Reference request: quantifier elimination testdense orders are saturatedAlgebras admitting quantifier elimination“Fraïssé limits” without amalgamationQuantifier elimination vs decidabilityWhere do uncountable models collapse to?Reference request on a notion of independence for families of [real-valued] functionsComplex L^1 spaces; reference requestPreservation results in abstract logicsQuantifier elimination, subgroups of modulesQuantifier elimination in uncountable elementary “Fraïssé classes”
Reference request: quantifier elimination test
dense orders are saturatedAlgebras admitting quantifier elimination“Fraïssé limits” without amalgamationQuantifier elimination vs decidabilityWhere do uncountable models collapse to?Reference request on a notion of independence for families of [real-valued] functionsComplex L^1 spaces; reference requestPreservation results in abstract logicsQuantifier elimination, subgroups of modulesQuantifier elimination in uncountable elementary “Fraïssé classes”
$begingroup$
I'm having difficulty finding this result in the standard texts.
Theorem. Let $T$ be a theory in a language $mathcalL$. TFAE:
1) $T$ has quantifier elimination,
2) Whenever $M, N$ are $aleph_1$-saturated models of $T$, $A subset
M$, $B subset N$ are countable nonempty substructures and $f : A
rightarrow B$ an $mathcalL$-isomorphism, then for any $a in M$
there exists an extension $f' supset f$ with $a$ in the domain of
$f'$ and $f'$ still an $mathcalL$-isomorphism. In addition, for any
$b in N$ there exists an extension $f'' supset f$ with $b$ in the
range of $f''$ and $f''$ still an $mathcalL$-isomorphism.
I'm obtaining this result from these notes by Chatzidakis (see 2.27): http://www.math.ens.fr/~zchatzid/papiers/CUP-MT.pdf. Any suggestions are appreciated.
reference-request lo.logic model-theory
edited Jul 7 at 14:15
YCor
30.2k4 gold badges91 silver badges146 bronze badges
asked Jul 7 at 14:14
user221330user221330
984 bronze badges
$endgroup$
add a comment |
$begingroup$
I'm having difficulty finding this result in the standard texts.
Theorem. Let $T$ be a theory in a language $mathcalL$. TFAE:
1) $T$ has quantifier elimination,
2) Whenever $M, N$ are $aleph_1$-saturated models of $T$, $A subset
M$, $B subset N$ are countable nonempty substructures and $f : A
rightarrow B$ an $mathcalL$-isomorphism, then for any $a in M$
there exists an extension $f' supset f$ with $a$ in the domain of
$f'$ and $f'$ still an $mathcalL$-isomorphism. In addition, for any
$b in N$ there exists an extension $f'' supset f$ with $b$ in the
range of $f''$ and $f''$ still an $mathcalL$-isomorphism.
I'm obtaining this result from these notes by Chatzidakis (see 2.27): http://www.math.ens.fr/~zchatzid/papiers/CUP-MT.pdf. Any suggestions are appreciated.
reference-request lo.logic model-theory
edited Jul 7 at 14:15
YCor
30.2k4 gold badges91 silver badges146 bronze badges
asked Jul 7 at 14:14
user221330user221330
984 bronze badges
$endgroup$
2
$begingroup$
You really mean to replace $aleph_1$-saturated by $|mathcal L|^+$-saturated. Also, it suffices to consider just the forth direction. That is, the last sentence beginning "In addition" may be dropped.
$endgroup$
– Thomas Scanlon
Jul 7 at 18:15
add a comment |
$begingroup$
I'm having difficulty finding this result in the standard texts.
Theorem. Let $T$ be a theory in a language $mathcalL$. TFAE:
1) $T$ has quantifier elimination,
2) Whenever $M, N$ are $aleph_1$-saturated models of $T$, $A subset
M$, $B subset N$ are countable nonempty substructures and $f : A
rightarrow B$ an $mathcalL$-isomorphism, then for any $a in M$
there exists an extension $f' supset f$ with $a$ in the domain of
$f'$ and $f'$ still an $mathcalL$-isomorphism. In addition, for any
$b in N$ there exists an extension $f'' supset f$ with $b$ in the
range of $f''$ and $f''$ still an $mathcalL$-isomorphism.
