A theory $T$ is model-complete if the union of $T$ with an atomic diagram is complete
$begingroup$
Let $T$ be a theory in first order logic over some language $L$. Let $mathfrak A$ be some structure over $L$ with $mathfrak A models T$ and with $A$ be its universe. Then consider every $a in A$ as a constant and look at the enriched language $L(A) = L cup A$ with the $L(A)$-structure $mathfrak A_A = (mathfrak A, a)_{ain A}$. A formula over $L(A)$ is called basic if it is an atomic formula. The set
$$
operatorname{Diag}(mathfrak A) = { varphi mbox{ is a basic $L(A)$-sentence } mid mathfrak A_A models varphi }
$$
is called the atomic diagram of $mathfrak A$.
A theory $T$ is called model-complete if every substructure relation between two models is actually an elementary embedding.
Then $T$ is model-complete if and only if for any $mathfrak A models T$ the theory $T cup operatorname{Diag}(mathfrak A)$ is complete.
These definitions are from A course in model theory by K.Tent/M.Zeigler.
I do not understand the quoted statement. For let $t_1 = t_2$ and $t_3 ne t_4$ be two atomic sentences for terms $t_1, t_2,t_3,t_4$ in $L(A)$. Then Set $varphi = (t_1 = t_2) land (t_3 ne t_4)$. Now suppose in the terms we have some constants from $A$. Then neither $varphi$ nor $neg varphi$ is in $T cup operatorname{Diag}(mathfrak A)$ as it is not in $operatorname{Diag}(mathfrak A)$ for it is not atomic, nor is it in $T$ as it is a statement over the enrichted language $L(A)$, but not over $L$. Could someone please explain the above statement (and what I oversee here...)?
logic first-order-logic model-theory
$endgroup$
add a comment |
$begingroup$
Let $T$ be a theory in first order logic over some language $L$. Let $mathfrak A$ be some structure over $L$ with $mathfrak A models T$ and with $A$ be its universe. Then consider every $a in A$ as a constant and look at the enriched language $L(A) = L cup A$ with the $L(A)$-structure $mathfrak A_A = (mathfrak A, a)_{ain A}$. A formula over $L(A)$ is called basic if it is an atomic formula. The set
$$
operatorname{Diag}(mathfrak A) = { varphi mbox{ is a basic $L(A)$-sentence } mid mathfrak A_A models varphi }
$$
is called the atomic diagram of $mathfrak A$.
A theory $T$ is called model-complete if every substructure relation between two models is actually an elementary embedding.
Then $T$ is model-complete if and only if for any $mathfrak A models T$ the theory $T cup operatorname{Diag}(mathfrak A)$ is complete.
These definitions are from A course in model theory by K.Tent/M.Zeigler.
I do not understand the quoted statement. For let $t_1 = t_2$ and $t_3 ne t_4$ be two atomic sentences for terms $t_1, t_2,t_3,t_4$ in $L(A)$. Then Set $varphi = (t_1 = t_2) land (t_3 ne t_4)$. Now suppose in the terms we have some constants from $A$. Then neither $varphi$ nor $neg varphi$ is in $T cup operatorname{Diag}(mathfrak A)$ as it is not in $operatorname{Diag}(mathfrak A)$ for it is not atomic, nor is it in $T$ as it is a statement over the enrichted language $L(A)$, but not over $L$. Could someone please explain the above statement (and what I oversee here...)?
logic first-order-logic model-theory
$endgroup$
add a comment |
$begingroup$
Let $T$ be a theory in first order logic over some language $L$. Let $mathfrak A$ be some structure over $L$ with $mathfrak A models T$ and with $A$ be its universe. Then consider every $a in A$ as a constant and look at the enriched language $L(A) = L cup A$ with the $L(A)$-structure $mathfrak A_A = (mathfrak A, a)_{ain A}$. A formula over $L(A)$ is called basic if it is an atomic formula. The set
$$
operatorname{Diag}(mathfrak A) = { varphi mbox{ is a basic $L(A)$-sentence } mid mathfrak A_A models varphi }
$$
is called the atomic diagram of $mathfrak A$.
A theory $T$ is called model-complete if every substructure relation between two models is actually an elementary embedding.
Then $T$ is model-complete if and only if for any $mathfrak A models T$ the theory $T cup operatorname{Diag}(mathfrak A)$ is complete.
