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Q. If for every sentence $\phi$ of a first-order language $\mathcal{L}$ is s.t. $\mathcal{A} \vDash \phi$ iff $\mathcal{A'} \vDash \phi $, then there is an isomorphism between two interpretations/structures/models $\mathcal{A}$ and $\mathcal{A'}$ of a first-order language $\mathcal{L}$. How do I show that this is false?

$\mathcal{A'}$ and $\mathcal{A'}$ are structures of $\mathcal{L}$ that make the same sentences $\phi$ of $\mathcal{L}$ true.

Yet they don't have to be isomorphic for this to be the case.

I'm not sure how to prove this.

Maybe we could use the fact that it's converse is a theorem, which is the following the theorem: If there is an isomorphism between two interpretations $\mathcal{A}$ and $\mathcal{A'}$ of a first-order language $\mathcal{L}$, then for every sentence $\phi$ of a first-order language $\mathcal{L}$ is s.t. $\mathcal{A} \vDash \phi$ iff $\mathcal{A'} \vDash \phi$.

Many thanks in advance!

1 Answers1

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The simplest counterexample would be to take $\mathcal{L}$ the language with only the equality symbol, and take $\mathcal{A}$ and $\mathcal{A'}$ to be infinite sets of different cardinalities. They will both satisfy the same first-order $\mathcal{L}$-sentences, but can never be isomorphic because of their different cardinality.

Mark Kamsma
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  • What do you mean by infinite sets of different cardinalities. Do you mean that one is countably infinite and the other non-countably infinite? –  Apr 10 '20 at 19:47
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    @HarryRothschild It means two infinite sets that are not in bijection. So any countable and uncountable set would work (e.g. $\mathbb{N}$ and $\mathbb{R}$). To be honest if you are not familiar with this kind thing, then you might want to learn that first. – Mark Kamsma Apr 10 '20 at 21:07
  • A set is enumerable if there's a surjection from the positive integers to that set. Then I'm wondering whether all denumerable sets have the same cardinality as the positive integers. When you take the power set of a denumerable set you get a non-denumerable set, then you can keep applying the power set to get different sizes of infinities. This I understand. –  Apr 11 '20 at 09:21
  • Wait, nevermind. All denumerable sets are equinumerous with the positive integers. We define a set to be enumerable through a surjection from the positive integers to allow for finite enumerable sets. –  Apr 11 '20 at 09:24