Consistency and Criticisms
According to Carnap's "Logicist Foundations of Mathematics", Russell wanted a theory that could plausibly be said to derive all of mathematics from purely logical axioms. However, Principia Mathematica required, in addition to the basic axioms of type theory, three further axioms that seemed to not be true as mere matters of logic, namely the axiom of infinity, the axiom of choice, and the axiom of reducibility. Since the first two were existential axioms, Russell phrased mathematical statements depending on them as conditionals. But reducibility was required to be sure that the formal statements even properly express statements of real analysis, so that statements depending on it could not be reformulated as conditionals. Frank P. Ramsey tried to argue that Russell's ramification of the theory of types was unnecessary, so that reducibility could be removed, but these arguments seemed inconclusive.
Beyond the status of the axioms as logical truths, the questions remained:
- whether a contradiction could be derived from the Principia's axioms (the question of inconsistency), and
- whether there exists a mathematical statement which could neither be proven nor disproven in the system (the question of completeness).
Propositional logic itself was known to be consistent, but the same had not been established for Principia's axioms of set theory. (See Hilbert's second problem.)
Read more about this topic: Principia Mathematica
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