Isomorphisms of models. (in logic).

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In summary, the conversation discusses the concept of isomorphism of models in logic, specifically in relation to a language containing a unary symbol function. The models given in the conversation are A=<Z,x+1>, B=<Z,x-1>, C=<Q,-x>, D=<Z,-x>, and E=<Q,x^2>, where Z represents the integer set and Q represents the rational set. The summary states that A is isomorphic to B and C is isomorphic to D, and these are the only isomorphisms in the given models. The speaker also requests for clarification and a basic definition or link to better understand the topic.
  • #1
MathematicalPhysicist
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suppose our language contains one unary syombol function.
we are given the next 5 models:
A=<Z,x+1> B=<Z,x-1> C=<Q,-x> D=<Z,-x> E=<Q,x^2>
write which of the models are isomorphic to each other.
where Z is the integer set, and Q is the ratioanls set.

my answer is that: A isomorphic to B, and C isomorphic to D, and these are the only isomorphisms.
 
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  • #2
I don't understand your question, but I am curious.
Could you explain it, or could you give me a link where I could find the basic definitions to understand you questions.

Thanks,

Michel
 
  • #3
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1. What is an isomorphism of models in logic?

An isomorphism of models in logic is a mapping between two models of a logical system that preserves the logical structure of the system. In other words, it is a one-to-one correspondence that preserves the truth values of logical statements between the two models.

2. How is an isomorphism of models different from a homomorphism?

While both isomorphisms and homomorphisms are mappings between models, isomorphisms preserve the entire logical structure of the system, including the relationships between elements, while homomorphisms only preserve the logical operations and their truth values.

3. Why are isomorphisms of models important in logic?

Isomorphisms of models are important in logic because they allow us to compare and relate different models of a logical system. This helps us gain a deeper understanding of the system and can also be useful in proving theorems and solving problems.

4. Can two non-isomorphic models have the same set of logical consequences?

No, two non-isomorphic models cannot have the same set of logical consequences. This is because isomorphisms preserve the truth values of logical statements, so if two models have different logical consequences, they must also be non-isomorphic.

5. How do you prove that two models are isomorphic?

To prove that two models are isomorphic, you must show that there exists a one-to-one correspondence between the elements of the models that preserves the logical structure. This can be done by constructing an isomorphism function and proving that it satisfies the definition of an isomorphism.

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