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August 8

I need a resource for studying the basics of convexity and relations

I am preparing for the upcoming school year. I was wondering if anyone had any recommendations for studying convexity and relations. Here are some practice problems about what is to be covered: http://www.econ.queensu.ca/pub/students/phds/mirza/methods/PS_ConvRel.pdf. I would prefer online pdf resources but I also have access to most well known textbooks. Thanks. 207.216.140.53 (talk) 15:57, 8 August 2011 (UTC)[reply]

Welcome to Queen's! I'm sure you'll enjoy your time there. Unfortunately, I do not know of any good reference books for the specific subject. I would recommend asking the relevant professor if you still don't have a good book to look to. Rosilisk (talk) 22:03, 10 August 2011 (UTC)[reply]

Categorical description of a nonstandard model of first order Peano arithmetic?

Does anyone know of a specific set of second order axioms satisfying all of the following conditions:

All models of the axioms are isomorphic.
No model of the axioms is isomorphic to the natural numbers.
All models of the axioms are models of first order Peano arithmetic.

I would like to examine a specific set of axioms of nonstandard arithmetic and compare them to the Peano axioms. Unfortunately, I do not personally know of any such axiomatization. Thank you in advance to anyone who can help! JamesMazur22 (talk) 19:59, 8 August 2011 (UTC)[reply]

Well, that would give you a definable nonstandard model of PA, so maybe I'd start from that side — see if you can come up with a definable such model. One example would be, note that L must have elements that are nonstandard models of PA, and pick the least one in the canonical wellorder on L. Now see if you can characterize that model internally, in second-order logic. It's not obvious to me that you can, but it also doesn't seem hopeless; you'd need to quantify over countable models, which are essentially reals, and second-order logic over the naturals lets you talk about reals. --Trovatore (talk) 20:12, 8 August 2011 (UTC)[reply]

Thanks. I wasn't sure if it had been done before. I'll be sure to look at that. JamesMazur22 (talk) 12:10, 9 August 2011 (UTC)[reply]