I'm obtaining this result from these notes by Chatzidakis (see 2.27): http://www.math.ens.fr/~zchatzid/papiers/CUP-MT.pdf. Any suggestions are appreciated.
reference-request lo.logic model-theory
edited Jul 7 at 14:15
YCor
30.2k4 gold badges91 silver badges146 bronze badges
asked Jul 7 at 14:14
user221330user221330
984 bronze badges
$endgroup$
I'm having difficulty finding this result in the standard texts.
Theorem. Let $T$ be a theory in a language $mathcalL$. TFAE:
1) $T$ has quantifier elimination,
2) Whenever $M, N$ are $aleph_1$-saturated models of $T$, $A subset
M$, $B subset N$ are countable nonempty substructures and $f : A
rightarrow B$ an $mathcalL$-isomorphism, then for any $a in M$
there exists an extension $f' supset f$ with $a$ in the domain of
$f'$ and $f'$ still an $mathcalL$-isomorphism. In addition, for any
$b in N$ there exists an extension $f'' supset f$ with $b$ in the
range of $f''$ and $f''$ still an $mathcalL$-isomorphism.
I'm obtaining this result from these notes by Chatzidakis (see 2.27): http://www.math.ens.fr/~zchatzid/papiers/CUP-MT.pdf. Any suggestions are appreciated.
reference-request lo.logic model-theory
reference-request lo.logic model-theory
edited Jul 7 at 14:15
YCor
30.2k4 gold badges91 silver badges146 bronze badges
asked Jul 7 at 14:14
user221330user221330
984 bronze badges
edited Jul 7 at 14:15
YCor
30.2k4 gold badges91 silver badges146 bronze badges
asked Jul 7 at 14:14
user221330user221330
984 bronze badges
edited Jul 7 at 14:15
YCor
30.2k4 gold badges91 silver badges146 bronze badges
edited Jul 7 at 14:15
YCor
30.2k4 gold badges91 silver badges146 bronze badges
edited Jul 7 at 14:15
YCor
30.2k4 gold badges91 silver badges146 bronze badges
30.2k4 gold badges91 silver badges146 bronze badges
asked Jul 7 at 14:14
user221330user221330
984 bronze badges
asked Jul 7 at 14:14
user221330user221330
984 bronze badges
asked Jul 7 at 14:14
user221330user221330
984 bronze badges
984 bronze badges
2
$begingroup$
You really mean to replace $aleph_1$-saturated by $|mathcal L|^+$-saturated. Also, it suffices to consider just the forth direction. That is, the last sentence beginning "In addition" may be dropped.
$endgroup$
– Thomas Scanlon
Jul 7 at 18:15
add a comment |
2
$begingroup$
You really mean to replace $aleph_1$-saturated by $|mathcal L|^+$-saturated. Also, it suffices to consider just the forth direction. That is, the last sentence beginning "In addition" may be dropped.
$endgroup$
– Thomas Scanlon
Jul 7 at 18:15
2
2
$begingroup$
You really mean to replace $aleph_1$-saturated by $|mathcal L|^+$-saturated. Also, it suffices to consider just the forth direction. That is, the last sentence beginning "In addition" may be dropped.
$endgroup$
– Thomas Scanlon
Jul 7 at 18:15
$begingroup$
You really mean to replace $aleph_1$-saturated by $|mathcal L|^+$-saturated. Also, it suffices to consider just the forth direction. That is, the last sentence beginning "In addition" may be dropped.
$endgroup$
– Thomas Scanlon
Jul 7 at 18:15
add a comment |
1 Answer
1
active
oldest
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$begingroup$
I'm not sure if this formulation with $omega_1$-saturated models is in any of the "main standard" texts. But there is a complete treatment in these course notes by Pillay (see Proposition 2.29).
If you're looking for something in a standard published text, Corollary 3.1.6 of Model Theory: An Introduction (Marker) is along the same lines. It says:
Let $T$ be an $L$-theory. Suppose that for all quantifier-free formulas $phi(barv,w)$, if $M,Nmodels T$, $A$ is a common substructure of $M$ and $N$, $barain A$, and there is $bin M$ such that $Mmodelsphi(bara,b)$, then there is $cin N$ such that $Nmodels phi(bara,c)$. Then $T$ has quantifier elimination.