These definitions are from A course in model theory by K.Tent/M.Zeigler.
I do not understand the quoted statement. For let $t_1 = t_2$ and $t_3 ne t_4$ be two atomic sentences for terms $t_1, t_2,t_3,t_4$ in $L(A)$. Then Set $varphi = (t_1 = t_2) land (t_3 ne t_4)$. Now suppose in the terms we have some constants from $A$. Then neither $varphi$ nor $neg varphi$ is in $T cup operatorname{Diag}(mathfrak A)$ as it is not in $operatorname{Diag}(mathfrak A)$ for it is not atomic, nor is it in $T$ as it is a statement over the enrichted language $L(A)$, but not over $L$. Could someone please explain the above statement (and what I oversee here...)?
logic first-order-logic model-theory
$endgroup$
Let $T$ be a theory in first order logic over some language $L$. Let $mathfrak A$ be some structure over $L$ with $mathfrak A models T$ and with $A$ be its universe. Then consider every $a in A$ as a constant and look at the enriched language $L(A) = L cup A$ with the $L(A)$-structure $mathfrak A_A = (mathfrak A, a)_{ain A}$. A formula over $L(A)$ is called basic if it is an atomic formula. The set
$$
operatorname{Diag}(mathfrak A) = { varphi mbox{ is a basic $L(A)$-sentence } mid mathfrak A_A models varphi }
$$
is called the atomic diagram of $mathfrak A$.
A theory $T$ is called model-complete if every substructure relation between two models is actually an elementary embedding.
Then $T$ is model-complete if and only if for any $mathfrak A models T$ the theory $T cup operatorname{Diag}(mathfrak A)$ is complete.
These definitions are from A course in model theory by K.Tent/M.Zeigler.
I do not understand the quoted statement. For let $t_1 = t_2$ and $t_3 ne t_4$ be two atomic sentences for terms $t_1, t_2,t_3,t_4$ in $L(A)$. Then Set $varphi = (t_1 = t_2) land (t_3 ne t_4)$. Now suppose in the terms we have some constants from $A$. Then neither $varphi$ nor $neg varphi$ is in $T cup operatorname{Diag}(mathfrak A)$ as it is not in $operatorname{Diag}(mathfrak A)$ for it is not atomic, nor is it in $T$ as it is a statement over the enrichted language $L(A)$, but not over $L$. Could someone please explain the above statement (and what I oversee here...)?
logic first-order-logic model-theory
logic first-order-logic model-theory
asked Dec 14 '18 at 17:34
StefanHStefanH
8,11152365
8,11152365
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Complete means for any sentence $varphi,$ either $Tvdash varphi$ or $Tvdash lnot varphi,$ not $varphiin T$ or $lnotvarphiin T.$
If $mathfrak A,mathfrak Bmodels T$ and $mathfrak Asubseteq mathfrak B,$ then both are models of $Tcup Diag(mathfrak A).$ If $Tcup Diag(mathfrak A)$ is complete, then they must agree on all $L(A)$-sentences, and hence the embedding is elementary. On the other hand if $Tcup Diag(mathfrak A)$ is not complete for some $mathfrak Amodels T,$ then we can find a $mathfrak Bmodels T$ with $mathfrak Asubseteq mathfrak B$ that differs from $mathfrak A$ on some $L(A)$-sentence (just let it be a model of the negation of some $varphi$ that is true in $mathfrak A$ but that $Tcup Diag(mathfrak A)$ does not decide). Hence this embedding is not elementary.
$endgroup$
$begingroup$
What makes you sure that such a model $mathfrak B$ of $T cup Diag(mathfrak A) cup {neg varphi}$ exists that contains $mathfrak A$ as an $L$-structure?
$endgroup$
– StefanH
Dec 15 '18 at 17:58
$begingroup$
@StefanH The whole point of the diagram of $A$ is that any model of it contains an isomorphic copy of $A$ as a substructure (via the interpretation of the constant symbols).
$endgroup$
– Alex Kruckman
Dec 15 '18 at 18:12
$begingroup$
Got it, just needs additionally that completeness of a theory is equivalent with the property that all models are elementary equivalent, then surely non-completeness lets us choose such a model.
