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About a year ago I realized that one could extend the method of truth tables to say verifying association for a finite table described by a binary operation, or helping to verify that a function satisfies the homomorphism property (given that such a function maps between sets, instead of algebras). Does anyone know of anyone else that has done this?

If and only if it's not clear what I mean and how it works, see here:

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Usually one is not interested in proving theorems for any specific group, though there are some exceptions (as I recall, computer calculations are needed to construct the sporadic simple groups). So your approach is not very helpful from a mathematical perspective.

Sometimes, though, one is interested in actually verifying a property for a specific group. Even in that case, usually (though not always) the group is given in some explicit manner, and it is not feasible to verify the property by "brute force", per your suggestion.

Finally, a similar approach to yours is used for brute-force proofs in the propositional calculus, for example using the proof search method known as "resolution". This idea breaks for the predicate calculus, as Gödel's incompleteness theorem shows.

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I think I followed you until the last part, since Godel actually proved predicate calculus complete ("the basic discovery" as some have written). The system of Principia Mathematica and similarly other related systems (e. g. Peano Arithmetic), come as incomplete. – Doug Spoonwood May 30 '11 at 1:21
Peano arithmetic is a special case of the predicate calculus, so there is in general no decision procedure for it. – Yuval Filmus May 30 '11 at 1:33

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