“A model is an interpretation of the sentences generated by a language. A model is a structure which assigns a truth value to each sentence generated by some language under some logic.”
I think this phrasing will be very misleading to anyone who tries to learn model theory from these posts. This is one thing a model DOES, but it isn’t what a model IS. As far as I can tell, you nowhere say what a model is, even approximately. Writing out precisely what a model is takes a lot of space (like in the book you’re reading!) so let me give an example.
Our alphabet will be the symbols of first order logic, plus as many variable names as we need, and the symbols +, =, 0.
Our axioms are
∀ x : x+0=0+x=x
∀ x,y: x+y=y+x
∀ x,y,z: (x+y)+z=x+(y+z)
∀ x ∃ y : x+y=y+x=0
Our THEORY is the set of all logical consequences of these statements, where “logical consequence” means “obtainable by the formal rules of first order logic . A MODEL of our theory is a specific set G, a specific element of G called 0 and a specific operation + taking two elements of G and returning a third element of G, such that all of these statements are true about G. In other words, a model of this theory is an abelian group.
One thing an abelian group can do is give us a way to assign a true or false value to any statement in our language. For example, consider the statement ∀ x ∃ y : y+y+y=x. This statement is true in the group of rational numbers, but false in the group of integers. If we choose a particular abelian group, that will force a specific choice as to whether this statement is true or false.
However, you shouldn’t identify an abelian group with a way of assigning truth values to statements about abelian groups. For example, the rational numbers and the real numbers are both abelian groups and, as it turns out, there is no statement using only +, 0, = and logical connectives whose truth value is different in these two groups. Nonetheless, they are different models.
Thanks! Good point, that distinction is useful. I’ve updated the post to make this more clear (under the “models” header).
Personally, I tend to view things the other way around. As far as I’m concerned, a model of abelian group theory is anything that interprets sentences appropriately (while obeying the rules of the logic), for some value of “interprets”.
It so happens that any model of group theory is isomorphic to some pointed set with an associative operator for which the point is an identity, but the model doesn’t have to be a pointed set with an associative operator for which point is an identity. It could also be operations on a rubix cube. From my point of view, you’ve got ‘IS’ and ‘DOES’ backwards :-)
It’s just perspective, I suppose. I don’t particularly view set theory as foundational; I view it as one formalization that happens to have high enough fidelity to represent the behavior of any given model.
Still, your view is definitely the more standard one.
As far as I can tell, you nowhere say what a model is, even approximately
Well, this post is “Context for Model Theory”: I didn’t intend to introduce models themselves here. Though your concerns probably apply to the follow-up post as well.
The operations on a Rubix cube aren’t abelian. Is that just a typo on your part, or am I missing some subtle point you are making?
I’m not sure what you are getting at when you say you don’t want to found math on sets. I definitely intended to use the word “set” in a naive sense, so that it is perfectly fine for sets to contain numbers, or rotations of a Rubix cube, or for that matter rocks and flowers. I wasn’t trying to imply that the elements of a model had to be recursively constructed from the nullset by the axioms of ZFC. If you prefer “collection of things”, I’d be glad to go with that. But I (and more to the point, model theorists) do want to think of a model as a bunch of objects with functions that take as inputs these objects and make other objects, and relations which do and do not hold between various pairs of the objects.
I’m retracting a bunch of the other things I wrote after this because, on reflection, the later material was replying to a misreading of what you wrote in your following post. I still think your de-emphasis on the fact that the model is the universe is very confusing, especially when you then talk about the cardinality of models. (What is the cardinality of a rule for assigning truth values?) But on careful reading, you weren’t actually saying something wrong.
Oops, typo. (The typo was that I said “commutative” when dereferencing “group”; notice that I said “any model of group theory” and not “any model of abelian group theory”.) Thanks for the tip.
I’m not sure what you are getting at when you say you don’t want to found math on sets … I wasn’t trying to imply that the elements of a model had to be recursively constructed from the nullset by the axioms of ZFC.
Ok, cool. I guess my point is that set theory is a formal representation of real things, but it is not the things themselves. The “model” is the real thing, which happens to be representable as a set. I tried to make this wording clear (especially in the next post), but I don’t think I succeeded.
