I Cardinality of a set of constant symbols (model theory)

nomadreid

Gold Member
1,288
103
First, I want to be pedantic here and underline the distinction between a set (in the model, or interpretation) and a sentence (in the theory) which is fulfilled by that set, and also constant symbols (in the theory) versus constants (in the universe of the model)
Given that, I would like to know if the following is a correct way to look at phrases such as "the cardinality of the set of constant symbols of a language."
(1) One can only make this phrase in reference to a new model in which the (previous) constant symbols become constants, so that this collection can be a set.
(2) The collection of constants in the universe of the new model must be larger than the universe of the original model, because there is nothing to stop one having one constant symbol (in the original theory) for every constant in the universe (of the original model), and a collection that is as large as the universe cannot be a set.

Please correct the faults in the above. Thanks.
 

stevendaryl

Staff Emeritus
Science Advisor
Insights Author
8,400
2,569
First, I want to be pedantic here and underline the distinction between a set (in the model, or interpretation) and a sentence (in the theory) which is fulfilled by that set, and also constant symbols (in the theory) versus constants (in the universe of the model)
Given that, I would like to know if the following is a correct way to look at phrases such as "the cardinality of the set of constant symbols of a language."
(1) One can only make this phrase in reference to a new model in which the (previous) constant symbols become constants, so that this collection can be a set.
(2) The collection of constants in the universe of the new model must be larger than the universe of the original model, because there is nothing to stop one having one constant symbol (in the original theory) for every constant in the universe (of the original model), and a collection that is as large as the universe cannot be a set.

Please correct the faults in the above. Thanks.
I would say that neither of those is correct. In model theory, everything is a set. You describe a language by giving:
  1. A set of constant symbols
  2. A set of function symbols
  3. A set of relation symbols
Then to talk about models of the language, you introduce more sets:
  1. A set of individuals, the domain of the model.
  2. For each constant symbol, there is a corresponding individual
  3. For each relation symbol with n arguments, there is a corresponding set of n-tuples of individuals
  4. For each function symbol with n arguments, there is a corresponding set of (n+1)-tuples
When people talk about the cardinality of the set of constant symbols, they mean the cardinality of a set. There is no issue of something being "too large" to be a set. By definition, models are sets.
 

nomadreid

Gold Member
1,288
103
Thanks, Stevendaryl. You have pointed out my error, albeit your statement that everything is a set is a bit hasty. To quote "Fiddler on the roof",
"He's right and he's right? They can't both be right." /"You know, you are also right." That is, you are looking at it top-down (as one should when talking about the class of constant symbols -- therein lies my error), and I was looking at it bottom-up (when you think of model on top and theory on the bottom). That is, it is a little much to say without qualification that "everything is a set": from the point of view of the theory, the universe of its model is definitely not a set, although it is a class -- this is one result of Russell's paradox. What you meant was that since the model relation is a first-order definition, one can make a model-theory construction within an existing theory, i.e., whereby the universe is an existing set. If one is looking at a model with an infinite universe from bottom-up, anything with the same cardinality as the universe is excluded from being describable as a set by the theory -- this is what I meant by "too big".
 

Want to reply to this thread?

"Cardinality of a set of constant symbols (model theory)" You must log in or register to reply here.

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top