Is categoricity retained when reducing the language?Why does an $varepsilon$-incomplete $Q$ remain so in the forcing extension of any $varepsilon$-complete $P$?Are all complete finitely axiomatizable first order theories $aleph_0$-categorical?Vaught conjecture for uncountable languagesDoes this property of a first-order structure imply categoricity?Uncountably categorical theories which are interpretable in a strongly minimalWhere do uncountable models collapse to?Applications of Morley's Categoricity TheoremHow many elementary embeddings can there be?Natural theories for the failure of gap-1 transfer principleA weakening of cardinal compactness - is it equivalent?
Is categoricity retained when reducing the language?
Why does an $varepsilon$-incomplete $Q$ remain so in the forcing extension of any $varepsilon$-complete $P$?Are all complete finitely axiomatizable first order theories $aleph_0$-categorical?Vaught conjecture for uncountable languagesDoes this property of a first-order structure imply categoricity?Uncountably categorical theories which are interpretable in a strongly minimalWhere do uncountable models collapse to?Applications of Morley's Categoricity TheoremHow many elementary embeddings can there be?Natural theories for the failure of gap-1 transfer principleA weakening of cardinal compactness - is it equivalent?
$begingroup$
Suppose $mathcal L subseteq mathcal L’$ are first-order languages, $kappa$ is a cardinal, and $T’$ is a theory in $mathcal L’$ that is $kappa$-categorical. Let $T = T’ restriction mathcal L$. Is $T$ $kappa$-categorical?
If $|mathcal L’| = kappa = aleph_0$, then I can show the answer is yes using the Ryll-Nardzewski Theorem, which says that a countable theory is $aleph_0$-categorical iff for each $n$, the number of $n$-types is finite.
lo.logic model-theory
asked May 22 at 6:07
Monroe EskewMonroe Eskew
8,18132568
$endgroup$
add a comment |
$begingroup$
Suppose $mathcal L subseteq mathcal L’$ are first-order languages, $kappa$ is a cardinal, and $T’$ is a theory in $mathcal L’$ that is $kappa$-categorical. Let $T = T’ restriction mathcal L$. Is $T$ $kappa$-categorical?
If $|mathcal L’| = kappa = aleph_0$, then I can show the answer is yes using the Ryll-Nardzewski Theorem, which says that a countable theory is $aleph_0$-categorical iff for each $n$, the number of $n$-types is finite.
lo.logic model-theory
asked May 22 at 6:07
Monroe EskewMonroe Eskew
8,18132568
$endgroup$
1
$begingroup$
I like this question quite a lot, since many of the categorical theories I considered at first do not seem directly to be counterexamples.
$endgroup$
– Joel David Hamkins
May 22 at 9:16
add a comment |
$begingroup$
Suppose $mathcal L subseteq mathcal L’$ are first-order languages, $kappa$ is a cardinal, and $T’$ is a theory in $mathcal L’$ that is $kappa$-categorical. Let $T = T’ restriction mathcal L$. Is $T$ $kappa$-categorical?
If $|mathcal L’| = kappa = aleph_0$, then I can show the answer is yes using the Ryll-Nardzewski Theorem, which says that a countable theory is $aleph_0$-categorical iff for each $n$, the number of $n$-types is finite.
lo.logic model-theory
asked May 22 at 6:07
Monroe EskewMonroe Eskew
8,18132568
$endgroup$
Suppose $mathcal L subseteq mathcal L’$ are first-order languages, $kappa$ is a cardinal, and $T’$ is a theory in $mathcal L’$ that is $kappa$-categorical. Let $T = T’ restriction mathcal L$. Is $T$ $kappa$-categorical?
If $|mathcal L’| = kappa = aleph_0$, then I can show the answer is yes using the Ryll-Nardzewski Theorem, which says that a countable theory is $aleph_0$-categorical iff for each $n$, the number of $n$-types is finite.
lo.logic model-theory
lo.logic model-theory
asked May 22 at 6:07
Monroe EskewMonroe Eskew
8,18132568
asked May 22 at 6:07
Monroe EskewMonroe Eskew
8,18132568
edited May 22 at 7:23
Monroe Eskew
asked May 22 at 6:07
Monroe EskewMonroe Eskew
8,18132568
asked May 22 at 6:07
Monroe EskewMonroe Eskew
8,18132568
asked May 22 at 6:07
Monroe EskewMonroe Eskew
8,18132568
8,18132568
1
$begingroup$
I like this question quite a lot, since many of the categorical theories I considered at first do not seem directly to be counterexamples.
$endgroup$
– Joel David Hamkins
May 22 at 9:16
add a comment |
1
$begingroup$
I like this question quite a lot, since many of the categorical theories I considered at first do not seem directly to be counterexamples.
$endgroup$
– Joel David Hamkins
May 22 at 9:16
1
1
$begingroup$
I like this question quite a lot, since many of the categorical theories I considered at first do not seem directly to be counterexamples.
