 An intermediate position is that defended by classical “predicativists” such as Poincaré and Weyl. The theory, presented in a satisfactory logical way by Feferman and others, accepts the excluded middle on the natural numbers (and as such it is arguably committed to the existence of the set of natural numbers and in any case to accepting bivalence on the natural numbers) but does not accept the existence of the power set of the natural numbers. According to predicativism (see Feferman 2005), sets exist only if they are definable in some non-circular linguistic way. In accepting the excluded middle on the natural numbers and in making the existence of sets depend on our definitional abilities, this position is a sort of a compromise between a classical and a constructive viewpoint. In 1918, Hermann Weyl (Weyl 1918; see Kaufmann 1930 for a related program) presented the foundations of analysis within this framework and showed that a great part of classical analysis can be carried out within the framework by replacing talk of arbitrary sets of real numbers with arithmetical sequences of real numbers. Feferman 1988 gave a detailed formal presentation of the theory and proved that, on a certain reconstruction, the theory is a conservative extension of Peano Arithmetic. In addition, he also used the theory to state an important conjecture concerning how much mathematics is needed in physics. In Feferman 1984 and 1987, he proposed that all of mathematics used in physical theory can be recaptured in a predicative system of analysis. Using the metatheoretical result of conservativity mentioned above, he also exploited the argument to claim that Quine and Putnam’s indispensability arguments (see the entry on  indispensability arguments in philosophy of mathematics)   at best commit us to what Peano Arithmetic commits us to.