how a theory can be categorical of a large cardinality?
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ehrenfeucht mostowski models eliminate types,also we can find saturated models of arbitrary large cardinalities, so I got confused how a theory can be categorical of that large cardinality?
logic first-order-logic model-theory
edited Dec 23 '18 at 8:54
Asaf Karagila♦
306k33437768
asked Dec 23 '18 at 7:42
user297564user297564
69039
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add a comment |
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ehrenfeucht mostowski models eliminate types,also we can find saturated models of arbitrary large cardinalities, so I got confused how a theory can be categorical of that large cardinality?
logic first-order-logic model-theory
edited Dec 23 '18 at 8:54
Asaf Karagila♦
306k33437768
asked Dec 23 '18 at 7:42
user297564user297564
69039
$endgroup$
add a comment |
$begingroup$
ehrenfeucht mostowski models eliminate types,also we can find saturated models of arbitrary large cardinalities, so I got confused how a theory can be categorical of that large cardinality?
logic first-order-logic model-theory
edited Dec 23 '18 at 8:54
Asaf Karagila♦
306k33437768
asked Dec 23 '18 at 7:42
user297564user297564
69039
$endgroup$
ehrenfeucht mostowski models eliminate types,also we can find saturated models of arbitrary large cardinalities, so I got confused how a theory can be categorical of that large cardinality?
logic first-order-logic model-theory
logic first-order-logic model-theory
edited Dec 23 '18 at 8:54
Asaf Karagila♦
306k33437768
asked Dec 23 '18 at 7:42
user297564user297564
69039
edited Dec 23 '18 at 8:54
Asaf Karagila♦
306k33437768
asked Dec 23 '18 at 7:42
user297564user297564
69039
edited Dec 23 '18 at 8:54
Asaf Karagila♦
306k33437768
edited Dec 23 '18 at 8:54
Asaf Karagila♦
306k33437768
edited Dec 23 '18 at 8:54
Asaf Karagila♦
306k33437768
306k33437768
asked Dec 23 '18 at 7:42
user297564user297564
69039
asked Dec 23 '18 at 7:42
user297564user297564
69039
asked Dec 23 '18 at 7:42
user297564user297564
69039
69039
add a comment |
add a comment |
2 Answers
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If all the types are principal, then you cannot omit them.
Consider the theory of a complete graph. This is categorical in every cardinality, because any two complete graphs with the same number of nodes are isomorphic.
There are many other such kind of theories. The theory of $=$ only. The theory of many disjoint copies of some finite graph. etc.
answered Dec 23 '18 at 13:24
JDHJDH
32.7k680146
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add a comment |
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Joel's answer suggests that uncountable categoricity can only happen when all types are principal. But this might be a bit misleading: given a countable theory $T$ (complete with infinite models), every type is principal relative to $T$ if and only if $T$ is $aleph_0$-categorical.
There are examples of theories (like the ones in Joel's answer) which are totally categorical, i.e. categorical in every infinite cardinality. But there are also many examples of uncountably categorical theories which are not $aleph_0$-categorical, such as the theory of algebraically closed fields, or the theory of $mathbb{Q}$-vector spaces.
When you say "Ehrenfeucht-Mostowski models eliminate types", I guess you're referring to the theorem that if $M$ is an Ehrenfeucht-Mostowski model of $T$ built on an indiscernible sequence $(a_i)_{iin I}$ where $(I,<)$ is well-ordered, then $M$ realizes only countably many types over any countable set.
This might seem like a problem if $T$ is $kappa$-categorical for some uncountable $kappa$, since given any uncountable set of types, we can realize them all in a model of size $kappa$, and hence in the Ehrenfeucht-Mostowski model $M$ built on an indiscernible sequence of order-type $kappa$.
But all we've done is proven that $T$ is $omega$-stable: for any countable subset $A$ of any model of $T$, there are only countably many types over $A$. This is the key first step in the proof of Morley's theorem on uncountably categorical theories.
answered Dec 24 '18 at 5:38
Alex KruckmanAlex Kruckman
27.7k32658
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2 Answers
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2 Answers
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$begingroup$
If all the types are principal, then you cannot omit them.
