Twins: non-isomorphic models forced to be isomorphic Part I

Abstract For which (first-order complete, usually countable) T do there exist non-isomorphic models of T which become isomorphic after forcing with a forcing notion $$\mathbb {P}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> ? Necessarily, $$\mathbb {P}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> is non-trivial; i.e it adds some new set of ordinals. It is best if we also demand that it collapses no cardinal. It is better if we demand on the one hand that the models are non-isomorphic, and even far from each other (in a suitable sense), but on the other hand, $$\mathscr {L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> -equivalent in some suitable logic $$\mathscr {L}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>L</mml:mi> </mml:math> . In this part we give sufficient conditions: for theories with the independence property, we prove this when $$\mathbb {P}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> adds no new $$\omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> -sequence. We may prove it “for some $$\mathbb {P}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> ," but better would be for some specific forcing notions, or a natural family. Best would be to characterize the pairs $$(T,\mathbb {P})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>,</mml:mo> <mml:mi>P</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> for which we have such models. The results say (e.g.) that there are models $$M_1,M_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> which are not isomorphic (and even far from being isomorphic, in a rigorous sense) which become isomorphic when we extend the universe by adding a new branch to the tree $$({}^{\theta &gt;}2,\lhd )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mmultiscripts> <mml:mrow/> <mml:mrow/> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>&gt;</mml:mo> </mml:mrow> </mml:mmultiscripts> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>⊲</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . We shall mention some specific choices of $$\mathbb {P}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> : mainly $$({}^{\theta &gt;}2,\lhd )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mmultiscripts> <mml:mrow/> <mml:mrow/> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>&gt;</mml:mo> </mml:mrow> </mml:mmultiscripts> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>⊲</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with $$\theta = \theta ^{&lt;\theta }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>θ</mml:mi> <mml:mrow> <mml:mo>&lt;</mml:mo> <mml:mi>θ</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . This work does not require any serious knowledge of forcings, nor of stability theory, though they form the motivation. Concerning forcing, the reader just has to agree that starting with a universe $${\textbf {V}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> of set t