difference between Axiom systems and a Model The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Is there a difference between a model and a representation?Difference between elementary logic and formal logicAxiom Systems and Formal SystemsDifference between “if $vdash P$, then $vdash Q$” and “$vdash(PRightarrow Q)$”?Existence of an axiom question in relation to $mathsfInfinity$Finitely axiomatized second-order theory without a model, which is consistent for reasonable deductive systemsCan Peano axioms be used to construct a model of Natural numbers?Difference(s) between an axiom scheme and an axiomConfusion over the definition of “model”How is mathematics formulated - with models of formal systems?
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difference between Axiom systems and a Model
The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Is there a difference between a model and a representation?Difference between elementary logic and formal logicAxiom Systems and Formal SystemsDifference between “if $vdash P$, then $vdash Q$” and “$vdash(PRightarrow Q)$”?Existence of an axiom question in relation to $mathsfInfinity$Finitely axiomatized second-order theory without a model, which is consistent for reasonable deductive systemsCan Peano axioms be used to construct a model of Natural numbers?Difference(s) between an axiom scheme and an axiomConfusion over the definition of “model”How is mathematics formulated - with models of formal systems?
$begingroup$
In my understanding, system of axioms are a set of axioms such as PA, ZF, ZFC. For example, we can write when ZFC proves something :
$ZFC vdash phi$
However, sometimes, in logic theory, it introduces a Model T and writes like:
$T vdash phi$
Here, the terminology "Model" is same as an axiom system? If not, what is difference between a system of Axioms and a Model.
logic model-theory
asked Mar 31 at 5:17
HoCheol SHINHoCheol SHIN
120211
$endgroup$
add a comment |
$begingroup$
In my understanding, system of axioms are a set of axioms such as PA, ZF, ZFC. For example, we can write when ZFC proves something :
$ZFC vdash phi$
However, sometimes, in logic theory, it introduces a Model T and writes like:
$T vdash phi$
Here, the terminology "Model" is same as an axiom system? If not, what is difference between a system of Axioms and a Model.
logic model-theory
asked Mar 31 at 5:17
HoCheol SHINHoCheol SHIN
120211
$endgroup$
add a comment |
$begingroup$
In my understanding, system of axioms are a set of axioms such as PA, ZF, ZFC. For example, we can write when ZFC proves something :
$ZFC vdash phi$
However, sometimes, in logic theory, it introduces a Model T and writes like:
$T vdash phi$
Here, the terminology "Model" is same as an axiom system? If not, what is difference between a system of Axioms and a Model.
logic model-theory
asked Mar 31 at 5:17
HoCheol SHINHoCheol SHIN
120211
$endgroup$
In my understanding, system of axioms are a set of axioms such as PA, ZF, ZFC. For example, we can write when ZFC proves something :
$ZFC vdash phi$
However, sometimes, in logic theory, it introduces a Model T and writes like:
$T vdash phi$
Here, the terminology "Model" is same as an axiom system? If not, what is difference between a system of Axioms and a Model.
logic model-theory
logic model-theory
asked Mar 31 at 5:17
HoCheol SHINHoCheol SHIN
120211
asked Mar 31 at 5:17
HoCheol SHINHoCheol SHIN
120211
asked Mar 31 at 5:17
HoCheol SHINHoCheol SHIN
120211
asked Mar 31 at 5:17
HoCheol SHINHoCheol SHIN
120211
asked Mar 31 at 5:17
HoCheol SHINHoCheol SHIN
120211
120211
add a comment |
add a comment |
1 Answer
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$begingroup$
Usually, $T$ is notation for a theory, not a model. A theory pretty similar to a system of axioms, arguably the same thing. It is just a set of sentences, and $Tvdash phi$ means there is a proof of $phi$ using the sentences of $T$ as premises.
(The difference, if any, is 'system of axioms' has the connotation that the set of sentences is effectively describable in some way, whereas a theory is just an arbitrary set of sentences. Some authors also require that a theory be closed in the sense that if $Tvdash phi$ then $phiin T$.)
Models are usually written as $mathcal M,$ or something like that, and usually the notation is $mathcal Mmodels phi$ for a sentence $phi$ or $mathcal Mmodels T$ for a theory $T.$ Models are not at all the same thing as theories, rather they are mathematical structures in which the language can be interpreted. For instance we can interpret the formal sentence $forall x exists y exists z(x+y=z)$ in the structure $(mathbb N, +)$ (where $+$ is the addition operation on $mathbb N)$ as "for every $xinmathbb N$ there are $yinmathbb N$ and $zinmathbb N$ such that $x+y=z$." The sentence is true in this interpretation.