It is not too difficult to deduce the theorem you quoted from this.
answered Jul 7 at 16:15
Gabe ConantGabe Conant
8467 silver badges16 bronze badges
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$begingroup$
I'm not sure if this formulation with $omega_1$-saturated models is in any of the "main standard" texts. But there is a complete treatment in these course notes by Pillay (see Proposition 2.29).
If you're looking for something in a standard published text, Corollary 3.1.6 of Model Theory: An Introduction (Marker) is along the same lines. It says:
Let $T$ be an $L$-theory. Suppose that for all quantifier-free formulas $phi(barv,w)$, if $M,Nmodels T$, $A$ is a common substructure of $M$ and $N$, $barain A$, and there is $bin M$ such that $Mmodelsphi(bara,b)$, then there is $cin N$ such that $Nmodels phi(bara,c)$. Then $T$ has quantifier elimination.
It is not too difficult to deduce the theorem you quoted from this.
answered Jul 7 at 16:15
Gabe ConantGabe Conant
8467 silver badges16 bronze badges
$endgroup$
add a comment |
$begingroup$
I'm not sure if this formulation with $omega_1$-saturated models is in any of the "main standard" texts. But there is a complete treatment in these course notes by Pillay (see Proposition 2.29).
If you're looking for something in a standard published text, Corollary 3.1.6 of Model Theory: An Introduction (Marker) is along the same lines. It says:
Let $T$ be an $L$-theory. Suppose that for all quantifier-free formulas $phi(barv,w)$, if $M,Nmodels T$, $A$ is a common substructure of $M$ and $N$, $barain A$, and there is $bin M$ such that $Mmodelsphi(bara,b)$, then there is $cin N$ such that $Nmodels phi(bara,c)$. Then $T$ has quantifier elimination.
It is not too difficult to deduce the theorem you quoted from this.
answered Jul 7 at 16:15
Gabe ConantGabe Conant
8467 silver badges16 bronze badges
$endgroup$
add a comment |
$begingroup$
I'm not sure if this formulation with $omega_1$-saturated models is in any of the "main standard" texts. But there is a complete treatment in these course notes by Pillay (see Proposition 2.29).
If you're looking for something in a standard published text, Corollary 3.1.6 of Model Theory: An Introduction (Marker) is along the same lines. It says:
Let $T$ be an $L$-theory. Suppose that for all quantifier-free formulas $phi(barv,w)$, if $M,Nmodels T$, $A$ is a common substructure of $M$ and $N$, $barain A$, and there is $bin M$ such that $Mmodelsphi(bara,b)$, then there is $cin N$ such that $Nmodels phi(bara,c)$. Then $T$ has quantifier elimination.
It is not too difficult to deduce the theorem you quoted from this.
answered Jul 7 at 16:15
Gabe ConantGabe Conant
8467 silver badges16 bronze badges
$endgroup$
I'm not sure if this formulation with $omega_1$-saturated models is in any of the "main standard" texts. But there is a complete treatment in these course notes by Pillay (see Proposition 2.29).
If you're looking for something in a standard published text, Corollary 3.1.6 of Model Theory: An Introduction (Marker) is along the same lines. It says:
Let $T$ be an $L$-theory. Suppose that for all quantifier-free formulas $phi(barv,w)$, if $M,Nmodels T$, $A$ is a common substructure of $M$ and $N$, $barain A$, and there is $bin M$ such that $Mmodelsphi(bara,b)$, then there is $cin N$ such that $Nmodels phi(bara,c)$. Then $T$ has quantifier elimination.
It is not too difficult to deduce the theorem you quoted from this.
answered Jul 7 at 16:15
Gabe ConantGabe Conant
8467 silver badges16 bronze badges
edited Jul 7 at 17:10
answered Jul 7 at 16:15
Gabe ConantGabe Conant
8467 silver badges16 bronze badges
answered Jul 7 at 16:15
Gabe ConantGabe Conant
8467 silver badges16 bronze badges
answered Jul 7 at 16:15
Gabe ConantGabe Conant
8467 silver badges16 bronze badges
8467 silver badges16 bronze badges
add a comment |
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You really mean to replace $aleph_1$-saturated by $|mathcal L|^+$-saturated. Also, it suffices to consider just the forth direction. That is, the last sentence beginning "In addition" may be dropped.
$endgroup$
– Thomas Scanlon
Jul 7 at 18:15