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| In the branch of [[mathematical logic]] called [[model theory]], an '''elementary class''' (or '''axiomatizable class''') is a [[class (set theory)|class]] consisting of all [[structure (mathematical logic)|structures]] satisfying a fixed [[first-order logic|first-order]] [[theory (mathematical logic)|theory]].
| | It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>Here's more regarding [http://www.youtube.com/watch?v=90z1mmiwNS8 dentist DC] have a look at the web site. |
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| == Definition ==
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| A [[class (set theory)|class]] ''K'' of [[structure (mathematical logic)|structures]] of a [[signature (logic)|signature]] σ is called an '''elementary class''' if there is a [[first-order logic|first-order]] [[theory (mathematical logic)|theory]] ''T'' of signature σ, such that ''K'' consists of all models of ''T'', i.e., of all σ-structures that satisfy ''T''. If ''T'' can be chosen as a theory consisting of a single first-order sentence, then ''K'' is called a '''basic elementary class'''.
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| More generally, ''K'' is a [[pseudoelementary class|pseudo-elementary class]] if there is a first-order theory ''T'' of a signature that extends σ, such that ''K'' consists of all σ-structures that are [[reduct]]s to σ of models of ''T''. In other words, a class ''K'' of σ-structures is pseudo-elementary [[iff]] there is an elementary class ''K<nowiki>'</nowiki>'' such that ''K'' consists of precisely the reducts to σ of the structures in ''K<nowiki>'</nowiki>''.
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| For obvious reasons, elementary classes are also called '''axiomatizable in first-order logic''', and basic elementary classes are called '''finitely axiomatizable in first-order logic'''. These definitions extend to other logics in the obvious way, but since the first-order case is by far the most important, '''axiomatizable''' implicitly refers to this case when no other logic is specified.
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| == Conflicting and alternative terminology ==
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| While the above is nowadays standard terminology in [[model theory|"infinite" model theory]], the slightly different earlier definitions are still in use in [[finite model theory]], where an elementary class may be called a '''Δ-elementary class''', and the terms '''elementary class''' and '''first-order axiomatizable class''' are reserved for basic elementary classes (Ebbinghaus et al. 1994, Ebbinghaus and Flum 2005). Hodges calls elementary classes '''axiomatizable classes''', and he refers to basic elementary classes as '''definable classes'''. He also uses the respective synonyms '''EC class''' and '''EC<math>_\Delta</math> class''' (Hodges, 1993).
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| There are good reasons for this diverging terminology. The [[signature (logic)|signature]]s that are considered in general model theory are often infinite, while a single [[first-order logic|first-order]] [[sentence (mathematical logic)|sentence]] contains only finitely many symbols. Therefore basic elementary classes are atypical in infinite model theory. Finite model theory, on the other hand, deals almost exclusively with finite signatures. It is easy to see that for every finite signature σ and for every class ''K'' of σ-structures closed under isomorphism there is an elementary class <math>K'</math> of σ-structures such that ''K'' and <math>K'</math> contain precisely the same finite structures. Hence elementary classes are not very interesting for finite model theorists.
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| == Easy relations between the notions ==
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| Clearly every basic elementary class is an elementary class, and every elementary class is a pseudo-elementary class. Moreover, as an easy consequence of the [[compactness theorem]], a class of σ-structures is basic elementary if and only if it is elementary and its complement is also elementary.
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| == Examples ==
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| === A basic elementary class ===
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| Let σ be a signature consisting only of a [[unary function]] symbol ''f''. The class ''K'' of σ-structures in which ''f'' is [[injection (mathematics)|one-to-one]] is a basic elementary class. This is witnessed by the theory ''T'', which consists only of the single sentence
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| :<math>\forall x\forall y( (f(x)=f(y)) \to (x=y) )</math>.
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| === An elementary, basic pseudoelementary class that is not basic elementary ===
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| Let σ be an arbitrary signature. The class ''K'' of all infinite σ-structures is elementary. To see this, consider the sentences
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| :<math>\rho_2={}</math> "<math>\exist x_1\exist x_2(x_1 \not =x_2)</math>",
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| :<math>\rho_3={}</math> "<math>\exist x_1\exist x_2\exist x_3((x_1 \not =x_2) \and (x_1 \not =x_3) \and (x_2 \not =x_3))</math>",
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| and so on. (So the sentence <math>\rho_n</math> says that there are at least ''n'' elements.) The infinite σ-structures are precisely the models of the theory
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| :<math>T_\infty=\{\rho_2, \rho_3, \rho_4, \dots\}</math>.
