Poincaré in a 1909 lecture in Göttingen proposed a solution to the apparent incompatibility of two results as viewed from a definitionist perspective: on the one hand, Richard’s proof that the definitions of real numbers form a countable set and, on the other, Cantor’s proof that the real numbers make up an uncountable class. Poincaré argues that, Richard’s result notwithstanding, there is no enumeration of all definable real numbers. We apply previous research by Luna and Taylor on Richard’s paradox, indefinite (...) extensibility and unrestricted quantification to evaluate Poincaré’s proposal. We emphasize that Poincaré’s solution involves an early recourse to indefinite extensibility and argue that his proposal, if it is to completely avoid Richard’s paradox, requires rejecting absolutely unrestricted quantification: Richard’s paradox provides a context in which paradox seems inescapable if unrestricted quantification is possible. In proposing his solution to the apparent conflict between Richard’s and Cantor’s results, Poincaré employs temporal expressions whose exact meaning he does not clarify. We suggest an interpretation of these expressions in terms of order of availability and briefly discuss its explanatory power in topics like paradoxes, limitation theorems and indefinite extensibility. (shrink)

In previous work in 2010 we have dealt with the problems arising from Cantor's theorem and the Richard paradox in a definable universe. We proposed indefinite extensibility as a solution. Now we address another definability paradox, the Berry paradox, and explore how Hartogs's cardinality theorem would behave in an indefinitely extensible definable universe where all sets are countable.

The Russellian argument against the possibility of absolutely unrestricted quantification can be answered by the partisan of that quantification in an apparently easy way, namely, arguing that the objects used in the argument do not exist because they are defined in a viciously circular fashion. We show that taking this contention along as a premise and relying on an extremely intuitive Principle of Determinacy, it is possible to devise a reductio of the possibility of absolutely unrestricted quantification. Therefore, there are (...) intuitive reasons to believe that the counter-argument fails to support the possibility of absolutely unrestricted quantification. (shrink)