  The observation that spacetime and quantum fields on it have to be
dynamically produced in any theory of quantum gravity implies that quantum
gravity should be defined on configuration space of fields rather than
spacetime. Such a theory that is defined on the configuration space of fields
rather than spacetime is a third quantized theory. So, both string theory and
group field theory are third-quantized theories. Thus, using axioms of string
field theory, we motivate similar axioms for group field theory. Then using the
structure of these axioms for string field theory and group field theory, we
identify general features of axioms for any such third quantized theory of
quantum gravity. Thus, we show that such third-quantized theories of quantum
gravity can be formulated as formal axiomatic systems. We then analyze the
consequences of Godel theorems on such third quantized theories. We thus
address problems of consistency and completeness of any third quantized
theories of quantum gravity.