Consider the theory of a complete graph. This is categorical in every cardinality, because any two complete graphs with the same number of nodes are isomorphic.
There are many other such kind of theories. The theory of $=$ only. The theory of many disjoint copies of some finite graph. etc.
answered Dec 23 '18 at 13:24
JDHJDH
32.7k680146
$endgroup$
add a comment |
$begingroup$
If all the types are principal, then you cannot omit them.
Consider the theory of a complete graph. This is categorical in every cardinality, because any two complete graphs with the same number of nodes are isomorphic.
There are many other such kind of theories. The theory of $=$ only. The theory of many disjoint copies of some finite graph. etc.
answered Dec 23 '18 at 13:24
JDHJDH
32.7k680146
$endgroup$
add a comment |
$begingroup$
If all the types are principal, then you cannot omit them.
Consider the theory of a complete graph. This is categorical in every cardinality, because any two complete graphs with the same number of nodes are isomorphic.
There are many other such kind of theories. The theory of $=$ only. The theory of many disjoint copies of some finite graph. etc.
answered Dec 23 '18 at 13:24
JDHJDH
32.7k680146
$endgroup$
If all the types are principal, then you cannot omit them.
Consider the theory of a complete graph. This is categorical in every cardinality, because any two complete graphs with the same number of nodes are isomorphic.
There are many other such kind of theories. The theory of $=$ only. The theory of many disjoint copies of some finite graph. etc.
answered Dec 23 '18 at 13:24
JDHJDH
32.7k680146
answered Dec 23 '18 at 13:24
JDHJDH
32.7k680146
answered Dec 23 '18 at 13:24
JDHJDH
32.7k680146
answered Dec 23 '18 at 13:24
JDHJDH
32.7k680146
32.7k680146
add a comment |
add a comment |
$begingroup$
Joel's answer suggests that uncountable categoricity can only happen when all types are principal. But this might be a bit misleading: given a countable theory $T$ (complete with infinite models), every type is principal relative to $T$ if and only if $T$ is $aleph_0$-categorical.
There are examples of theories (like the ones in Joel's answer) which are totally categorical, i.e. categorical in every infinite cardinality. But there are also many examples of uncountably categorical theories which are not $aleph_0$-categorical, such as the theory of algebraically closed fields, or the theory of $mathbb{Q}$-vector spaces.
When you say "Ehrenfeucht-Mostowski models eliminate types", I guess you're referring to the theorem that if $M$ is an Ehrenfeucht-Mostowski model of $T$ built on an indiscernible sequence $(a_i)_{iin I}$ where $(I,<)$ is well-ordered, then $M$ realizes only countably many types over any countable set.
This might seem like a problem if $T$ is $kappa$-categorical for some uncountable $kappa$, since given any uncountable set of types, we can realize them all in a model of size $kappa$, and hence in the Ehrenfeucht-Mostowski model $M$ built on an indiscernible sequence of order-type $kappa$.
But all we've done is proven that $T$ is $omega$-stable: for any countable subset $A$ of any model of $T$, there are only countably many types over $A$. This is the key first step in the proof of Morley's theorem on uncountably categorical theories.
answered Dec 24 '18 at 5:38
Alex KruckmanAlex Kruckman
27.7k32658
$endgroup$
add a comment |
$begingroup$
Joel's answer suggests that uncountable categoricity can only happen when all types are principal. But this might be a bit misleading: given a countable theory $T$ (complete with infinite models), every type is principal relative to $T$ if and only if $T$ is $aleph_0$-categorical.
There are examples of theories (like the ones in Joel's answer) which are totally categorical, i.e. categorical in every infinite cardinality. But there are also many examples of uncountably categorical theories which are not $aleph_0$-categorical, such as the theory of algebraically closed fields, or the theory of $mathbb{Q}$-vector spaces.
When you say "Ehrenfeucht-Mostowski models eliminate types", I guess you're referring to the theorem that if $M$ is an Ehrenfeucht-Mostowski model of $T$ built on an indiscernible sequence $(a_i)_{iin I}$ where $(I,<)$ is well-ordered, then $M$ realizes only countably many types over any countable set.
