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.
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.