Philosophy of Mathematics

*Why is Russell forced to introduce the axiom of reducibility into his type theory? A good answer will need to bring in the need for logicists to define the real numbers, and an explanation of why Cantor's proof that the reals are more numerous than the naturals is blocked by Russell's rejection of all impredicative definitions.
*Why is is crucial for the Logicist to define the real numbers within logic? Why can't she just leave the real numbers out of mathematics?
* What are the differences between Frege's and Russell's treatments of definition of the numbers?
* Explain how the real numbers can be defined once the natural numbers are available.