Slutsky equation: Difference between revisions

Revision as of 00:26, 27 February 2013

In mathematical logic, a logical theory $T_{2}$ is a (proof theoretic) conservative extension of a theory $T_{1}$ if the language of $T_{2}$ extends the language of $T_{1}$ ; every theorem of $T_{1}$ is a theorem of $T_{2}$ ; and any theorem of $T_{2}$ that is in the language of $T_{1}$ is already a theorem of $T_{1}$ .

More generally, if Γ is a set of formulas in the common language of $T_{1}$ and $T_{2}$ , then $T_{2}$ is Γ-conservative over $T_{1}$ if every formula from Γ provable in $T_{2}$ is also provable in $T_{1}$ .

To put it informally, the new theory may possibly be more convenient for proving theorems, but it proves no new theorems about the language of the old theory.

Note that a conservative extension of a consistent theory is consistent. Hence, conservative extensions do not bear the risk of introducing new inconsistencies. This can also be seen as a methodology for writing and structuring large theories: start with a theory, $T_{0}$ , that is known (or assumed) to be consistent, and successively build conservative extensions $T_{1}$ , $T_{2}$ , ... of it.

The theorem provers Isabelle and ACL2 adopt this methodology by providing a language for conservative extensions by definition.

Recently, conservative extensions have been used for defining a notion of module for ontologies: if an ontology is formalized as a logical theory, a subtheory is a module if the whole ontology is a conservative extension of the subtheory.

An extension which is not conservative may be called a proper extension.

Examples

ACA₀ (a subsystem of second-order arithmetic) is a conservative extension of first-order Peano arithmetic.
Von Neumann–Bernays–Gödel set theory is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice (ZFC).
Internal set theory is a conservative extension of Zermelo–Fraenkel set theory with the axiom of choice (ZFC).
Extensions by definitions are conservative.
Extensions by unconstrained predicate or function symbols are conservative.
IΣ₁ (a subsystem of Peano arithmetic with induction only for Σ⁰₁-formulas) is a Π⁰₂-conservative extension of the primitive recursive arithmetic (PRA).
ZFC is a Π¹₃-conservative extension of ZF by Shoenfield's absoluteness theorem.
ZFC with the continuum hypothesis is a Π²₁-conservative extension of ZFC.

Model-theoretic conservative extension

With model-theoretic means, a stronger notion is obtained: an extension $T_{2}$ of a theory $T_{1}$ is model-theoretically conservative if every model of $T_{1}$ can be expanded to a model of $T_{2}$ . It is straightforward to see that each model-theoretic conservative extension also is a (proof-theoretic) conservative extension in the above sense. The model theoretic notion has the advantage over the proof theoretic one that it does not depend so much on the language at hand; on the other hand, it is usually harder to establish model theoretic conservativity.

Template:Further2

External links

The importance of conservative extensions for the foundations of mathematics

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+In [[mathematical logic]], a [[logical theory]] <math>T_2</math> is a ([[Proof theory|proof theoretic]]) '''conservative extension''' of a theory <math>T_1</math> if the language of <math>T_2</math> extends the language of <math>T_1</math>; every theorem of <math>T_1</math> is a theorem of <math>T_2</math>; and any theorem of <math>T_2</math> that is in the language of <math>T_1</math> is already a theorem of <math>T_1</math>.
+More generally, if Γ is a set of formulas in the common language of <math>T_1</math> and <math>T_2</math>, then <math>T_2</math> is '''Γ-conservative''' over <math>T_1</math> if every formula from Γ provable in <math>T_2</math> is also provable in <math>T_1</math>.
+To put it informally, the new theory may possibly be more convenient for proving [[theorem]]s, but it proves no new theorems about the language of the old theory.
+Note that a conservative extension of a [[consistent]] theory is consistent. Hence, conservative extensions do not bear the risk of introducing new inconsistencies. This can also be seen as a [[methodology]] for writing and structuring large theories: start with a theory, <math>T_0</math>, that is known (or assumed) to be consistent, and successively build conservative extensions <math>T_1</math>, <math>T_2</math>, ... of it.
+The theorem provers [[Isabelle (theorem prover)|Isabelle]] and [[ACL2]] adopt this methodology by providing a language for conservative extensions by definition.
+Recently, conservative extensions have been used for defining a notion of [[ontology modularization|module]] for [[Ontology (computer science)|ontologies]]: if an ontology is formalized as a logical theory, a subtheory is a module if the whole ontology is a conservative extension of the subtheory.
+An extension which is not conservative may be called a '''proper extension'''.
+==Examples==
+* ACA<sub>0</sub> (a subsystem of [[second-order arithmetic]]) is a conservative extension of first-order [[Peano arithmetic]].
+* [[Von Neumann–Bernays–Gödel set theory]] is a conservative extension of [[Zermelo–Fraenkel set theory]] with the [[axiom of choice]] (ZFC).
+* [[Internal set theory]] is a conservative extension of [[Zermelo–Fraenkel set theory]] with the [[axiom of choice]] (ZFC).
+* [[Extension by definitions|Extensions by definitions]] are conservative.
+* Extensions by unconstrained predicate or function symbols are conservative.
+* IΣ<sub>1</sub> (a subsystem of Peano arithmetic with induction only for [[arithmetical hierarchy|Σ<sup>0</sup><sub style="margin-left:-0.65em">1</sub>-formulas]]) is a Π<sup>0</sup><sub style="margin-left:-0.65em">2</sub>-conservative extension of the [[primitive recursive arithmetic]] (PRA).
+* ZFC is a [[analytical hierarchy|Π<sup>1</sup><sub style="margin-left:-0.65em">3</sub>]]-conservative extension of ZF by [[absoluteness (mathematical logic)|Shoenfield's absoluteness theorem]].
+* ZFC with the [[continuum hypothesis]] is a Π<sup>2</sup><sub style="margin-left:-0.65em">1</sub>-conservative extension of ZFC.
+==Model-theoretic conservative extension==
+With [[Model theory|model-theoretic]] means, a stronger notion is obtained: an extension <math>T_2</math> of a theory <math>T_1</math> is '''model-theoretically conservative''' if every model of <math>T_1</math> can be expanded to a model of <math>T_2</math>. It is straightforward to see that each model-theoretic conservative extension also is a (proof-theoretic) conservative extension in the above sense. The model theoretic notion has the advantage over the proof theoretic one that it does not depend so much on the language at hand; on the other hand, it is usually harder to establish model theoretic conservativity.
+{{Further2|[[Conservativity theorem]]}}
+<!--* [[relative consistency]]-->
+==External links==
+*[http://www.cs.nyu.edu/pipermail/fom/1998-October/002306.html The importance of conservative extensions for the foundations of mathematics]
+[[Category:Proof theory]]
+[[Category:Model theory]]

Slutsky equation: Difference between revisions

Revision as of 00:26, 27 February 2013

Examples

Model-theoretic conservative extension

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