$endgroup$
– Joel David Hamkins
May 22 at 9:16
$begingroup$
I like this question quite a lot, since many of the categorical theories I considered at first do not seem directly to be counterexamples.
$endgroup$
– Joel David Hamkins
May 22 at 9:16
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The answer is no, one can lose categoricity in a reduct of a theory. Consider the following example.
Consider the theory $T$ describing a bijection between two disjoint infinite
predicates $f:Ato B$. So a model consists of two disjoint parts, the
$A$-part and the $B$-part, and a bijection $f$ between them. The
language is $f,A,B$.
This theory is categorical in every cardinality. But if we restrict
the theory to its consequences in the language with the two predicates $A,B$
and without the bijection, then it is just the theory of two disjoint infinite predicates, which is no longer categorical in
uncountable powers, since one predicate could have a different
cardinality than the other.
edited May 22 at 7:24
Monroe Eskew
8,18132568
answered May 22 at 7:18
Joel David HamkinsJoel David Hamkins
166k27511888
$endgroup$
add a comment |
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1 Answer
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$begingroup$
The answer is no, one can lose categoricity in a reduct of a theory. Consider the following example.
Consider the theory $T$ describing a bijection between two disjoint infinite
predicates $f:Ato B$. So a model consists of two disjoint parts, the
$A$-part and the $B$-part, and a bijection $f$ between them. The
language is $f,A,B$.
This theory is categorical in every cardinality. But if we restrict
the theory to its consequences in the language with the two predicates $A,B$
and without the bijection, then it is just the theory of two disjoint infinite predicates, which is no longer categorical in
uncountable powers, since one predicate could have a different
cardinality than the other.
edited May 22 at 7:24
Monroe Eskew
8,18132568
answered May 22 at 7:18
Joel David HamkinsJoel David Hamkins
166k27511888
$endgroup$
add a comment |
$begingroup$
The answer is no, one can lose categoricity in a reduct of a theory. Consider the following example.
Consider the theory $T$ describing a bijection between two disjoint infinite
predicates $f:Ato B$. So a model consists of two disjoint parts, the
$A$-part and the $B$-part, and a bijection $f$ between them. The
language is $f,A,B$.
This theory is categorical in every cardinality. But if we restrict
the theory to its consequences in the language with the two predicates $A,B$
and without the bijection, then it is just the theory of two disjoint infinite predicates, which is no longer categorical in
uncountable powers, since one predicate could have a different
cardinality than the other.
edited May 22 at 7:24
Monroe Eskew
8,18132568
answered May 22 at 7:18
Joel David HamkinsJoel David Hamkins
166k27511888
$endgroup$
add a comment |
$begingroup$
The answer is no, one can lose categoricity in a reduct of a theory. Consider the following example.
Consider the theory $T$ describing a bijection between two disjoint infinite
predicates $f:Ato B$. So a model consists of two disjoint parts, the
$A$-part and the $B$-part, and a bijection $f$ between them. The
language is $f,A,B$.
This theory is categorical in every cardinality. But if we restrict
the theory to its consequences in the language with the two predicates $A,B$
and without the bijection, then it is just the theory of two disjoint infinite predicates, which is no longer categorical in
uncountable powers, since one predicate could have a different
cardinality than the other.
edited May 22 at 7:24
Monroe Eskew
8,18132568
answered May 22 at 7:18
Joel David HamkinsJoel David Hamkins
166k27511888
$endgroup$
The answer is no, one can lose categoricity in a reduct of a theory. Consider the following example.
Consider the theory $T$ describing a bijection between two disjoint infinite
predicates $f:Ato B$. So a model consists of two disjoint parts, the
$A$-part and the $B$-part, and a bijection $f$ between them. The
language is $f,A,B$.
This theory is categorical in every cardinality. But if we restrict
the theory to its consequences in the language with the two predicates $A,B$
and without the bijection, then it is just the theory of two disjoint infinite predicates, which is no longer categorical in
uncountable powers, since one predicate could have a different
cardinality than the other.
edited May 22 at 7:24
Monroe Eskew
8,18132568
answered May 22 at 7:18
Joel David HamkinsJoel David Hamkins
166k27511888
edited May 22 at 7:24
Monroe Eskew
8,18132568
edited May 22 at 7:24
Monroe Eskew
8,18132568
edited May 22 at 7:24
Monroe Eskew
8,18132568
8,18132568
answered May 22 at 7:18
Joel David HamkinsJoel David Hamkins
166k27511888
answered May 22 at 7:18
Joel David HamkinsJoel David Hamkins
166k27511888
answered May 22 at 7:18
Joel David HamkinsJoel David Hamkins
166k27511888
166k27511888
add a comment |
add a comment |
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$begingroup$
I like this question quite a lot, since many of the categorical theories I considered at first do not seem directly to be counterexamples.
$endgroup$
– Joel David Hamkins
May 22 at 9:16