But I (and more to the point, model theorists) do want to think of a model as a bunch of objects with functions that take as inputs these objects and make other objects, and relations which do and do not hold between various pairs of the objects.
Me too! But mostly because my “implicit” formal system is set theory. If we were working with different foundations (let’s say type theory, because that’s the only other potentially-foundational system I know) then I would want to think of a model as elements of a type, and function symbols would need to be typed, and so on.
This is why I defined the model as an in interpretation which follows certain rules, rather than as a set+function specifically: In my head, the concept of a model is separate from the system I use to represent them.
At this point, it’s a matter of perspective, and I acknowledge that my viewpoint is non-standard. You’re definitely correct that I should have used more concrete examples (“these axioms are group theory; actual groups are models” etc.) from the get-go.
I still think your de-emphasis on the fact that the model is the universe is very confusing, especially when you then talk about the cardinality of models.
Thanks, I’ve edited the post to make this a bit more clear.
But on careful reading, you weren’t actually saying something wrong.
I very much appreciate the critiques. I admit that the next post is pretty sloppy; it was somewhat rushed and I couldn’t go into the depth I wanted. I far underestimated how much must be taught before you can express even the easy parts of model theory. I skimped on formally defining quite a few things, power-of-a-model among them.
However, you shouldn’t identify an abelian group with a way of assigning truth values to statements about abelian groups. For example, the rational numbers and the real numbers are both abelian groups and, as it turns out, there is no statement using only +, 0, = and logical connectives whose truth value is different in these two groups. Nonetheless, they are different models.
Hmm, but the axiom sets are different for rationals and reals, since the latter require Dedekind-completeness, which selects a different theory from the language+logic (in So8res’s terms). Why would one try to compare/distinguish models in different theories based on a subset of the logic and a subset of axioms?
The reals can be studied as models of many theories. They (with the operation +, relation = and element 0) are a model of the axioms of an abelian group. They are also a model of the axioms of a group. The reals with (+, , 0, 1, =) are a model of the axioms of a field. The reals with (+, , 0, 1, =, <) are a model of the axioms of an ordered field. Etcetera…
Models are things. Theories are collections of statements about things. A model can satisfy many theories; a theory can have many models. I agree completely with So8res statement that it is important to keep the two straight.
In addition, your example of Dedekind completeness is an awkward one because the Dedekind completeness axiom is a good example of the kind of thing you can’t say in first order logic. (There are partial ways around this, but I’m trying to keep my replies on the introductory level of this post.) But I can just imagine that you had distinguished the reals and the rationals by saying that, in R, ∃ x : x^2=1+1 is true and in Q it is false, so I don’t need to focus on that.
“A model is an interpretation of the sentences generated by a language. A model is a structure which assigns a truth value to each sentence generated by some language under some logic.”
I think this phrasing will be very misleading to anyone who tries to learn model theory from these posts. This is one thing a model DOES, but it isn’t what a model IS. As far as I can tell, you nowhere say what a model is, even approximately. Writing out precisely what a model is takes a lot of space (like in the book you’re reading!) so let me give an example.
Our alphabet will be the symbols of first order logic, plus as many variable names as we need, and the symbols +, =, 0.
Our axioms are
∀ x : x+0=0+x=x
∀ x,y: x+y=y+x
∀ x,y,z: (x+y)+z=x+(y+z)
∀ x ∃ y : x+y=y+x=0
Our THEORY is the set of all logical consequences of these statements, where “logical consequence” means “obtainable by the formal rules of first order logic . A MODEL of our theory is a specific set G, a specific element of G called 0 and a specific operation + taking two elements of G and returning a third element of G, such that all of these statements are true about G. In other words, a model of this theory is an abelian group.
One thing an abelian group can do is give us a way to assign a true or false value to any statement in our language. For example, consider the statement ∀ x ∃ y : y+y+y=x. This statement is true in the group of rational numbers, but false in the group of integers. If we choose a particular abelian group, that will force a specific choice as to whether this statement is true or false.
However, you shouldn’t identify an abelian group with a way of assigning truth values to statements about abelian groups. For example, the rational numbers and the real numbers are both abelian groups and, as it turns out, there is no statement using only +, 0, = and logical connectives whose truth value is different in these two groups. Nonetheless, they are different models.
Thanks! Good point, that distinction is useful. I’ve updated the post to make this more clear (under the “models” header).
