# James Taylor

Final report: http://images.xeny.net/jason/d/187-1/Taylor+-+Computable+reals+and+constructible+sets.pdf

—-

Notes:

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

Normal J. Wildberger: - http://web.maths.unsw.edu.au/~norman/papers/Ordinals.pdf

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/settheory-alternative/ — 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