Consistency and Completeness of PM
When we are confronted with an axiomatic system, the two great questions are: Is it consistent? Is it complete? As we saw in the discussion of axiomatic systems in Chapter 13, consistency and completeness each involve a relation between the notions of provability and validity; and it is because of this that consistency gives us a guarantee against triviality (at least against triviality of the most extreme sort), and completeness is a guarantee of adequacy. Several senses of consistency and completeness can be distinguished, so the main questions give rise to more special questions whether the system is consistent (complete) in this or that sense.