—-

Notes:

http://en.wikipedia.org/wiki/Ordinal_number#Transfinite_induction

Normal J. Wildberger:

Paradoxes of naive set theory — basically only three? (Philosophers argue a lot about what counts as a set-theoretic paradox and what counts as a "semantic" paradox.)

- Cantor's paradox: 187?, the largest cardinal

- Burali-Forti's paradox: 1897?, the largest ordinal

- Russell's paradox: the set of sets that are not members of themselves, 1900

# Questions to follow up

None of these compulsory (i.e. do none or more):

- Are Wildburger's infinitesimals helpful with the original problem about Cantor's infinities not being fine-grained enough?

- How are other types of infinitesimal related to Wildburger's? Are they more helpful?

- Do alternative set theories look like they might be helpful? See

— http://plato.stanford.edu/entries/quine-nf/

- The author of the latter page will be in Australia briefly after our next meeting, so maybe think of questions to ask him.

Alternative Set Theory:

- Pure sets are constructed by induction

- Automatically avoids the paradoxes

- Is simple and doesnt rely on 9 axioms

- Similar in many ways to Wildberger's set-up and also to hyperreals, and natural numbers

—-

Additional topic 3 September: internal and external proofs; e.g. applications of http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem

—-

Computable reals:

http://www.thocp.net/biographies/papers/turing_oncomputablenumbers_1936.pdf