This is not Godel's incompleteness theorem(s). This deals with a completely different notion of completeness. Godel's completeness theorem is the theoretical j

This is not Godel's incompleteness theorem(s). This deals with a completely different notion of completeness. Godel's completeness theorem is the theoretical justification for why first-order logic occupies such an important position in mathematical logic. It basically establishes that two different methods of proving a logical statement happen to coincide for first-order logic. Let's say that I come up with a long list of axioms that define what it means for an object to be a "cow" and what it means to have a "spot." Then I state "all cows have spots." There are two ways of proving this. One way, the "semantic" way, is to take every example of cow and prove that it has a spot. This is hard; I not only have to examine every cow that exists, I have to examine every possible imaginary entity that satisfies the definition of "cow" and prove that it has a spot. The second way, the "syntactic" way, is to play the string manipulation game that people often associate with the notion of a "logical proof." I start out with the axioms of a "cow," and I'm given some rules for how I can manipulate these axioms. Example of this include the string "A and B" allows me to immediately substitute "A" instead. Given "A implies C" I can substitute all instances of "A" with "C" and so on and so forth. Hopefully I can work out a series of string manipulation steps that eventually end with the statement "All cows have spots." It is not obvious that these two methods should coincide in what they are able to prove. The rule set that I get to manipulate my strings by could be a really wacky set of rules. Maybe it says something like if you see "A implies B," you get to substitute any arbitrary string "C." It's also not apparent how you might choose to interpret the axioms that define a "cow." One of the axioms might say "a cow has four legs" and you might say, in my world, I've decided the string "four" corresponds to the natural number 5, and I'm going to look at all entities with 5 legs. Godel's completeness theorem states that for first-order logic these two proof techniques coincide: every statement you can prove or disprove the first way can also be proved or disproved the second way. More specifically, if you stipulate that rules of string manipulation are exactly the rules of deduction in first-order logic, and that you interpret your axioms when constructing semantic entities according to the standard semantics of first-order logic (e.g. you don't get to arbitrarily stipulate that the string "and" actually means logical disjunction, i.e. or, in your world), then any property that is true of every entity you can conjure up that satisfies your axioms has a "string manipulation" proof that proves that property. Conversely, every statement you can prove with your string manipulation game also holds true of every entity you can conjure up that satisfies your axioms (this isn't strictly speaking part of the completeness theorem, but is also true and usually stated hand-in-hand with it).