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Usual definition of domain of interpretation which I have found in wikipedia here uses set "structure" but not state clearly that domain of discourse has to be first order set from the beginning! It is something which is not clear for me here. Should I believe that some "interpretations" of theory may be inconsistent whilst other may be consistent when we drop interpretation part as a definition and we leave it free?

Note 1. As I do not want to be found very speculative, I will point to blog of Terence Tao here where You may find some information about nonfirstorderisability and specially this sentence:

Note 2. In wikipedia here I found the following remark:

Usual definition of domain of interpretation which I have found in wikipedia here uses set "structure" but not state clearly that domain of discourse has to be first order set from the beginning! It is something which is not clear for me here. Should I believe that some "interpretations" of theory may be inconsistent whilst other may be consistent when we drop interpretation part as a definition and we leave it free?

Note 1. As I do not want to be found very speculative, I will point to blog of Terence Tao here where You may find some information about nonfirstorderisability and specially this sentence:

Note 2. In wikipedia here I found the following remark:

Thank everybody for answering my previous questions: first, and second.

I may think about theory which when interpreted in set universe defined by some formulas, relations whatever may be consistent and have models, whilst in other universes do not. Is this a case? For example in answer about possible definition of domain of discourse for first order theories here Joel David Hamkins wrote: "If V is the universe of all sets, we can define certain classes in V, such as { x | φ(x) }, where φ is any property."

2. If we construct theory in first order logic ( for example as in my second question), and then we will try to use it on domain of objects defined by means of higher order logic, do we end with higher order theory, or rather first order theory which is saying something about second order logic objects? Or maybe the answer is" "it depends" and there should be stated additional requirements?

Thank everybody for answering my previous questions: first, and second.

I may think about theory which when interpreted in set universe defined by some formulas, relations whatever may be consistent and have models, whilst in other universes do not. Is this a case? For example in answer about possible definition of domain of discourse for first order theories here Joel David Hamkins wrote: "If V is the universe of all sets, we can define certain classes in V, such as { x | φ(x) }, where φ is any property."

2. If we construct theory in first order logic ( for example as in my second question), and then we will try to use it on domain of objects defined by means of higher order logic, do we end with higher order theory, or rather first order theory which is saying something about second order logic objects? Or maybe the answer is" "it depends" and there should be stated additional requirements?

As I do not want to be found very speculative, I will point to blog of Terence Tao here where You may find some information about nonfirstorderisability and specially this sentence:

So there are pretty useful practical statement, not very abstract, in normal mathematics which cannot be expressed in first order theory language.

Note 1. As I do not want to be found very speculative, I will point to blog of Terence Tao here where You may find some information about nonfirstorderisability and specially this sentence:

So there are pretty useful practical statement, not very abstract, in normal mathematics which cannot be expressed in first order theory language.

Note 2. In wikipedia here I found the following remark:

"MK can be confused with second-order ZFC, ZFC with second-order logic (representing second-order objects in set rather than predicate language) as its background logic. The language of second-order ZFC is similar to that of MK (although a set and a class having the same extension can no longer be identified), and their syntactical resources for practical proof are almost identical (and are identical if MK includes the strong form of Limitation of Size). But the semantics of second-order ZFC are quite different from those of MK. For example, if MK is consistent then it has a countable first-order model, while second-order ZFC has no countable models."