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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">P</mi> </math></EquationSource> </InlineEquation>? Necessarily, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">P</mi> </math></EquationSource> </InlineEquation> 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, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathscr {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>-equivalent in some suitable logic <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathscr {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation>. In this part we give sufficient conditions: for theories with the independence property, we prove this when <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">P</mi> </math></EquationSource> </InlineEquation> adds no new <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation>-sequence. We may prove it “for some <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathbb {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">P</mi> </math></EquationSource> </InlineEquation>," but better would be for some specific forcing notions, or a natural family. Best would be to characterize the pairs <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$(T,\mathbb {P})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi mathvariant="double-struck">P</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for which we have such models. The results say (e.g.) that there are models <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$M_1,M_2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>M</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> which are not isomorphic (and even far from being isomorphic, in a rigorous sense) which become isomorphic when we extend the universe by adding a new branch to the tree <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$({}^{\theta >}2,\lhd )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mmultiscripts> <mrow /> <mrow /> <mrow> <mi>θ</mi> <mo>></mo> </mrow> </mmultiscripts> <mn>2</mn> <mo>,</mo> <mo>⊲</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We shall mention some specific choices of <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathbb {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">P</mi> </math></EquationSource> </InlineEquation>: mainly <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$({}^{\theta >}2,\lhd )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mmultiscripts> <mrow /> <mrow /> <mrow> <mi>θ</mi> <mo>></mo> </mrow> </mmultiscripts> <mn>2</mn> <mo>,</mo> <mo>⊲</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\theta = \theta ^{<\theta }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>=</mo> <msup> <mi>θ</mi> <mrow> <mo><</mo> <mi>θ</mi> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation>. This work does not require any serious knowledge of forcings, nor of stability theory, though they form the motivation. Concerning forcing, the reader just has to agree that starting with a universe <InlineEquation ID="IEq15"> <EquationSource Format="TEX">${\textbf {V}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">V</mi> </math></EquationSource> </InlineEquation> of set theory (i.e. a model of <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\textsf {ZFC}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ZFC</mi> </math></EquationSource> </InlineEquation>) and a quasiorder <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\mathbb {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">P</mi> </math></EquationSource> </InlineEquation>, there are a new directed <InlineEquation ID="IEq18"> <EquationSource Format="TEX">${\textbf {G}}\subseteq \mathbb {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">G</mi> <mo>⊆</mo> <mi mathvariant="double-struck">P</mi> </mrow> </math></EquationSource> </InlineEquation> meeting every dense subset D of <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$\mathbb {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">P</mi> </math></EquationSource> </InlineEquation> and a universe <InlineEquation ID="IEq20"> <EquationSource Format="TEX">${\textbf {V}}[{\textbf {G}}]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">V</mi> <mo stretchy="false">[</mo> <mi mathvariant="bold">G</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> (so it satisfies <InlineEquation ID="IEq21"> <EquationSource Format="TEX">$\textsf {ZFC}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="sans-serif">ZFC</mi> </math></EquationSource> </InlineEquation>) of which the original <InlineEquation ID="IEq22"> <EquationSource Format="TEX">${\textbf {V}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">V</mi> </math></EquationSource> </InlineEquation> is a transitive subclass. We may say that <InlineEquation ID="IEq23"> <EquationSource Format="TEX">${\textbf {V}}[{\textbf {G}}]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">V</mi> <mo stretchy="false">[</mo> <mi mathvariant="bold">G</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> (also denoted <InlineEquation ID="IEq24"> <EquationSource Format="TEX">${\textbf {V}}^\mathbb {P}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="bold">V</mi> </mrow> <mi mathvariant="double-struck">P</mi> </msup> </math></EquationSource> </InlineEquation>) is the universe obtained by forcing with <InlineEquation ID="IEq25"> <EquationSource Format="TEX">$\mathbb {P}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">P</mi> </math></EquationSource> </InlineEquation>. This is part of the classification and so-called Main Gap programs.

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

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2025年10月4日

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Saharon Shelah

摘要

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 $\mathbb {P}$ $P$ ? Necessarily, $\mathbb {P}$ $P$ 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, $\mathscr {L}$ $L$ -equivalent in some suitable logic $\mathscr {L}$ $L$ . In this part we give sufficient conditions: for theories with the independence property, we prove this when $\mathbb {P}$ $P$ adds no new $\omega $ $ω$ -sequence. We may prove it “for some $\mathbb {P}$ $P$ ," but better would be for some specific forcing notions, or a natural family. Best would be to characterize the pairs $(T,\mathbb {P})$ $(T, P)$ for which we have such models. The results say (e.g.) that there are models $M_1,M_2$ $M_{1}, M_{2}$ 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 $({}^{\theta >}2,\lhd )$ $(^{θ >} 2, ⊲)$ . We shall mention some specific choices of $\mathbb {P}$ $P$ : mainly $({}^{\theta >}2,\lhd )$ $(^{θ >} 2, ⊲)$ with $\theta = \theta ^{<\theta }$ $θ = θ^{< θ}$ . 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 ${\textbf {V}}$ $V$ of set theory (i.e. a model of $\textsf {ZFC}$ $ZFC$ ) and a quasiorder $\mathbb {P}$ $P$ , there are a new directed ${\textbf {G}}\subseteq \mathbb {P}$ $G \subseteq P$ meeting every dense subset D of $\mathbb {P}$ $P$ and a universe ${\textbf {V}}[{\textbf {G}}]$ $V [G]$ (so it satisfies $\textsf {ZFC}$ $ZFC$ ) of which the original ${\textbf {V}}$ $V$ is a transitive subclass. We may say that ${\textbf {V}}[{\textbf {G}}]$ $V [G]$ (also denoted ${\textbf {V}}^\mathbb {P}$ $V^{P}$ ) is the universe obtained by forcing with $\mathbb {P}$ $P$ . This is part of the classification and so-called Main Gap programs.