What does the word “extend” mean in the context of model theory?
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Consider the following two problems:
(1) Let $L={E}$ be a language consisting one binary relation symbol. Let $T$ be the $L$-theory saying that $E$ is an equivalence relation with infinitely many classes. Prove that there are infinitely many inequivalent complete theories extending $T$.
(2) Prove that an ultrafilter on an infinite set is non-principal if and only if it extends to the Frechet filter.
In both problems, I don't understand what the word "extend" means.
Can someone explain to me? Thanks!
terminology first-order-logic model-theory filters
edited yesterday
bof
48.7k450115
asked yesterday
bbw
36917
add a comment |
up vote
1
down vote
favorite
Consider the following two problems:
(1) Let $L={E}$ be a language consisting one binary relation symbol. Let $T$ be the $L$-theory saying that $E$ is an equivalence relation with infinitely many classes. Prove that there are infinitely many inequivalent complete theories extending $T$.
(2) Prove that an ultrafilter on an infinite set is non-principal if and only if it extends to the Frechet filter.
In both problems, I don't understand what the word "extend" means.
Can someone explain to me? Thanks!
terminology first-order-logic model-theory filters
edited yesterday
bof
48.7k450115
asked yesterday
bbw
36917
@MauroALLEGRANZA I understand what it means to extend a structure or a language. But what does it mean by extending a theory? And what does it mean by extending an filter to the frechet filter?
– bbw
yesterday
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Consider the following two problems:
(1) Let $L={E}$ be a language consisting one binary relation symbol. Let $T$ be the $L$-theory saying that $E$ is an equivalence relation with infinitely many classes. Prove that there are infinitely many inequivalent complete theories extending $T$.
(2) Prove that an ultrafilter on an infinite set is non-principal if and only if it extends to the Frechet filter.
In both problems, I don't understand what the word "extend" means.
Can someone explain to me? Thanks!
terminology first-order-logic model-theory filters
edited yesterday
bof
48.7k450115
asked yesterday
bbw
36917
Consider the following two problems:
(1) Let $L={E}$ be a language consisting one binary relation symbol. Let $T$ be the $L$-theory saying that $E$ is an equivalence relation with infinitely many classes. Prove that there are infinitely many inequivalent complete theories extending $T$.
(2) Prove that an ultrafilter on an infinite set is non-principal if and only if it extends to the Frechet filter.
In both problems, I don't understand what the word "extend" means.
Can someone explain to me? Thanks!
terminology first-order-logic model-theory filters
terminology first-order-logic model-theory filters
edited yesterday
bof
48.7k450115
asked yesterday
bbw
36917
edited yesterday
bof
48.7k450115
asked yesterday
bbw
36917
edited yesterday
bof
48.7k450115
edited yesterday
bof
48.7k450115
edited yesterday
bof
48.7k450115
48.7k450115
asked yesterday
bbw
36917
asked yesterday
bbw
36917
asked yesterday
bbw
36917
36917
@MauroALLEGRANZA I understand what it means to extend a structure or a language. But what does it mean by extending a theory? And what does it mean by extending an filter to the frechet filter?
– bbw
yesterday
add a comment |
@MauroALLEGRANZA I understand what it means to extend a structure or a language. But what does it mean by extending a theory? And what does it mean by extending an filter to the frechet filter?
– bbw
yesterday
@MauroALLEGRANZA I understand what it means to extend a structure or a language. But what does it mean by extending a theory? And what does it mean by extending an filter to the frechet filter?
– bbw
yesterday
@MauroALLEGRANZA I understand what it means to extend a structure or a language. But what does it mean by extending a theory? And what does it mean by extending an filter to the frechet filter?
– bbw
yesterday
add a comment |
1 Answer
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"Extend" just means "be a superset of" in this context. So a theory $T'$ extends a theory $T$ if $Tsubseteq T'$, and a filter $F'$ extends a filter $F$ if $Fsubseteq F'$.
(The phrasing "extends to" in statement (2) is an error and should be just "extends". Indeed, saying "$F$ extends to $F'$" would normally mean that $F'$ extends $F$, which is the opposite of the intended meaning here: it means to say that the ultrafilter extends the Frechet filter.)
answered yesterday
Eric Wofsey
174k12201326
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
"Extend" just means "be a superset of" in this context. So a theory $T'$ extends a theory $T$ if $Tsubseteq T'$, and a filter $F'$ extends a filter $F$ if $Fsubseteq F'$.
(The phrasing "extends to" in statement (2) is an error and should be just "extends". Indeed, saying "$F$ extends to $F'$" would normally mean that $F'$ extends $F$, which is the opposite of the intended meaning here: it means to say that the ultrafilter extends the Frechet filter.)
answered yesterday
Eric Wofsey
174k12201326
add a comment |
up vote
3
down vote
accepted
"Extend" just means "be a superset of" in this context. So a theory $T'$ extends a theory $T$ if $Tsubseteq T'$, and a filter $F'$ extends a filter $F$ if $Fsubseteq F'$.
(The phrasing "extends to" in statement (2) is an error and should be just "extends". Indeed, saying "$F$ extends to $F'$" would normally mean that $F'$ extends $F$, which is the opposite of the intended meaning here: it means to say that the ultrafilter extends the Frechet filter.)
answered yesterday
Eric Wofsey
174k12201326
add a comment |
up vote
3
down vote
accepted
up vote
3
down vote
accepted
"Extend" just means "be a superset of" in this context. So a theory $T'$ extends a theory $T$ if $Tsubseteq T'$, and a filter $F'$ extends a filter $F$ if $Fsubseteq F'$.
(The phrasing "extends to" in statement (2) is an error and should be just "extends". Indeed, saying "$F$ extends to $F'$" would normally mean that $F'$ extends $F$, which is the opposite of the intended meaning here: it means to say that the ultrafilter extends the Frechet filter.)
answered yesterday
Eric Wofsey
174k12201326
"Extend" just means "be a superset of" in this context. So a theory $T'$ extends a theory $T$ if $Tsubseteq T'$, and a filter $F'$ extends a filter $F$ if $Fsubseteq F'$.
(The phrasing "extends to" in statement (2) is an error and should be just "extends". Indeed, saying "$F$ extends to $F'$" would normally mean that $F'$ extends $F$, which is the opposite of the intended meaning here: it means to say that the ultrafilter extends the Frechet filter.)
answered yesterday
Eric Wofsey
174k12201326
answered yesterday
Eric Wofsey
174k12201326
answered yesterday
Eric Wofsey
174k12201326
answered yesterday
Eric Wofsey
174k12201326
174k12201326
add a comment |
add a comment |
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@MauroALLEGRANZA I understand what it means to extend a structure or a language. But what does it mean by extending a theory? And what does it mean by extending an filter to the frechet filter?
– bbw
yesterday