# prove that any two isomorphic structures are elementarily equivalent

Imagine we have two L-structures $M$ and $N$. For each L-sentence $\phi$ , $M$ models $\phi$ iff $N$ models $\phi$. We call $M$ and $N$ two elementary equivalent L-structures.

We say $M$ and $N$ are isomorphic if there exists a function from $U_M$ to $U_N$ which is 1 to 1 and onto and preserves symbols. ( i mean an embedding which is also onto )

Now i want to prove that if two structures are isomorphic, then they are elementary equivalent.

Note : When we speak about L-formulas, we prove these kind of questions by induction on the complexity of the formulas. but in this case, we have sentences not formulas.

Choose an isomorphism $f$. Prove the more general statement: for any formula $\phi$, if we consider any assignment of values in $M$ to the variables, and the corresponding (via $f$) assignment in $N$, the formula $\phi$ will be satisfied by the one assignment if and only if it is satisfied by the other.

• would you please explain it more ? – Arman Malekzadeh Jan 7 '16 at 14:11
• @ArmanMalekzade In order to prove it for all formulas, you can induct on the recursive definition of a formula. The base case, for example, includes proving that it holds for formulas of the form $t_1 = t_2$ where $t_1$ and $t_2$ are terms. – 6005 Jan 7 '16 at 14:53
• @6005 i found something like that in here : cs.rice.edu/~vardi/comp409/lec20.pdf ( page 3 ) but i can't understand what are the functions mentioned as h and $\alpha$ – Arman Malekzadeh Jan 7 '16 at 14:56

Your notion of isomorphism is incomplete and does not suffice to prove the property you're after.

For the structures $M$ and $N$ to be "isomorphic" in the usual sense, you need not only a bijection between the universes, but the bijection also needs to preserve the meanings of function and predicate symbols.

• i fixed that one too... is that correct now ? :) i meant an embedding . i just forgot to write it :) i'm sorry dude – Arman Malekzadeh Jan 7 '16 at 14:53
• This is true, but not an answer. – YoTengoUnLCD Jan 7 '16 at 14:59
• @YoTengoUnLCD: It was an answer to the original question. When the question is "how do I prove A from B" and A doesn't actually follow from B, a perfectly good answer is "you don't, because such-and-such". This holds even if the question uses a non-standard (and confusing) name for its B. – Henning Makholm Jan 7 '16 at 15:25

The structures (2Z, 0,+) and (Z,0,+) are isomorphic in the language of gropus but are not elementary equivalent. E.g. the closed formula "There exists v such that (v+v=2)" is not satisfied in the first but is satisfied in the second.

• $\exists v\, (v+v=2)$ is not a sentence in the language $\{0,+\}$, since it uses the symbol $2$. If you add the constant $2$ to the language, then you have correctly observed that $(2\mathbb{Z},0,2,+)$ is not elementarily equivalent to $(\mathbb{Z},0,2+)$.... but these structures are not isomorphic, either! – Alex Kruckman Dec 13 '16 at 15:58
• Alex, you are absolutely right. I confused the notion of elementary substructures with elementary equivalent structures! Thanks a lot – George Chailos Dec 14 '16 at 21:25
• Alex, You are absolutely right and thanks a lot. I somehow confused the concept of elementary substructure wilth elementary equivalent structures. The structure (2Z,0, +) is a substructure of (Z,0,+) and isomorphic to it, though it is not an elementary substructure of it (for the reason I provided). But as you pointed out is an elementary equivalent structure to (Z,0,+) since elementary equivalence concerns only closed formulas of the underlying languagey – George Chailos Dec 14 '16 at 21:32