[SolitaryWalker]

Originally Posted by Jon

(1) Russell's paradox is generated by the combination of (a) the Naive Comprehension Axiom (every concept has an extension), and (b) the Rule of Substitution (every open sentence which defines a condition on objects corresponds to a concept). .

This does not define Russell's paradox and I fail to see the difference between your conception of the notion and mine. Clarify.

Originally Posted by Jon

((2) Responses to the Paradox: The two solutions are (a) The Theory of Types (which tries to find a non ad hoc way to restrict what counts as a genuine property), and (b) Zermelo-Frankel Set Theory (denies that there is a set for every property)..

I am familiar with the theory of types, yet you should explain what Zermelo-Frankel Set theory is.

Originally Posted by Jon

(((3) The Fate of Frege: Two respects in which the elements which generate the paradox are deeply entrenched in Frege's entire programme. (a) getting big enough classes (he needs there to be an infinite number of objects in order to secure the result that every number has a successor), and (b) Hume's Principle (the fact that Frege derives Hume's Principle from Basic Law 5))..

Why must there be an infinite number of objects in order for every number to have a successor? Numbers are not objects, but merely abstract figures of objects. Here we encounter the distinction between existing mathematical entities, finite sets and mathematical possibilities. Numbers inhere within mathematical possibilities and there is no reason why every number could not have a successor owing to the fact that a set of natural numbers is infinite.

Originally Posted by Jon

(((I'll elaborate if asked.))..

Yes you should? Which of Hume's principles are you talking about? His epistemology is very extensive.