$endgroup$
– StefanH
Dec 16 '18 at 21:09
add a comment |
$begingroup$
You have two errors here:
"$T$ is complete" means that for every sentence $varphi$, $Tmodels varphi$ or $Tmodels lnotvarphi$. It does not mean that $varphiin T$ or $lnot varphiin T$.
A basic formula is not just an atomic formula. It is an atomic formula or a negation of an atomic formula.
In your example, if $mathfrak{A}models varphi$, then $t_1 = t_2in text{Diag}(mathfrak{A})$ and $t_3neq t_4in text{Diag}(mathfrak{A})$, so $Tcup text{Diag}(mathfrak{A})models varphi$. Otherwise, if $mathfrak{A}notmodels varphi$, then either $t_1neq t_2in text{Diag}(mathfrak{A})$ or $t_3 = t_4in text{Diag}(mathfrak{A})$, and in either case $Tcup text{Diag}(mathfrak{A})models lnot varphi$.
It's an easy exercise to show that for any structure $mathfrak{A}$ and any quantifier-free $L(A)$-sentence $varphi$, either $text{Diag}(mathfrak{A})models varphi$ or $text{Diag}(mathfrak{A})models lnot varphi$. The interesting fact is that $T$ is model complete if and only if $Tcuptext{Diag}(mathfrak{A})$ decides the truth of all $L(A)$-formulas, even those with quantifiers. The answer by spaceisdarkgreen explains the easy proof of this fact.
$endgroup$
$begingroup$
Guess you mean $T cup operatorname{Diag}(mathfrak A)$ decides the truth of all $L(A)$-formulas, not just $operatorname{Diag}(mathfrak A)$?
$endgroup$
– StefanH
Dec 15 '18 at 17:52
1
$begingroup$
@StefanH "All extensions of $mathfrak A$ are elementary" means exactly the same thing as "$Diag(mathfrak A)$ is complete." If this holds and $mathfrak Amodels T,$ then all the extensions are models of $T$ as well. "$T$ is model complete" means all models of $T$ have this property.
$endgroup$
– spaceisdarkgreen
Dec 15 '18 at 23:54
$begingroup$
@StefanH Yes, that was a typo! I misunderstood what you were referring to at first. Fixed.
$endgroup$
– Alex Kruckman
Dec 16 '18 at 1:52
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3039687%2fa-theory-t-is-model-complete-if-the-union-of-t-with-an-atomic-diagram-is-com%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Complete means for any sentence $varphi,$ either $Tvdash varphi$ or $Tvdash lnot varphi,$ not $varphiin T$ or $lnotvarphiin T.$
If $mathfrak A,mathfrak Bmodels T$ and $mathfrak Asubseteq mathfrak B,$ then both are models of $Tcup Diag(mathfrak A).$ If $Tcup Diag(mathfrak A)$ is complete, then they must agree on all $L(A)$-sentences, and hence the embedding is elementary. On the other hand if $Tcup Diag(mathfrak A)$ is not complete for some $mathfrak Amodels T,$ then we can find a $mathfrak Bmodels T$ with $mathfrak Asubseteq mathfrak B$ that differs from $mathfrak A$ on some $L(A)$-sentence (just let it be a model of the negation of some $varphi$ that is true in $mathfrak A$ but that $Tcup Diag(mathfrak A)$ does not decide). Hence this embedding is not elementary.
$endgroup$
$begingroup$
What makes you sure that such a model $mathfrak B$ of $T cup Diag(mathfrak A) cup {neg varphi}$ exists that contains $mathfrak A$ as an $L$-structure?
$endgroup$
– StefanH
Dec 15 '18 at 17:58
$begingroup$
@StefanH The whole point of the diagram of $A$ is that any model of it contains an isomorphic copy of $A$ as a substructure (via the interpretation of the constant symbols).
$endgroup$
– Alex Kruckman
Dec 15 '18 at 18:12
$begingroup$
Got it, just needs additionally that completeness of a theory is equivalent with the property that all models are elementary equivalent, then surely non-completeness lets us choose such a model.
