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Are there exist any aleph-one categorical theories which are not strongly minimal?
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Is every aleph-one categorical theory strongly minimal?Are there exist any aleph-one categorical theories which are not strongly minimal?
logic model-theory
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Is every aleph-one categorical theory strongly minimal?Are there exist any aleph-one categorical theories which are not strongly minimal?
logic model-theory
asked 9 hours ago
user297564
59629
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Is every aleph-one categorical theory strongly minimal?Are there exist any aleph-one categorical theories which are not strongly minimal?
logic model-theory
asked 9 hours ago
user297564
59629
Is every aleph-one categorical theory strongly minimal?Are there exist any aleph-one categorical theories which are not strongly minimal?
logic model-theory
logic model-theory
asked 9 hours ago
user297564
59629
asked 9 hours ago
user297564
59629
asked 9 hours ago
user297564
59629
asked 9 hours ago
user297564
59629
asked 9 hours ago
user297564
59629
59629
add a comment |Â
add a comment |Â
2 Answers
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up vote
6
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accepted
Consider the theory of two equal-cardinality infinite sets:
The language has two unary predicates $U$ and $V$ and a binary relation $E$.
The theory says that $U$ and $V$ partition the universe, that $E$ defines a bijection between $U$ and $V$, and that the universe is infinite.
This is in fact totally categorical, but not strongly minimal - indeed, every model has infinite coinfinite definable sets!
Similarly, you can "glue a pure set" to any uncountably categorical theory to completely break minimality.
answered 9 hours ago
Noah Schweber
115k9143270
Cool example! :-)
– Asaf Karagila♦
8 hours ago
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up vote
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Many examples of $aleph_1$-categorical theories which are not strongly minimal are still almost strongly minimal. That is, there is formula $varphi(x)$ (possibly with parameters satisfying an isolated type over $emptyset$, so that a realization exists in every model) such that in any model $Mmodels T$, $X = varphi(M)$ is a strongly minimal set and $M = textacl(X)$. Intuitively, the elements of an arbitrary $M$ can be coordinatized by elements of a strongly minimal subset of that model. For a reference, see Section 4.7 of Model Theory by Hodges.
The example in Noah's answer is almost strongly minimal (with either $U$ or $V$ serving as the strongly minimal set). And many other examples with a similar flavor (e.g. the theory of two vector spaces with an isomorphism between them) are almost strongly minimal.
For some more "natural mathematical examples", consider the following:
Zilber showed that if $G$ is an infinite simple algebraic group over an algebraically closed field, then $textTh(G)$ is almost strongly minimal. So for example the theory of $textSL_2(mathbbC)$ is $aleph_1$-categorical and almost strongly minimal but not strongly minimal.
Let $K$ be an algebraically closed field. Consider the structure consisting of a set $P$ containing the points of $K^2$, a set $L$ containing the lines in $K^2$, and a relation $Isubseteq Ptimes L$ such that $I(p,l)$ if and only if point $p$ is on line $l$. This structure is $aleph_1$-categorical and almost strongly minimal but not strongly minimal.
Examples which are not almost strongly minimal are a little harder to find. Maybe the most natural is the theory of the abelian group $(mathbbZ/4mathbbZ)^omega$ (feel free to replace $4$ by $p^2$ for any prime $p$). The set picked out by $x+x=0$ is strongly minimal (the induced structure on it is that of an $mathbbF_2$-vector space), but the rest of the structure is not algebraic over it.
answered 7 hours ago
Alex Kruckman
24.6k22455
This is great! (Although I don't think you've shown that $(mathbbZ/4mathbbZ)^omega$ is not almost strongly minimal, just that $x+x=0$ doesn't witness that.)
– Noah Schweber
7 hours ago
@NoahSchweber That's right, I haven't. It turns out that this subgroup (and the set you get by removing $0$) are the only two strongly minimal sets definable without parameters. And then with parameters you only get cosets of this subgroup, plus or minus finitely many points.
– Alex Kruckman
6 hours ago
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
down vote
accepted
Consider the theory of two equal-cardinality infinite sets:
The language has two unary predicates $U$ and $V$ and a binary relation $E$.
The theory says that $U$ and $V$ partition the universe, that $E$ defines a bijection between $U$ and $V$, and that the universe is infinite.
This is in fact totally categorical, but not strongly minimal - indeed, every model has infinite coinfinite definable sets!
