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Full Idea
Unfortunately, Russell's new logic, as well as preventing the deduction of paradoxes, also prevented the deduction of mathematics, so he supplemented it with additional axioms, of Infinity, of Choice, and of Reducibility.
Gist of Idea
Russell's improvements blocked mathematics as well as paradoxes, and needed further axioms
Source
report of Bertrand Russell (Mathematical logic and theory of types [1908]) by Alan Musgrave - Logicism Revisited §2
Book Ref
-: 'British Soc for the Philosophy of Science' [-], p.102
A Reaction
The first axiom seems to be an empirical hypothesis, and the second has turned out to be independent of logic and set theory.
| 11064 | Classes can be reduced to propositional functions [Russell, by Hanna] |
| 6407 | The class of classes which lack self-membership leads to a contradiction [Russell, by Grayling] |
| 10418 | Type theory seems an extreme reaction, since self-exemplification is often innocuous [Swoyer on Russell] |
| 10047 | Russell's improvements blocked mathematics as well as paradoxes, and needed further axioms [Russell, by Musgrave] |
| 23478 | Type theory means that features shared by different levels cannot be expressed [Morris,M on Russell] |
| 21718 | Ramified types can be defended as a system of intensional logic, with a 'no class' view of sets [Russell, by Linsky,B] |
| 18126 | A set does not exist unless at least one of its specifications is predicative [Russell, by Bostock] |
| 18128 | Russell is a conceptualist here, saying some abstracta only exist because definitions create them [Russell, by Bostock] |
| 18124 | Vicious Circle says if it is expressed using the whole collection, it can't be in the collection [Russell, by Bostock] |