In this analogy, the map is playing the role of “model” (as it assigns sentences to the analog of truth values).
Hmm, I thought only the preimage of 1 is the model.
in your construction, the preimage of 1 would be the theory of the map (the largest theory that the map is a model of).
But… a theory can include sentences not in the preimage of 1 (undecidable?)… I am confused.
I would instead say that the preimage of 1 is the largest model of the map, and is the model of any “large enough” theory.
Note that in order for the map to be a model, the map must have certain behavior (whenever both φ and ψ map to 1, φ∧ψ also maps to 1, etc.). In model theory we restrict consideration to maps obeying these laws; if your map strays outside these boundaries then model theory has nothing to say about it.
Where are these laws defined? In the logic? In the logic+language? Then the only “valid” maps are those which are homomorphisms (in some sense) from logic(?) to the Z_2 subset (true/false) of the codomain.
I’ve missed something in my explanation of models. Allow me to define them more precisely.
Intuitively, we want an “interpretation” of the sentences generated by a logic+language which assigns each to a truth value. Model theory formalizes this as an object+relation, but we can also look at it as a map from sentences onto Z_2.
Any such means of assigning a sentence to {true, false} (or equivalent) is an “interpretation” of sorts, but not necessarily a model. We reserve the term “model” specifically for not-stupid interpretations (ones where the interpretation maps “x” and “not x” to different values, etc.)
In your construction, when you consider a surjection from sentences to some set and pick one element of the range to be “truth”, you’ve essentially defined an interpretation in a roundabout way. (Every sentence mapped to 1 is true in that interpretation, every sentence mapped elsewhere is false in that interpretation.)
If your interpretation obeys the rules of logic (“x*y” maps to 1 whenever “x” maps to 1 and “y” maps to 1) then it’s a model. Otherwise, model theory doesn’t have much to say about it.
I’m not sure I understand the construct you’re describing: does the above help at all? I’m not sure if I’m answering the right questions.
Don’t worry about decidability in this context, I think it might be confusing things somewhat. The point I was making earlier about completeness is this:
If you consider the set of all sentences as the domain of your map, then a “theory” T is just a subset of the domain of your map. If there are multiple models (functions which obey the rules of the logic) from the set of all sentences onto Z_2 which map the subset of all sentences (theory) T to 1, then T is “incomplete”.
Thank you for your patience!
Hmm, I thought only the preimage of 1 is the model.
But… a theory can include sentences not in the preimage of 1 (undecidable?)… I am confused.
I would instead say that the preimage of 1 is the largest model of the map, and is the model of any “large enough” theory.
Where are these laws defined? In the logic? In the logic+language? Then the only “valid” maps are those which are homomorphisms (in some sense) from logic(?) to the Z_2 subset (true/false) of the codomain.
Thanks again!
I’ve missed something in my explanation of models. Allow me to define them more precisely.
Intuitively, we want an “interpretation” of the sentences generated by a logic+language which assigns each to a truth value. Model theory formalizes this as an object+relation, but we can also look at it as a map from sentences onto Z_2.
Any such means of assigning a sentence to {true, false} (or equivalent) is an “interpretation” of sorts, but not necessarily a model. We reserve the term “model” specifically for not-stupid interpretations (ones where the interpretation maps “x” and “not x” to different values, etc.)
In your construction, when you consider a surjection from sentences to some set and pick one element of the range to be “truth”, you’ve essentially defined an interpretation in a roundabout way. (Every sentence mapped to 1 is true in that interpretation, every sentence mapped elsewhere is false in that interpretation.)
If your interpretation obeys the rules of logic (“x*y” maps to 1 whenever “x” maps to 1 and “y” maps to 1) then it’s a model. Otherwise, model theory doesn’t have much to say about it.
I’m not sure I understand the construct you’re describing: does the above help at all? I’m not sure if I’m answering the right questions.
Don’t worry about decidability in this context, I think it might be confusing things somewhat. The point I was making earlier about completeness is this:
If you consider the set of all sentences as the domain of your map, then a “theory” T is just a subset of the domain of your map. If there are multiple models (functions which obey the rules of the logic) from the set of all sentences onto Z_2 which map the subset of all sentences (theory) T to 1, then T is “incomplete”.
OK, I think I have a clearer picture now, between yours and DavidS’s explanations. I only wish I had a chance to learn it in school. Thanks!