$endgroup$
– StefanH
Dec 16 '18 at 21:09
add a comment |
$begingroup$
Complete means for any sentence $varphi,$ either $Tvdash varphi$ or $Tvdash lnot varphi,$ not $varphiin T$ or $lnotvarphiin T.$
If $mathfrak A,mathfrak Bmodels T$ and $mathfrak Asubseteq mathfrak B,$ then both are models of $Tcup Diag(mathfrak A).$ If $Tcup Diag(mathfrak A)$ is complete, then they must agree on all $L(A)$-sentences, and hence the embedding is elementary. On the other hand if $Tcup Diag(mathfrak A)$ is not complete for some $mathfrak Amodels T,$ then we can find a $mathfrak Bmodels T$ with $mathfrak Asubseteq mathfrak B$ that differs from $mathfrak A$ on some $L(A)$-sentence (just let it be a model of the negation of some $varphi$ that is true in $mathfrak A$ but that $Tcup Diag(mathfrak A)$ does not decide). Hence this embedding is not elementary.
$endgroup$
$begingroup$
What makes you sure that such a model $mathfrak B$ of $T cup Diag(mathfrak A) cup {neg varphi}$ exists that contains $mathfrak A$ as an $L$-structure?
$endgroup$
– StefanH
Dec 15 '18 at 17:58
$begingroup$
@StefanH The whole point of the diagram of $A$ is that any model of it contains an isomorphic copy of $A$ as a substructure (via the interpretation of the constant symbols).
$endgroup$
– Alex Kruckman
Dec 15 '18 at 18:12
$begingroup$
Got it, just needs additionally that completeness of a theory is equivalent with the property that all models are elementary equivalent, then surely non-completeness lets us choose such a model.
$endgroup$
– StefanH
Dec 16 '18 at 21:09
add a comment |
$begingroup$
Complete means for any sentence $varphi,$ either $Tvdash varphi$ or $Tvdash lnot varphi,$ not $varphiin T$ or $lnotvarphiin T.$
If $mathfrak A,mathfrak Bmodels T$ and $mathfrak Asubseteq mathfrak B,$ then both are models of $Tcup Diag(mathfrak A).$ If $Tcup Diag(mathfrak A)$ is complete, then they must agree on all $L(A)$-sentences, and hence the embedding is elementary. On the other hand if $Tcup Diag(mathfrak A)$ is not complete for some $mathfrak Amodels T,$ then we can find a $mathfrak Bmodels T$ with $mathfrak Asubseteq mathfrak B$ that differs from $mathfrak A$ on some $L(A)$-sentence (just let it be a model of the negation of some $varphi$ that is true in $mathfrak A$ but that $Tcup Diag(mathfrak A)$ does not decide). Hence this embedding is not elementary.
$endgroup$
Complete means for any sentence $varphi,$ either $Tvdash varphi$ or $Tvdash lnot varphi,$ not $varphiin T$ or $lnotvarphiin T.$
If $mathfrak A,mathfrak Bmodels T$ and $mathfrak Asubseteq mathfrak B,$ then both are models of $Tcup Diag(mathfrak A).$ If $Tcup Diag(mathfrak A)$ is complete, then they must agree on all $L(A)$-sentences, and hence the embedding is elementary. On the other hand if $Tcup Diag(mathfrak A)$ is not complete for some $mathfrak Amodels T,$ then we can find a $mathfrak Bmodels T$ with $mathfrak Asubseteq mathfrak B$ that differs from $mathfrak A$ on some $L(A)$-sentence (just let it be a model of the negation of some $varphi$ that is true in $mathfrak A$ but that $Tcup Diag(mathfrak A)$ does not decide). Hence this embedding is not elementary.
answered Dec 14 '18 at 18:36
spaceisdarkgreenspaceisdarkgreen
32.8k21753
32.8k21753
$begingroup$
What makes you sure that such a model $mathfrak B$ of $T cup Diag(mathfrak A) cup {neg varphi}$ exists that contains $mathfrak A$ as an $L$-structure?
$endgroup$
– StefanH
Dec 15 '18 at 17:58
$begingroup$
@StefanH The whole point of the diagram of $A$ is that any model of it contains an isomorphic copy of $A$ as a substructure (via the interpretation of the constant symbols).
$endgroup$
– Alex Kruckman
Dec 15 '18 at 18:12
$begingroup$
Got it, just needs additionally that completeness of a theory is equivalent with the property that all models are elementary equivalent, then surely non-completeness lets us choose such a model.
$endgroup$
– StefanH
Dec 16 '18 at 21:09
add a comment |
$begingroup$
What makes you sure that such a model $mathfrak B$ of $T cup Diag(mathfrak A) cup {neg varphi}$ exists that contains $mathfrak A$ as an $L$-structure?
