| consistency of ZF, even if we allow the use of the axiom
| of choice. Similarly, for any of the other set theories
| T we have introduced in the Appendix, we cannot prove the
| consistency of T if our intuitive set theory is a replica
| of T. These assertions follow from the Gödel incompleteness
| theorem. On the other hand, in Bernays-Morse set theory we
| can prove the consistency of Bernays set theory and of ZF.
| In ZF we can prove the consistency of Zermelo set theory.
| If we assume the existence of an inaccessible cardinal,
| then we can prove that Bernays-Morse set theory as well
| as ZF are consistent. Bernays set theory and ZF are
| very close to each other, and we can prove that one
| is consistent if and only if the other is. We shall
| leave the last three results above for exercises.
|
| Neither Zermelo set theory, nor ZF, nor Bernays-Morse
| set theory is finitely axiomatizable (assuming that they
| are consistent). But, surprisingly, Bernays set theory is
| finitely axiomatizable (Bernays, 1937). With its finite
| axiomatization it is sometimes called Bernays-Gödel set
| theory. Each of the four set theories in our discussion,
| like number theory, has the following property: if the
| theory is consistent, then it is not complete, and no
| finite extension of it is complete. This is another
| consequence of the Gödel incompleteness theorem.
|
| There is no completely satisfactory notion of a "standard" model of set theory.
| The closest thing to it is the notion of a 'natural model'. Natural models,
| roughly, are models of the form <M, in>, where M is a set of sets formed
| by starting with the empty set and repeating the operations of union
| and power set, while 'in' is the [epsilon]-relation restricted to M.
| More precisely, we define for each ordinal !a! the set R(!a!) by:
|
| R(0) = 0,
|
| R(!a! + 1) = S(R(!a!)),
|
| R(!a!) = |_|^(!b! < !a!) R(!b!), if !a! is a limit ordinal.
|
| Then a 'natural model' of ZF (or of Zermelo set theory) is a model of the form
| <R(!a!), in>. A natural model of Bernays set theory is a model of the form
| <R(!a! + 1), in, R(!a!)>.
|
| None of our set theories has any countable natural models. For this
| reason, a somewhat weaker notion of "standard" model is also important.
| A model <M, in> is said to be a 'transitive model' if and only if 'in' is
| the [epsilon]-relation restricted to M and every element of an element of
| M is an element of M. For models of the language $L$’ = {in, V} we make a
| similar definition. The countable transitive models are the most important
| models for Cohen's forcing construction.
|
| Since number theory has just one standard model and is not complete, it
| has consistent extensions which have no standard models. If ZF has any
| transitive model at all, then it has many nonequivalent transitive models.
| Nevertheless, if ZF is consistent, then it has consistent extensions which
| have no transitive models at all. Moreover, in ZF plus the axiom of choice,
| we cannot prove the following: If ZF has a model, then ZF has a transitive
| model.
|
| Chang & Keisler, 'Model Theory', pages 43-45.
|
| C.C. Chang and H.J. Keisler, 'Model Theory',
| North-Holland, Amsterdam, Netherlands, 1973.
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