Talk:Existence of Non-Standard Models of Arithmetic

I'm starting to worry about how much we're implicitly assuming first-order systems in logic stuff here. I'm guilty of this via not having been specific in a few places as well. Our recent conversation about geometry stuff and stumbling across this page is reminding me though. For example, second-order axiomatizations of arithmetic make it essentially unique. The non-standard models like on this page are a phenomenon in first-order versions.

I know I've been trying to bring up some stuff that might not be familiar to other active members here at the moment, so I guess I'm throwing this out there for the record to see if there's any comments. Is it something we should worry about now, or wait until later and just go back and fix it in the future if it becomes an issue?

-- Qedetc 15:07, 14 June 2011 (CDT)


 * (Good grief - did I write this page? I don't remember! It's not even in my usual style!)


 * This is one of the reasons I deliberately built the contents of this site up from ground level: basic first-order logic by natural deduction from an "intuitive" set of axioms, then the ZFC axioms, then abstract algebra via naturally ordered semigroups - deliberately skipping past Peano arithmetic, I always planned to come back to it later) so as to define the natural numbers, integers, rationals, etc. It was always possible to trace back to establish exactly which axioms any particular result depended upon. Thus we know which ones rest on AoC, CH, LEM even. Jumping in at the deep end with a result whose axioms one would "eventually" expect to establish by working backwards is an approach I was not prepared to attack.


 * Others have different views. Some want to put the "popular" results up (e.g. FLT and all that), hoping they'll be linked to in the end. Others decide on an area they are interested in, and go into it in some depth but without establishing the bedrock of the assumptions on which the discipline rests.


 * I understand it can be tedious establishing the really simple, basic results (I ought to, I did the most of set theory, 1st-order logic, mapping and relation theory, abstract algebra, basic point-set topology, analysis, calculus and graph theory, and possibly other areas as well that I've forgotten about now). But from my OCD-fuelled viewpoint it's the only way to do it. Otherwise the assumptions being made may just not hold water. Take note from that vexed result in Topology that's been wrangled over the last few days because the OP didn't establish the original axioms. --prime mover 15:57, 14 June 2011 (CDT)