Re:Computing the Uncomputable
by wolfgke on Wednesday April 06, 2016 @05:40PM
(#11442369)
Why doesn't one simply use higher order logic to solve this problem? (serious question, I'm a mathematician, but not a set theorist).
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It turns out this only pushes back the problem. You can write down second-order axioms for arithmetic which have a unique model, but in a model of a second-order theory predicates are interpreted as sets. So, you need to choose a version of set theory, to know what a model actually is. Suppose you choose ZFC (the usual axioms of set theory). Unfortunately this itself has infinitely many models!
I explained this in a bit more detail here:
https://johncarlosbaez.wordpress.com/2016/04/02/computing-th...
Thanks, this looks plausible.
A lot of properties that you'd want in a logic stop holding once you hit second-order logic: https://en.wikipedia.org/wiki/Second-order_logic#Metalogical...