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(1) $T = \{(\forall x) (x=c_1 \lor x=c_2 ) \}$

(2) $T =\{c_1\neq c_2 \}$,

$T' = \{(\exists x)(\exists y)((x \neq y) \land (\forall z)(z=y \lor z=x))\}$

I am not sure, how to determine that T' is or isn't an extension/conservative extension of T.

In first theory, there is an axiom that is valid in structure, which has at most 2 elements. T' is valid for structures, which universe has just two elements. So I think the T' cannot be extension of first theory, because T' axiom cannot be valid in structures, which has one-element universe.

In the second example there is a theory, which axiom is valid in only two-element structure. Both theories are valid in same models, so T' is extension of theory T.

Extension is conservative, if we can prove any formula in language of theory T from axioms of theory T'. I would say, it si conservative extension, because we can prove any formula in language of theory T, which has two constants, in theory T'.

Mafi
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  • I don't understand your question. The fact that $T'$ is a conservative extension of $T$ entails that the language of $T'$ extends the language of $T$. In your question it is the contrary, since the constants $c_1$ and $c_2$ are in the language of $T$, not of $T'$. See the definition of conservative extension on Wikipedia. – Taroccoesbrocco Jan 31 '18 at 13:38
  • If $T$ has axioms (1) and (2), they says that every model has exactly two elements: for ax (1), at most two, but for ax (2) at least two. – Mauro ALLEGRANZA Jan 31 '18 at 13:39
  • How can (2) be valid in structures with at least two? – Mafi Jan 31 '18 at 13:58
  • I knew that theory, which extends another theory has to also extends the language. But I wasn't sure, if the constants has some role in a language in the dependence with theories. So if the constants arent part of language of theory T', then theory T' cannot extend any of these two theories? – Mafi Jan 31 '18 at 14:03

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