What is the axiom of reducibility? And what philosophical controversies did it incite?

Question

Trying to come to terms with basics concerning philosophy of logic, and wish to ask about some particular issue: What is in simple words the axiom of reducibility put forward by Russell? And what is its philosophical context? Among which philosophers controversy regarding this axiom took place? And over what philosophical issues?

Answer 1

To put it in simple words we have to describe in a couple of words the project of Principia Mathematica, which Russell inherited from Frege: reconstructing mathematics from logic alone. For a broader context see What is the philosophical ground for distinguishing logic and mathematics?

Frege himself could not complete the project because Russell discovered his famous paradox, where a set of all sets not belonging to themselves leads to a contradiction. To pre-empt such definitional circularity, or impredicativity as Poincare called it, Russell eventually introduced the so-called "ramified hierarchy": first-order predicates can not refer in their definitions to any totalities of predicates, and objects of the first order are the ones fulfilling only the first-order predicates. The second-order predicates can refer to collections of first-order predicates but no higher, we then get objects of the second order, etc.

One can see how this would prevent something like "the smallest positive integer not definable in fewer than twelve words" from leading to a paradox. Unfortunately, it also prevents many standard mathematical constructions, e.g. the least upper bound of a set. Zermelo pointed out that requiring all definitions to be predicative would erase most of classical analysis. Moreover, the stratification of integers into orders turned the ordinary number-theoretic relations among them into a conundrum. Originally Russell attempted to show that all orders of the hierarchy above some finite order are in fact identical, the hierarchy "collapses" to a finite level. If that were the case it would deliver the holy grail: mathematics in its original impredicative form but with a built-in structural guarantee against the paradoxes of self-reference. That did not work out however, so Russell had a brilliant idea... of making it an axiom. The axiom of reducibility: for any predicate of any order, there is a predicate of the first order which is equivalent to it. Ergo, all integers are first order. Problem solved.

To quote Russell himself from another context this solution had all “the advantages of theft over honest toil”. Russell considered it simply a matter of having "enough" first order predicates, "a generalised form of Leibniz's identity of indiscernibles", a kind of completeness axiom. Others begged to differ. Weyl for example described it as

"a bold, an almost fantastic axiom; there is little justification for it in the real world in which we live, and none at all in the evidence on which our mind bases its constructions".

In any case, it did not look like a self-evident "law of thought" belonging to pure logic upon which mathematics was to be built. In 1919 Russell himself admitted as much in Introduction to Mathematical Philosophy:

"I do not see any reason to believe that the axiom of reducibility is logically necessary, which is what would be meant by saying that it is true in all possible worlds. The admission of this axiom into a system of logic is therefore a defect...".

Later Chestwick and Ramsey pointed out that with the axiom of reducibility the ramified hierarchy becomes pointless, we might as well admit impredicative definitions and hope for the best.

In the end, the axiom was a link in a long chain that led to the abandonment of logicism after Gödel and Quine, see Friedman's Logical Truth and Analyticity in Carnap s "Logical Syntax of Language" for that chapter. But it had a long afterlife in mathematical logic, namely in the proof theory. Gödel extended the ramified hierarchy from finite to transfinite orders, and was able to prove that it does collapse at level ω1. This led to his constructible universe and proving the consistency of the continuum hypothesis. The work of Takeuti, Schütte, Tait and Prawitz showed that the axiom of reducibility implies the consistency of the second order arithmetic, a very strong formal theory, and despite its bold and fantastic embrace of circularity no contradictions were found there yet.