$endgroup$
– StefanH
Dec 15 '18 at 17:58
$begingroup$
@StefanH The whole point of the diagram of $A$ is that any model of it contains an isomorphic copy of $A$ as a substructure (via the interpretation of the constant symbols).
$endgroup$
– Alex Kruckman
Dec 15 '18 at 18:12
$begingroup$
Got it, just needs additionally that completeness of a theory is equivalent with the property that all models are elementary equivalent, then surely non-completeness lets us choose such a model.
$endgroup$
– StefanH
Dec 16 '18 at 21:09
$begingroup$
What makes you sure that such a model $mathfrak B$ of $T cup Diag(mathfrak A) cup {neg varphi}$ exists that contains $mathfrak A$ as an $L$-structure?
$endgroup$
– StefanH
Dec 15 '18 at 17:58
$begingroup$
What makes you sure that such a model $mathfrak B$ of $T cup Diag(mathfrak A) cup {neg varphi}$ exists that contains $mathfrak A$ as an $L$-structure?
$endgroup$
– StefanH
Dec 15 '18 at 17:58
$begingroup$
@StefanH The whole point of the diagram of $A$ is that any model of it contains an isomorphic copy of $A$ as a substructure (via the interpretation of the constant symbols).
$endgroup$
– Alex Kruckman
Dec 15 '18 at 18:12
$begingroup$
@StefanH The whole point of the diagram of $A$ is that any model of it contains an isomorphic copy of $A$ as a substructure (via the interpretation of the constant symbols).
$endgroup$
– Alex Kruckman
Dec 15 '18 at 18:12
$begingroup$
Got it, just needs additionally that completeness of a theory is equivalent with the property that all models are elementary equivalent, then surely non-completeness lets us choose such a model.
$endgroup$
– StefanH
Dec 16 '18 at 21:09
$begingroup$
Got it, just needs additionally that completeness of a theory is equivalent with the property that all models are elementary equivalent, then surely non-completeness lets us choose such a model.
$endgroup$
– StefanH
Dec 16 '18 at 21:09
add a comment |
$begingroup$
You have two errors here:
"$T$ is complete" means that for every sentence $varphi$, $Tmodels varphi$ or $Tmodels lnotvarphi$. It does not mean that $varphiin T$ or $lnot varphiin T$.
A basic formula is not just an atomic formula. It is an atomic formula or a negation of an atomic formula.
In your example, if $mathfrak{A}models varphi$, then $t_1 = t_2in text{Diag}(mathfrak{A})$ and $t_3neq t_4in text{Diag}(mathfrak{A})$, so $Tcup text{Diag}(mathfrak{A})models varphi$. Otherwise, if $mathfrak{A}notmodels varphi$, then either $t_1neq t_2in text{Diag}(mathfrak{A})$ or $t_3 = t_4in text{Diag}(mathfrak{A})$, and in either case $Tcup text{Diag}(mathfrak{A})models lnot varphi$.
It's an easy exercise to show that for any structure $mathfrak{A}$ and any quantifier-free $L(A)$-sentence $varphi$, either $text{Diag}(mathfrak{A})models varphi$ or $text{Diag}(mathfrak{A})models lnot varphi$. The interesting fact is that $T$ is model complete if and only if $Tcuptext{Diag}(mathfrak{A})$ decides the truth of all $L(A)$-formulas, even those with quantifiers. The answer by spaceisdarkgreen explains the easy proof of this fact.
$endgroup$
$begingroup$
Guess you mean $T cup operatorname{Diag}(mathfrak A)$ decides the truth of all $L(A)$-formulas, not just $operatorname{Diag}(mathfrak A)$?
$endgroup$
– StefanH
Dec 15 '18 at 17:52
1
$begingroup$
@StefanH "All extensions of $mathfrak A$ are elementary" means exactly the same thing as "$Diag(mathfrak A)$ is complete." If this holds and $mathfrak Amodels T,$ then all the extensions are models of $T$ as well. "$T$ is model complete" means all models of $T$ have this property.
$endgroup$
– spaceisdarkgreen
Dec 15 '18 at 23:54
$begingroup$
@StefanH Yes, that was a typo! I misunderstood what you were referring to at first. Fixed.