The notation $mathcal Mmodels phi$ just means the sentence $phi$ is true in the interpretation $mathcal M.$ For theory $T$ (which recall is a set of sentences) $mathcal Mmodels T$ means that all the sentences of $T$ are true in the interpretation $mathcal M.$ In this case, we say "$mathcal M$ is a model for the theory $T$."
One also sees the notation $Tmodels phi,$ and this means that the sentence $phi$ is true in every interpretation that is a model of the theory $T.$ Sometimes, when it's clear from context that we aren't talking about formal proofs, $Tvdash phi,$ is also used to denote this. (In first order logic, the completeness theorem says $phi$ is provable from $T$ if and only if it is true in all models of $T,$ so the two notions are equivalent anyway.)
answered Mar 31 at 5:49
spaceisdarkgreenspaceisdarkgreen
34k21754
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add a comment |
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$begingroup$
Usually, $T$ is notation for a theory, not a model. A theory pretty similar to a system of axioms, arguably the same thing. It is just a set of sentences, and $Tvdash phi$ means there is a proof of $phi$ using the sentences of $T$ as premises.
(The difference, if any, is 'system of axioms' has the connotation that the set of sentences is effectively describable in some way, whereas a theory is just an arbitrary set of sentences. Some authors also require that a theory be closed in the sense that if $Tvdash phi$ then $phiin T$.)
Models are usually written as $mathcal M,$ or something like that, and usually the notation is $mathcal Mmodels phi$ for a sentence $phi$ or $mathcal Mmodels T$ for a theory $T.$ Models are not at all the same thing as theories, rather they are mathematical structures in which the language can be interpreted. For instance we can interpret the formal sentence $forall x exists y exists z(x+y=z)$ in the structure $(mathbb N, +)$ (where $+$ is the addition operation on $mathbb N)$ as "for every $xinmathbb N$ there are $yinmathbb N$ and $zinmathbb N$ such that $x+y=z$." The sentence is true in this interpretation.
The notation $mathcal Mmodels phi$ just means the sentence $phi$ is true in the interpretation $mathcal M.$ For theory $T$ (which recall is a set of sentences) $mathcal Mmodels T$ means that all the sentences of $T$ are true in the interpretation $mathcal M.$ In this case, we say "$mathcal M$ is a model for the theory $T$."
One also sees the notation $Tmodels phi,$ and this means that the sentence $phi$ is true in every interpretation that is a model of the theory $T.$ Sometimes, when it's clear from context that we aren't talking about formal proofs, $Tvdash phi,$ is also used to denote this. (In first order logic, the completeness theorem says $phi$ is provable from $T$ if and only if it is true in all models of $T,$ so the two notions are equivalent anyway.)
answered Mar 31 at 5:49
spaceisdarkgreenspaceisdarkgreen
34k21754
$endgroup$
add a comment |
$begingroup$
Usually, $T$ is notation for a theory, not a model. A theory pretty similar to a system of axioms, arguably the same thing. It is just a set of sentences, and $Tvdash phi$ means there is a proof of $phi$ using the sentences of $T$ as premises.
(The difference, if any, is 'system of axioms' has the connotation that the set of sentences is effectively describable in some way, whereas a theory is just an arbitrary set of sentences. Some authors also require that a theory be closed in the sense that if $Tvdash phi$ then $phiin T$.)
Models are usually written as $mathcal M,$ or something like that, and usually the notation is $mathcal Mmodels phi$ for a sentence $phi$ or $mathcal Mmodels T$ for a theory $T.$ Models are not at all the same thing as theories, rather they are mathematical structures in which the language can be interpreted. For instance we can interpret the formal sentence $forall x exists y exists z(x+y=z)$ in the structure $(mathbb N, +)$ (where $+$ is the addition operation on $mathbb N)$ as "for every $xinmathbb N$ there are $yinmathbb N$ and $zinmathbb N$ such that $x+y=z$." The sentence is true in this interpretation.
The notation $mathcal Mmodels phi$ just means the sentence $phi$ is true in the interpretation $mathcal M.$ For theory $T$ (which recall is a set of sentences) $mathcal Mmodels T$ means that all the sentences of $T$ are true in the interpretation $mathcal M.$ In this case, we say "$mathcal M$ is a model for the theory $T$."