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| But ''K'' is not a basic elementary class. Otherwise the infinite σ-structures would be precisely those that satisfy a certain first-order sentence τ. But then the set
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| <math>\{\neg\tau, \rho_2, \rho_3, \rho_4, \dots\}</math> would be inconsistent. By the [[compactness theorem]], for some natural number ''n'' the set <math>\{\neg\tau, \rho_2, \rho_3, \rho_4, \dots, \rho_n\}</math> would be inconsistent. But this is absurd, because this theory is satisfied by any σ-structure with <math>n+1</math> or more elements. | |
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| However, there is a basic elementary class ''K<nowiki>'</nowiki>'' in the signature σ' = σ <math>\cup</math> {''f''}, where ''f'' is a unary function symbol, such that ''K'' consists exactly of the reducts to σ of σ'-structures in ''K<nowiki>'</nowiki>''. ''K<nowiki>'</nowiki>'' is axiomatised by the single sentence <math>(\forall x\forall y(f(x) = f(y) \rightarrow x=y) \land \exists y\neg\exists x(y = f(x))),</math>, which expresses that ''f'' is injective but not surjective. Therefore ''K'' is elementary and what could be called basic pseudo-elementary, but not basic elementary.
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| === Pseudo-elementary class that is non-elementary ===
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| Finally, consider the signature σ consisting of a single unary relation symbol ''P''. Every σ-structure is [[partition of a set|partitioned]] into two subsets: Those elements for which ''P'' holds, and the rest. Let ''K'' be the class of all σ-structures for which these two subsets have the same [[cardinality]], i.e., there is a bijection between them. This class is not elementary, because a σ-structure in which both the set of realisations of ''P'' and its complement are countably infinite satisfies precisely the same first-order sentences as a σ-structure in which one of the sets is countably infinite and the other is uncountable.
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| Now consider the signature <math>\sigma'</math>, which consists of ''P'' along with a unary function symbol ''f''. Let <math>K'</math> be the class of all <math>\sigma'</math>-structures such that ''f'' is a bijection and ''P'' holds for ''x'' [[iff]] ''P'' does not hold for ''f(x)''. <math>K'</math> is clearly an elementary class, and therefore ''K'' is an example of a pseudo-elementary class that is not elementary.
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| === Non-pseudo-elementary class===
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| Let σ be an arbitrary signature. The class ''K'' of all finite σ-structures is not elementary, because (as shown above) its complement is elementary but not basic elementary. Since this is also true for every signature extending σ, ''K'' is not even a pseudo-elementary class.
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| This example demonstrates the limits of expressive power inherent in [[first-order logic]] as opposed to the far more expressive [[second-order logic]]. Second-order logic, however, fails to retain many desirable properties of first-order logic, such as the compactness theorem.
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| == References ==
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| * {{Citation | last1=Chang | first1=Chen Chung | last2=Keisler | first2=H. Jerome | author2-link=Howard Jerome Keisler | title=Model Theory | origyear=1973 | publisher=[[Elsevier]] | edition=3rd | series=Studies in Logic and the Foundations of Mathematics | isbn=978-0-444-88054-3 | year=1990}}
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| * {{Citation | last1=Ebbinghaus | first1=Heinz-Dieter | last2=Flum | first2=Jörg | title=Finite model theory | origyear=1995 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-28787-2 | year=2005 | pages=360}}
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| * {{Citation | last1=Ebbinghaus | first1=Heinz-Dieter | last2=Flum | first2=Jörg | last3=Thomas | first3=Wolfgang | title=Mathematical Logic | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-0-387-94258-2 | year=1994}}
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| * {{Citation | last1=Hodges | first1=Wilfrid | author1-link=Wilfrid Hodges | title=A shorter model theory | publisher=[[Cambridge University Press]] | isbn=978-0-521-58713-6 | year=1997}}
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| * {{Citation | last1=Poizat | first1=Bruno | title=A Course in Model Theory: An Introduction to Contemporary Mathematical Logic | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-98655-5 | year=2000}}
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| {{DEFAULTSORT:Elementary Class}}
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| [[Category:Model theory]]
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It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.
Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.
At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.
Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.
Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.
Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.
Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.
Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.
In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.
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