Personally, I tend to view things the other way around. As far as I’m concerned, a model of abelian group theory is anything that interprets sentences appropriately (while obeying the rules of the logic), for some value of “interprets”.
It so happens that any model of group theory is isomorphic to some pointed set with an associative operator for which the point is an identity, but the model doesn’t have to be a pointed set with an associative operator for which point is an identity. It could also be operations on a rubix cube. From my point of view, you’ve got ‘IS’ and ‘DOES’ backwards :-)
It’s just perspective, I suppose. I don’t particularly view set theory as foundational; I view it as one formalization that happens to have high enough fidelity to represent the behavior of any given model.
Still, your view is definitely the more standard one.
Well, this post is “Context for Model Theory”: I didn’t intend to introduce models themselves here. Though your concerns probably apply to the follow-up post as well.
The operations on a Rubix cube aren’t abelian. Is that just a typo on your part, or am I missing some subtle point you are making?
I’m not sure what you are getting at when you say you don’t want to found math on sets. I definitely intended to use the word “set” in a naive sense, so that it is perfectly fine for sets to contain numbers, or rotations of a Rubix cube, or for that matter rocks and flowers. I wasn’t trying to imply that the elements of a model had to be recursively constructed from the nullset by the axioms of ZFC. If you prefer “collection of things”, I’d be glad to go with that. But I (and more to the point, model theorists) do want to think of a model as a bunch of objects with functions that take as inputs these objects and make other objects, and relations which do and do not hold between various pairs of the objects.
I’m retracting a bunch of the other things I wrote after this because, on reflection, the later material was replying to a misreading of what you wrote in your following post. I still think your de-emphasis on the fact that the model is the universe is very confusing, especially when you then talk about the cardinality of models. (What is the cardinality of a rule for assigning truth values?) But on careful reading, you weren’t actually saying something wrong.
Oops, typo. (The typo was that I said “commutative” when dereferencing “group”; notice that I said “any model of group theory” and not “any model of abelian group theory”.) Thanks for the tip.
Ok, cool. I guess my point is that set theory is a formal representation of real things, but it is not the things themselves. The “model” is the real thing, which happens to be representable as a set. I tried to make this wording clear (especially in the next post), but I don’t think I succeeded.
Me too! But mostly because my “implicit” formal system is set theory. If we were working with different foundations (let’s say type theory, because that’s the only other potentially-foundational system I know) then I would want to think of a model as elements of a type, and function symbols would need to be typed, and so on.
This is why I defined the model as an in interpretation which follows certain rules, rather than as a set+function specifically: In my head, the concept of a model is separate from the system I use to represent them.
At this point, it’s a matter of perspective, and I acknowledge that my viewpoint is non-standard. You’re definitely correct that I should have used more concrete examples (“these axioms are group theory; actual groups are models” etc.) from the get-go.
Thanks, I’ve edited the post to make this a bit more clear.
I very much appreciate the critiques. I admit that the next post is pretty sloppy; it was somewhat rushed and I couldn’t go into the depth I wanted. I far underestimated how much must be taught before you can express even the easy parts of model theory. I skimped on formally defining quite a few things, power-of-a-model among them.
Hmm, but the axiom sets are different for rationals and reals, since the latter require Dedekind-completeness, which selects a different theory from the language+logic (in So8res’s terms). Why would one try to compare/distinguish models in different theories based on a subset of the logic and a subset of axioms?
The reals can be studied as models of many theories. They (with the operation +, relation = and element 0) are a model of the axioms of an abelian group. They are also a model of the axioms of a group. The reals with (+, , 0, 1, =) are a model of the axioms of a field. The reals with (+, , 0, 1, =, <) are a model of the axioms of an ordered field. Etcetera…
Models are things. Theories are collections of statements about things. A model can satisfy many theories; a theory can have many models. I agree completely with So8res statement that it is important to keep the two straight.
In addition, your example of Dedekind completeness is an awkward one because the Dedekind completeness axiom is a good example of the kind of thing you can’t say in first order logic. (There are partial ways around this, but I’m trying to keep my replies on the introductory level of this post.) But I can just imagine that you had distinguished the reals and the rationals by saying that, in R, ∃ x : x^2=1+1 is true and in Q it is false, so I don’t need to focus on that.