$endgroup$
– Alex Kruckman
Dec 16 '18 at 1:52
add a comment |
$begingroup$
You have two errors here:
"$T$ is complete" means that for every sentence $varphi$, $Tmodels varphi$ or $Tmodels lnotvarphi$. It does not mean that $varphiin T$ or $lnot varphiin T$.
A basic formula is not just an atomic formula. It is an atomic formula or a negation of an atomic formula.
In your example, if $mathfrak{A}models varphi$, then $t_1 = t_2in text{Diag}(mathfrak{A})$ and $t_3neq t_4in text{Diag}(mathfrak{A})$, so $Tcup text{Diag}(mathfrak{A})models varphi$. Otherwise, if $mathfrak{A}notmodels varphi$, then either $t_1neq t_2in text{Diag}(mathfrak{A})$ or $t_3 = t_4in text{Diag}(mathfrak{A})$, and in either case $Tcup text{Diag}(mathfrak{A})models lnot varphi$.
It's an easy exercise to show that for any structure $mathfrak{A}$ and any quantifier-free $L(A)$-sentence $varphi$, either $text{Diag}(mathfrak{A})models varphi$ or $text{Diag}(mathfrak{A})models lnot varphi$. The interesting fact is that $T$ is model complete if and only if $Tcuptext{Diag}(mathfrak{A})$ decides the truth of all $L(A)$-formulas, even those with quantifiers. The answer by spaceisdarkgreen explains the easy proof of this fact.
$endgroup$
$begingroup$
Guess you mean $T cup operatorname{Diag}(mathfrak A)$ decides the truth of all $L(A)$-formulas, not just $operatorname{Diag}(mathfrak A)$?
$endgroup$
– StefanH
Dec 15 '18 at 17:52
1
$begingroup$
@StefanH "All extensions of $mathfrak A$ are elementary" means exactly the same thing as "$Diag(mathfrak A)$ is complete." If this holds and $mathfrak Amodels T,$ then all the extensions are models of $T$ as well. "$T$ is model complete" means all models of $T$ have this property.
$endgroup$
– spaceisdarkgreen
Dec 15 '18 at 23:54
$begingroup$
@StefanH Yes, that was a typo! I misunderstood what you were referring to at first. Fixed.
$endgroup$
– Alex Kruckman
Dec 16 '18 at 1:52
add a comment |
$begingroup$
You have two errors here:
"$T$ is complete" means that for every sentence $varphi$, $Tmodels varphi$ or $Tmodels lnotvarphi$. It does not mean that $varphiin T$ or $lnot varphiin T$.
A basic formula is not just an atomic formula. It is an atomic formula or a negation of an atomic formula.
In your example, if $mathfrak{A}models varphi$, then $t_1 = t_2in text{Diag}(mathfrak{A})$ and $t_3neq t_4in text{Diag}(mathfrak{A})$, so $Tcup text{Diag}(mathfrak{A})models varphi$. Otherwise, if $mathfrak{A}notmodels varphi$, then either $t_1neq t_2in text{Diag}(mathfrak{A})$ or $t_3 = t_4in text{Diag}(mathfrak{A})$, and in either case $Tcup text{Diag}(mathfrak{A})models lnot varphi$.
It's an easy exercise to show that for any structure $mathfrak{A}$ and any quantifier-free $L(A)$-sentence $varphi$, either $text{Diag}(mathfrak{A})models varphi$ or $text{Diag}(mathfrak{A})models lnot varphi$. The interesting fact is that $T$ is model complete if and only if $Tcuptext{Diag}(mathfrak{A})$ decides the truth of all $L(A)$-formulas, even those with quantifiers. The answer by spaceisdarkgreen explains the easy proof of this fact.
$endgroup$
You have two errors here:
"$T$ is complete" means that for every sentence $varphi$, $Tmodels varphi$ or $Tmodels lnotvarphi$. It does not mean that $varphiin T$ or $lnot varphiin T$.
A basic formula is not just an atomic formula. It is an atomic formula or a negation of an atomic formula.
In your example, if $mathfrak{A}models varphi$, then $t_1 = t_2in text{Diag}(mathfrak{A})$ and $t_3neq t_4in text{Diag}(mathfrak{A})$, so $Tcup text{Diag}(mathfrak{A})models varphi$. Otherwise, if $mathfrak{A}notmodels varphi$, then either $t_1neq t_2in text{Diag}(mathfrak{A})$ or $t_3 = t_4in text{Diag}(mathfrak{A})$, and in either case $Tcup text{Diag}(mathfrak{A})models lnot varphi$.