Similarly, you can "glue a pure set" to any uncountably categorical theory to completely break minimality.
answered 9 hours ago
Noah Schweber
115k9143270
Cool example! :-)
– Asaf Karagila♦
8 hours ago
add a comment |Â
up vote
6
down vote
accepted
Consider the theory of two equal-cardinality infinite sets:
The language has two unary predicates $U$ and $V$ and a binary relation $E$.
The theory says that $U$ and $V$ partition the universe, that $E$ defines a bijection between $U$ and $V$, and that the universe is infinite.
This is in fact totally categorical, but not strongly minimal - indeed, every model has infinite coinfinite definable sets!
Similarly, you can "glue a pure set" to any uncountably categorical theory to completely break minimality.
answered 9 hours ago
Noah Schweber
115k9143270
Cool example! :-)
– Asaf Karagila♦
8 hours ago
add a comment |Â
up vote
6
down vote
accepted
up vote
6
down vote
accepted
Consider the theory of two equal-cardinality infinite sets:
The language has two unary predicates $U$ and $V$ and a binary relation $E$.
The theory says that $U$ and $V$ partition the universe, that $E$ defines a bijection between $U$ and $V$, and that the universe is infinite.
This is in fact totally categorical, but not strongly minimal - indeed, every model has infinite coinfinite definable sets!
Similarly, you can "glue a pure set" to any uncountably categorical theory to completely break minimality.
answered 9 hours ago
Noah Schweber
115k9143270
Consider the theory of two equal-cardinality infinite sets:
The language has two unary predicates $U$ and $V$ and a binary relation $E$.
The theory says that $U$ and $V$ partition the universe, that $E$ defines a bijection between $U$ and $V$, and that the universe is infinite.
This is in fact totally categorical, but not strongly minimal - indeed, every model has infinite coinfinite definable sets!
Similarly, you can "glue a pure set" to any uncountably categorical theory to completely break minimality.
answered 9 hours ago
Noah Schweber
115k9143270
answered 9 hours ago
Noah Schweber
115k9143270
answered 9 hours ago
Noah Schweber
115k9143270
answered 9 hours ago
Noah Schweber
115k9143270
115k9143270
Cool example! :-)
– Asaf Karagila♦
8 hours ago
add a comment |Â
Cool example! :-)
– Asaf Karagila♦
8 hours ago
Cool example! :-)
– Asaf Karagila♦
8 hours ago
Cool example! :-)
– Asaf Karagila♦
8 hours ago
add a comment |Â
up vote
4
down vote
Many examples of $aleph_1$-categorical theories which are not strongly minimal are still almost strongly minimal. That is, there is formula $varphi(x)$ (possibly with parameters satisfying an isolated type over $emptyset$, so that a realization exists in every model) such that in any model $Mmodels T$, $X = varphi(M)$ is a strongly minimal set and $M = textacl(X)$. Intuitively, the elements of an arbitrary $M$ can be coordinatized by elements of a strongly minimal subset of that model. For a reference, see Section 4.7 of Model Theory by Hodges.
The example in Noah's answer is almost strongly minimal (with either $U$ or $V$ serving as the strongly minimal set). And many other examples with a similar flavor (e.g. the theory of two vector spaces with an isomorphism between them) are almost strongly minimal.
For some more "natural mathematical examples", consider the following:
Zilber showed that if $G$ is an infinite simple algebraic group over an algebraically closed field, then $textTh(G)$ is almost strongly minimal. So for example the theory of $textSL_2(mathbbC)$ is $aleph_1$-categorical and almost strongly minimal but not strongly minimal.
Let $K$ be an algebraically closed field. Consider the structure consisting of a set $P$ containing the points of $K^2$, a set $L$ containing the lines in $K^2$, and a relation $Isubseteq Ptimes L$ such that $I(p,l)$ if and only if point $p$ is on line $l$. This structure is $aleph_1$-categorical and almost strongly minimal but not strongly minimal.
Examples which are not almost strongly minimal are a little harder to find. Maybe the most natural is the theory of the abelian group $(mathbbZ/4mathbbZ)^omega$ (feel free to replace $4$ by $p^2$ for any prime $p$). The set picked out by $x+x=0$ is strongly minimal (the induced structure on it is that of an $mathbbF_2$-vector space), but the rest of the structure is not algebraic over it.
answered 7 hours ago
Alex Kruckman
24.6k22455
This is great! (Although I don't think you've shown that $(mathbbZ/4mathbbZ)^omega$ is not almost strongly minimal, just that $x+x=0$ doesn't witness that.)
