  Let us now consider some examples. Consider first the conception of natural number that underlies the theory of Peano Arithmetic (PA). This conception of natural number not only justifies mathematical induction for the language of PA but also for any mathematical extension of the language of PA that is  meaningful.[7]   For example, if we extend the language of PA by adding the Tarski truth predicate and we extend the axioms of PA by adding the Tarski truth axioms, then, on the basis of the conception of natural number, we are justified in accepting instances of mathematical induction involving the truth predicate. In the resulting theory, one can prove Con(PA). This process can then be iterated (for more on this subject see Feferman (1991) and the references therein) In contrast, the Π01-statement Con(ZFC + “There are ω-many Woodin cardinals”) (a statement to which we shall return below) is undoubtedly not intrinsically justified on the basis of the conception of natural number; rather its justification flows from an intricate network of theorems in contemporary set theory, as we shall argue below. 