The reader of Part VI will have noticed that among the set-theoretic models considered there some models were missing which were announced in Part II for certain proofs of independence. These models will be supplied now.Mainly two models have to be constructed: one with the property that there exists a set which is its own only element, and another in which the axioms I–III and VII, but not Va, are satisfied. In either case we need not satisfy the axiom of (...) infinity. Thereby it becomes possible to set up the models on the basis of only I–III, and either VII or Va, a basis from which number theory can be obtained as we saw in Part II.On both these bases the Π0-system of Part VI, which satisfies the axioms I–V and VII, but not VI, can be constructed, as we stated there. An isomorphic model can also be obtained on that basis, by first setting up number theory as in Part II, and then proceeding as Ackermann did.Let us recall the main points of this procedure.For the sake of clarity in the discussion of this and the subsequent models, it will be necessary to distinguish precisely between the concepts which are relative to the basic set-theoretic system, and those which are relative to the model to be defined. (shrink)

This paper relates to a formal statement of the mechanisms that are thought minimally necessary to underpin consciousness. This is expressed in the form of axioms. We deem this to be useful if there is ever to be clarity in answering questions about whether this or the other organism is or is not conscious. As usual, axioms are ways of making formal statements of intuitive beliefs and looking, again formally, at the consequences of such beliefs. The use of this style (...) of exposition does not entail a claim to provide a mathematically rigorous formal deductive system. Conventional mathematical notation is used to achieve clarity, although this is elaborated with natural language in an attempt to reduce terseness. In our view, making the approach axiomatic is synonymous with building clear usable tests for consciousness and is therefore a central feature of the paper. The extended scope of this approach is to lay down some essential properties that should be considered when designing machines that could be said to be conscious. In the broader discussion about the nature of consciousness and its neurological mechanisms, it may seem to some that axiomatisation is premature and continues to beg many questions. However, the approach is meant to be open- ended so that others can build further axiomatic clarifications that address the very large number of questions which, in the search for a formal basis for consciousness, still remain to be answered. Of course, in discussions about consciousness many will also argue that the subject is not one that may ever be formally addressed by means of axioms. The view taken in this paper is 'let's try to do it and see how far it gets'. (shrink)