– Noah Schweber
7 hours ago
@NoahSchweber That's right, I haven't. It turns out that this subgroup (and the set you get by removing $0$) are the only two strongly minimal sets definable without parameters. And then with parameters you only get cosets of this subgroup, plus or minus finitely many points.
– Alex Kruckman
6 hours ago
add a comment |Â
up vote
4
down vote
Many examples of $aleph_1$-categorical theories which are not strongly minimal are still almost strongly minimal. That is, there is formula $varphi(x)$ (possibly with parameters satisfying an isolated type over $emptyset$, so that a realization exists in every model) such that in any model $Mmodels T$, $X = varphi(M)$ is a strongly minimal set and $M = textacl(X)$. Intuitively, the elements of an arbitrary $M$ can be coordinatized by elements of a strongly minimal subset of that model. For a reference, see Section 4.7 of Model Theory by Hodges.
The example in Noah's answer is almost strongly minimal (with either $U$ or $V$ serving as the strongly minimal set). And many other examples with a similar flavor (e.g. the theory of two vector spaces with an isomorphism between them) are almost strongly minimal.
For some more "natural mathematical examples", consider the following:
Zilber showed that if $G$ is an infinite simple algebraic group over an algebraically closed field, then $textTh(G)$ is almost strongly minimal. So for example the theory of $textSL_2(mathbbC)$ is $aleph_1$-categorical and almost strongly minimal but not strongly minimal.
Let $K$ be an algebraically closed field. Consider the structure consisting of a set $P$ containing the points of $K^2$, a set $L$ containing the lines in $K^2$, and a relation $Isubseteq Ptimes L$ such that $I(p,l)$ if and only if point $p$ is on line $l$. This structure is $aleph_1$-categorical and almost strongly minimal but not strongly minimal.
Examples which are not almost strongly minimal are a little harder to find. Maybe the most natural is the theory of the abelian group $(mathbbZ/4mathbbZ)^omega$ (feel free to replace $4$ by $p^2$ for any prime $p$). The set picked out by $x+x=0$ is strongly minimal (the induced structure on it is that of an $mathbbF_2$-vector space), but the rest of the structure is not algebraic over it.
answered 7 hours ago
Alex Kruckman
24.6k22455
This is great! (Although I don't think you've shown that $(mathbbZ/4mathbbZ)^omega$ is not almost strongly minimal, just that $x+x=0$ doesn't witness that.)
– Noah Schweber
7 hours ago
@NoahSchweber That's right, I haven't. It turns out that this subgroup (and the set you get by removing $0$) are the only two strongly minimal sets definable without parameters. And then with parameters you only get cosets of this subgroup, plus or minus finitely many points.
– Alex Kruckman
6 hours ago
add a comment |Â
up vote
4
down vote
up vote
4
down vote
Many examples of $aleph_1$-categorical theories which are not strongly minimal are still almost strongly minimal. That is, there is formula $varphi(x)$ (possibly with parameters satisfying an isolated type over $emptyset$, so that a realization exists in every model) such that in any model $Mmodels T$, $X = varphi(M)$ is a strongly minimal set and $M = textacl(X)$. Intuitively, the elements of an arbitrary $M$ can be coordinatized by elements of a strongly minimal subset of that model. For a reference, see Section 4.7 of Model Theory by Hodges.
The example in Noah's answer is almost strongly minimal (with either $U$ or $V$ serving as the strongly minimal set). And many other examples with a similar flavor (e.g. the theory of two vector spaces with an isomorphism between them) are almost strongly minimal.
For some more "natural mathematical examples", consider the following:
Zilber showed that if $G$ is an infinite simple algebraic group over an algebraically closed field, then $textTh(G)$ is almost strongly minimal. So for example the theory of $textSL_2(mathbbC)$ is $aleph_1$-categorical and almost strongly minimal but not strongly minimal.
Let $K$ be an algebraically closed field. Consider the structure consisting of a set $P$ containing the points of $K^2$, a set $L$ containing the lines in $K^2$, and a relation $Isubseteq Ptimes L$ such that $I(p,l)$ if and only if point $p$ is on line $l$. This structure is $aleph_1$-categorical and almost strongly minimal but not strongly minimal.