One also sees the notation $Tmodels phi,$ and this means that the sentence $phi$ is true in every interpretation that is a model of the theory $T.$ Sometimes, when it's clear from context that we aren't talking about formal proofs, $Tvdash phi,$ is also used to denote this. (In first order logic, the completeness theorem says $phi$ is provable from $T$ if and only if it is true in all models of $T,$ so the two notions are equivalent anyway.)
answered Mar 31 at 5:49
spaceisdarkgreenspaceisdarkgreen
34k21754
$endgroup$
add a comment |
$begingroup$
Usually, $T$ is notation for a theory, not a model. A theory pretty similar to a system of axioms, arguably the same thing. It is just a set of sentences, and $Tvdash phi$ means there is a proof of $phi$ using the sentences of $T$ as premises.
(The difference, if any, is 'system of axioms' has the connotation that the set of sentences is effectively describable in some way, whereas a theory is just an arbitrary set of sentences. Some authors also require that a theory be closed in the sense that if $Tvdash phi$ then $phiin T$.)
Models are usually written as $mathcal M,$ or something like that, and usually the notation is $mathcal Mmodels phi$ for a sentence $phi$ or $mathcal Mmodels T$ for a theory $T.$ Models are not at all the same thing as theories, rather they are mathematical structures in which the language can be interpreted. For instance we can interpret the formal sentence $forall x exists y exists z(x+y=z)$ in the structure $(mathbb N, +)$ (where $+$ is the addition operation on $mathbb N)$ as "for every $xinmathbb N$ there are $yinmathbb N$ and $zinmathbb N$ such that $x+y=z$." The sentence is true in this interpretation.
The notation $mathcal Mmodels phi$ just means the sentence $phi$ is true in the interpretation $mathcal M.$ For theory $T$ (which recall is a set of sentences) $mathcal Mmodels T$ means that all the sentences of $T$ are true in the interpretation $mathcal M.$ In this case, we say "$mathcal M$ is a model for the theory $T$."
One also sees the notation $Tmodels phi,$ and this means that the sentence $phi$ is true in every interpretation that is a model of the theory $T.$ Sometimes, when it's clear from context that we aren't talking about formal proofs, $Tvdash phi,$ is also used to denote this. (In first order logic, the completeness theorem says $phi$ is provable from $T$ if and only if it is true in all models of $T,$ so the two notions are equivalent anyway.)
answered Mar 31 at 5:49
spaceisdarkgreenspaceisdarkgreen
34k21754
$endgroup$
Usually, $T$ is notation for a theory, not a model. A theory pretty similar to a system of axioms, arguably the same thing. It is just a set of sentences, and $Tvdash phi$ means there is a proof of $phi$ using the sentences of $T$ as premises.
(The difference, if any, is 'system of axioms' has the connotation that the set of sentences is effectively describable in some way, whereas a theory is just an arbitrary set of sentences. Some authors also require that a theory be closed in the sense that if $Tvdash phi$ then $phiin T$.)
Models are usually written as $mathcal M,$ or something like that, and usually the notation is $mathcal Mmodels phi$ for a sentence $phi$ or $mathcal Mmodels T$ for a theory $T.$ Models are not at all the same thing as theories, rather they are mathematical structures in which the language can be interpreted. For instance we can interpret the formal sentence $forall x exists y exists z(x+y=z)$ in the structure $(mathbb N, +)$ (where $+$ is the addition operation on $mathbb N)$ as "for every $xinmathbb N$ there are $yinmathbb N$ and $zinmathbb N$ such that $x+y=z$." The sentence is true in this interpretation.
The notation $mathcal Mmodels phi$ just means the sentence $phi$ is true in the interpretation $mathcal M.$ For theory $T$ (which recall is a set of sentences) $mathcal Mmodels T$ means that all the sentences of $T$ are true in the interpretation $mathcal M.$ In this case, we say "$mathcal M$ is a model for the theory $T$."
One also sees the notation $Tmodels phi,$ and this means that the sentence $phi$ is true in every interpretation that is a model of the theory $T.$ Sometimes, when it's clear from context that we aren't talking about formal proofs, $Tvdash phi,$ is also used to denote this. (In first order logic, the completeness theorem says $phi$ is provable from $T$ if and only if it is true in all models of $T,$ so the two notions are equivalent anyway.)
answered Mar 31 at 5:49
spaceisdarkgreenspaceisdarkgreen
34k21754
answered Mar 31 at 5:49
spaceisdarkgreenspaceisdarkgreen
34k21754
answered Mar 31 at 5:49
spaceisdarkgreenspaceisdarkgreen
34k21754
answered Mar 31 at 5:49
spaceisdarkgreenspaceisdarkgreen
34k21754
34k21754
add a comment |
add a comment |
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