This might seem like a problem if $T$ is $kappa$-categorical for some uncountable $kappa$, since given any uncountable set of types, we can realize them all in a model of size $kappa$, and hence in the Ehrenfeucht-Mostowski model $M$ built on an indiscernible sequence of order-type $kappa$.
But all we've done is proven that $T$ is $omega$-stable: for any countable subset $A$ of any model of $T$, there are only countably many types over $A$. This is the key first step in the proof of Morley's theorem on uncountably categorical theories.
answered Dec 24 '18 at 5:38
Alex KruckmanAlex Kruckman
27.7k32658
$endgroup$
add a comment |
$begingroup$
Joel's answer suggests that uncountable categoricity can only happen when all types are principal. But this might be a bit misleading: given a countable theory $T$ (complete with infinite models), every type is principal relative to $T$ if and only if $T$ is $aleph_0$-categorical.
There are examples of theories (like the ones in Joel's answer) which are totally categorical, i.e. categorical in every infinite cardinality. But there are also many examples of uncountably categorical theories which are not $aleph_0$-categorical, such as the theory of algebraically closed fields, or the theory of $mathbb{Q}$-vector spaces.
When you say "Ehrenfeucht-Mostowski models eliminate types", I guess you're referring to the theorem that if $M$ is an Ehrenfeucht-Mostowski model of $T$ built on an indiscernible sequence $(a_i)_{iin I}$ where $(I,<)$ is well-ordered, then $M$ realizes only countably many types over any countable set.
This might seem like a problem if $T$ is $kappa$-categorical for some uncountable $kappa$, since given any uncountable set of types, we can realize them all in a model of size $kappa$, and hence in the Ehrenfeucht-Mostowski model $M$ built on an indiscernible sequence of order-type $kappa$.
But all we've done is proven that $T$ is $omega$-stable: for any countable subset $A$ of any model of $T$, there are only countably many types over $A$. This is the key first step in the proof of Morley's theorem on uncountably categorical theories.
answered Dec 24 '18 at 5:38
Alex KruckmanAlex Kruckman
27.7k32658
$endgroup$
Joel's answer suggests that uncountable categoricity can only happen when all types are principal. But this might be a bit misleading: given a countable theory $T$ (complete with infinite models), every type is principal relative to $T$ if and only if $T$ is $aleph_0$-categorical.
There are examples of theories (like the ones in Joel's answer) which are totally categorical, i.e. categorical in every infinite cardinality. But there are also many examples of uncountably categorical theories which are not $aleph_0$-categorical, such as the theory of algebraically closed fields, or the theory of $mathbb{Q}$-vector spaces.
When you say "Ehrenfeucht-Mostowski models eliminate types", I guess you're referring to the theorem that if $M$ is an Ehrenfeucht-Mostowski model of $T$ built on an indiscernible sequence $(a_i)_{iin I}$ where $(I,<)$ is well-ordered, then $M$ realizes only countably many types over any countable set.
This might seem like a problem if $T$ is $kappa$-categorical for some uncountable $kappa$, since given any uncountable set of types, we can realize them all in a model of size $kappa$, and hence in the Ehrenfeucht-Mostowski model $M$ built on an indiscernible sequence of order-type $kappa$.
But all we've done is proven that $T$ is $omega$-stable: for any countable subset $A$ of any model of $T$, there are only countably many types over $A$. This is the key first step in the proof of Morley's theorem on uncountably categorical theories.
answered Dec 24 '18 at 5:38
Alex KruckmanAlex Kruckman
27.7k32658
edited Dec 24 '18 at 5:47
answered Dec 24 '18 at 5:38
Alex KruckmanAlex Kruckman
27.7k32658
answered Dec 24 '18 at 5:38
Alex KruckmanAlex Kruckman
27.7k32658
answered Dec 24 '18 at 5:38
Alex KruckmanAlex Kruckman
27.7k32658
27.7k32658
add a comment |
add a comment |
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