Examples which are not almost strongly minimal are a little harder to find. Maybe the most natural is the theory of the abelian group $(mathbbZ/4mathbbZ)^omega$ (feel free to replace $4$ by $p^2$ for any prime $p$). The set picked out by $x+x=0$ is strongly minimal (the induced structure on it is that of an $mathbbF_2$-vector space), but the rest of the structure is not algebraic over it.
answered 7 hours ago
Alex Kruckman
24.6k22455
Many examples of $aleph_1$-categorical theories which are not strongly minimal are still almost strongly minimal. That is, there is formula $varphi(x)$ (possibly with parameters satisfying an isolated type over $emptyset$, so that a realization exists in every model) such that in any model $Mmodels T$, $X = varphi(M)$ is a strongly minimal set and $M = textacl(X)$. Intuitively, the elements of an arbitrary $M$ can be coordinatized by elements of a strongly minimal subset of that model. For a reference, see Section 4.7 of Model Theory by Hodges.
The example in Noah's answer is almost strongly minimal (with either $U$ or $V$ serving as the strongly minimal set). And many other examples with a similar flavor (e.g. the theory of two vector spaces with an isomorphism between them) are almost strongly minimal.
For some more "natural mathematical examples", consider the following:
Zilber showed that if $G$ is an infinite simple algebraic group over an algebraically closed field, then $textTh(G)$ is almost strongly minimal. So for example the theory of $textSL_2(mathbbC)$ is $aleph_1$-categorical and almost strongly minimal but not strongly minimal.
Let $K$ be an algebraically closed field. Consider the structure consisting of a set $P$ containing the points of $K^2$, a set $L$ containing the lines in $K^2$, and a relation $Isubseteq Ptimes L$ such that $I(p,l)$ if and only if point $p$ is on line $l$. This structure is $aleph_1$-categorical and almost strongly minimal but not strongly minimal.
Examples which are not almost strongly minimal are a little harder to find. Maybe the most natural is the theory of the abelian group $(mathbbZ/4mathbbZ)^omega$ (feel free to replace $4$ by $p^2$ for any prime $p$). The set picked out by $x+x=0$ is strongly minimal (the induced structure on it is that of an $mathbbF_2$-vector space), but the rest of the structure is not algebraic over it.
answered 7 hours ago
Alex Kruckman
24.6k22455
answered 7 hours ago
Alex Kruckman
24.6k22455
answered 7 hours ago
Alex Kruckman
24.6k22455
answered 7 hours ago
Alex Kruckman
24.6k22455
24.6k22455
This is great! (Although I don't think you've shown that $(mathbbZ/4mathbbZ)^omega$ is not almost strongly minimal, just that $x+x=0$ doesn't witness that.)
– Noah Schweber
7 hours ago
@NoahSchweber That's right, I haven't. It turns out that this subgroup (and the set you get by removing $0$) are the only two strongly minimal sets definable without parameters. And then with parameters you only get cosets of this subgroup, plus or minus finitely many points.
– Alex Kruckman
6 hours ago
add a comment |Â
This is great! (Although I don't think you've shown that $(mathbbZ/4mathbbZ)^omega$ is not almost strongly minimal, just that $x+x=0$ doesn't witness that.)
– Noah Schweber
7 hours ago
@NoahSchweber That's right, I haven't. It turns out that this subgroup (and the set you get by removing $0$) are the only two strongly minimal sets definable without parameters. And then with parameters you only get cosets of this subgroup, plus or minus finitely many points.
– Alex Kruckman
6 hours ago
This is great! (Although I don't think you've shown that $(mathbbZ/4mathbbZ)^omega$ is not almost strongly minimal, just that $x+x=0$ doesn't witness that.)
– Noah Schweber
7 hours ago
This is great! (Although I don't think you've shown that $(mathbbZ/4mathbbZ)^omega$ is not almost strongly minimal, just that $x+x=0$ doesn't witness that.)
– Noah Schweber
7 hours ago
@NoahSchweber That's right, I haven't. It turns out that this subgroup (and the set you get by removing $0$) are the only two strongly minimal sets definable without parameters. And then with parameters you only get cosets of this subgroup, plus or minus finitely many points.
– Alex Kruckman
6 hours ago
@NoahSchweber That's right, I haven't. It turns out that this subgroup (and the set you get by removing $0$) are the only two strongly minimal sets definable without parameters. And then with parameters you only get cosets of this subgroup, plus or minus finitely many points.
– Alex Kruckman
6 hours ago
add a comment |Â
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