MHB Is this a correct application of the reduction method?

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The discussion focuses on the application of the reduction method to prove the undecidability of a positive existential theory T by interpreting another known undecidable theory T'. The initial formulation describes translating formulas from T' to T, establishing that if T' is undecidable, then T must also be undecidable. An alternative approach is presented, assuming T is decidable and leading to a contradiction by interpreting T in T', reinforcing the undecidability of T. The revised formulation emphasizes the need for a computable mapping between the theories, suggesting that the terms "positive" and "existential" may not be crucial for the construction. Overall, the participants seek clarity and potential improvements in their formulation of the reduction method.
mathmari
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Hey! :o

Is the following formulation of the reduction correct? Undecidability of an (positive) existential theory $T$ is proved often by reduction, i.e., by interpreting another (positive) existential theory $T'$, known to be undecidable, in $T$.

Each formula $\phi'$ of the language of $T'$ is translated into a formula $\phi$ of the language of $T$.

So, the (positive) existential theory $T'$ is reduced to the (positive) existential theory $T$.

Since $T'$ is undecidable, $T$ is also undecidable.
An other way to apply the reduction is the following: We suppose that the (positive) existential theory $T$ is decidable. We want to interpret $T$ in another (positive) existential theory $T'$, known to be undecidable.

Each formula $\phi$ of the language of $T$ is translated into a formula $\phi'$ of the language of $T'$.

So, the (positive) existential theory $T$ is reduced to the (positive) existential theory $T'$.

Since $T'$ is undecidable and we have supposed that $T$ is decidable, we aget a contradiction.

So, $T$ is also undecidable.

Is everything correct? Could I improve something at the formulation?
 
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I changed the formulation...

Undecidability of an (positive) existential theory $T$ is proved often by reducing an other (positive) existential theory $T'$, which is known to be undecidable,to $T$, i.e., by a mapping from the (positive) existential sentences in the language of $T'$ to the (positive) existential sentences in the language of $T$, $$\phi' \mapsto \phi$$ such that $T'$ proves $\phi'$ iff $T$ proves $\phi$.
Is the formulation now correct? Could I improve something?
 
I agree. Of course, the mapping $\phi'\mapsto\phi$ has to be computable. I also don't think that "positive" and "existential" are important here; this construction works for any fragment of $T'$ and $T$.
 
Ok... Thanks a lot! (Smile)
 
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