# An introduction to mathematical logic by Wolfram Pohlers (author), Thomas Glaß (editor) By Wolfram Pohlers (author), Thomas Glaß (editor)

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Trans nite recursion, which (in a special form) says that there is a uniquely determined function F from the ordinals into the sets satisfying the following recursion equations F(0) = X F( + 1) = G( F( )) F( ) = H( fF( ) : < g) for limit ordinals where G and H are known functions and X is a set. Thus trans nite recursion is nothing but a straightforward generalisation of the principle of de nition by primitive recursion as it is known from the natural numbers (cf. 3). We give a short introduction to the theory of ordinals in the appendix.

F entails M ` :G ! F and M ` H ! F c) If M ` (9xF ! F) ! G and x 2= FV(M fGg), then M ` G: Proof. a) The formula (G ! F) ! ((:G ! F) ! F) is boolean valid. 3. b) Both formulas ((G ! H) ! F) ! (:G ! F) as well as ((G ! H) ! F) ! (H ! F ) are boolean valid. 3. c) M ` (9xF ! F) ! G entails M ` :9xF ! G and M ` F ! G by b) Thus we have by an application of the 9-rule also M ` 9xF ! G and obtain M ` G by a). Here we are able to give a positive answer to the rst question: the calculus is complete. For a calculus of a similar type such a result has been observed by K.

Thus two sets M0 M1 have the same cardinality if there is a one-one map from M0 onto M1 . The in nite cardinals are numbered by the @-function. 4. Propositional Properties of First Order Logic 33 the smallest in nite cardinal. @1 is the next one and so on. Sets whose cardinality is @0 are called countable. More details concerning cardinals and their arithmetic can be found in the appendix. Let us return to the propositional properties of rst order languages. We want to show that we can extend a nitely sententially consistent set to a maximally nitely sententially consistent set.

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