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moshez comments on Godel's Completeness and Incompleteness Theorems - Less Wrong
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Given these recent logic-related posts, I'm curious how others "visualize" this part of math, e.g. what do you "see" when you try to understand Goedel's incompleteness theorem?
(And don't tell me it's kittens all the way down.)
Things like derivatives or convex functions are really easy in this regard, but when someone starts talking about models, proofs and formal systems, my mental paintbrush starts doing some pretty weird stuff. In addition to ordinary imagery like bubbles of half-imagined objects, there is also something machine-like in the concept of a formal system, for example, like it was imbued with a potential to produce a specific universe of various thingies in a larger multiverse (another mental image)...
Anyway, this is becoming quite hard to describe - and it's not all due to me being a non-native speaker, so... if anyone is prepared to share her mind's roundabouts, that would be really nice, but apart from that - is there a book, by a professional mathematician if possible, where one can find such revelations?
Here's how I visualize Goedel's incompleteness theorem (I'm not sure how "visual" this is, but bear with me): I imagine the Goedel construction over the axioms of first-order Peano arithmetic. Clearly, in the standard model, the Goedel sentence is true, so we add G to the axioms. Now we construct G' a Goedel sentence in this new set, and add G'' as an axiom. We go on and on, G''', etc. Luckily that construction is computable, so we add G^w as a Goedel sentence in this new set. We continue on and on, until we reach the first uncomputable countable ordinal, at which point we stop, because we have an uncomputable axiom set. Note that Goedel is fine with that -- you can have a complete first-order Peano arithmetic (it would have non-standard models, but it would be complete!) -- as long as you are willing to live with the fact that you cannot know if something is a proof or not with a mere machine (and yes, Virginia, humans are also mere machines).