It's an easy exercise to show that for any structure $mathfrak{A}$ and any quantifier-free $L(A)$-sentence $varphi$, either $text{Diag}(mathfrak{A})models varphi$ or $text{Diag}(mathfrak{A})models lnot varphi$. The interesting fact is that $T$ is model complete if and only if $Tcuptext{Diag}(mathfrak{A})$ decides the truth of all $L(A)$-formulas, even those with quantifiers. The answer by spaceisdarkgreen explains the easy proof of this fact.
edited Dec 16 '18 at 1:51
answered Dec 14 '18 at 18:52
Alex KruckmanAlex Kruckman
27.2k22557
27.2k22557
$begingroup$
Guess you mean $T cup operatorname{Diag}(mathfrak A)$ decides the truth of all $L(A)$-formulas, not just $operatorname{Diag}(mathfrak A)$?
$endgroup$
– StefanH
Dec 15 '18 at 17:52
1
$begingroup$
@StefanH "All extensions of $mathfrak A$ are elementary" means exactly the same thing as "$Diag(mathfrak A)$ is complete." If this holds and $mathfrak Amodels T,$ then all the extensions are models of $T$ as well. "$T$ is model complete" means all models of $T$ have this property.
$endgroup$
– spaceisdarkgreen
Dec 15 '18 at 23:54
$begingroup$
@StefanH Yes, that was a typo! I misunderstood what you were referring to at first. Fixed.
$endgroup$
– Alex Kruckman
Dec 16 '18 at 1:52
add a comment |
$begingroup$
Guess you mean $T cup operatorname{Diag}(mathfrak A)$ decides the truth of all $L(A)$-formulas, not just $operatorname{Diag}(mathfrak A)$?
$endgroup$
– StefanH
Dec 15 '18 at 17:52
1
$begingroup$
@StefanH "All extensions of $mathfrak A$ are elementary" means exactly the same thing as "$Diag(mathfrak A)$ is complete." If this holds and $mathfrak Amodels T,$ then all the extensions are models of $T$ as well. "$T$ is model complete" means all models of $T$ have this property.
$endgroup$
– spaceisdarkgreen
Dec 15 '18 at 23:54
$begingroup$
@StefanH Yes, that was a typo! I misunderstood what you were referring to at first. Fixed.
$endgroup$
– Alex Kruckman
Dec 16 '18 at 1:52
$begingroup$
Guess you mean $T cup operatorname{Diag}(mathfrak A)$ decides the truth of all $L(A)$-formulas, not just $operatorname{Diag}(mathfrak A)$?
$endgroup$
– StefanH
Dec 15 '18 at 17:52
$begingroup$
Guess you mean $T cup operatorname{Diag}(mathfrak A)$ decides the truth of all $L(A)$-formulas, not just $operatorname{Diag}(mathfrak A)$?
$endgroup$
– StefanH
Dec 15 '18 at 17:52
1
1
$begingroup$
@StefanH "All extensions of $mathfrak A$ are elementary" means exactly the same thing as "$Diag(mathfrak A)$ is complete." If this holds and $mathfrak Amodels T,$ then all the extensions are models of $T$ as well. "$T$ is model complete" means all models of $T$ have this property.
$endgroup$
– spaceisdarkgreen
Dec 15 '18 at 23:54
$begingroup$
@StefanH "All extensions of $mathfrak A$ are elementary" means exactly the same thing as "$Diag(mathfrak A)$ is complete." If this holds and $mathfrak Amodels T,$ then all the extensions are models of $T$ as well. "$T$ is model complete" means all models of $T$ have this property.
$endgroup$
– spaceisdarkgreen
Dec 15 '18 at 23:54
$begingroup$
@StefanH Yes, that was a typo! I misunderstood what you were referring to at first. Fixed.
$endgroup$
– Alex Kruckman
Dec 16 '18 at 1:52
$begingroup$
@StefanH Yes, that was a typo! I misunderstood what you were referring to at first. Fixed.
$endgroup$
– Alex Kruckman
Dec 16 '18 at 1:52
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3039687%2fa-theory-t-is-model-complete-if-the-union-of-t-with-an-atomic-diagram-is-com%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown