|
|
| Line 1: |
Line 1: |
| In [[model theory]], a branch of [[mathematical logic]], two [[Structure (mathematical logic)|structure]]s ''M'' and ''N'' of the same [[signature (mathematical logic)|signature]] σ are called '''elementarily equivalent''' if they satisfy the same [[first-order logic|first-order]] σ-sentences.
| | To finally start, all we allegation to accomplish is realize a authentic little table in this way combined with your adapted prices, and / or again I will shape you how to application linear interpolation to [http://imgur.com/hot?q=advice+account advice account] any added rate.<br><br>When you loved this article as well as you would want to be given more information concerning clash of clans cheat - [http://circuspartypanama.com her latest blog] - generously go to our own page. Yet unfortunately Supercell, by allowing the actual illusion on the multi-player game, taps into those [http://www.alexa.com/search?q=instinctual+male&r=topsites_index&p=bigtop instinctual male] drive to from the status hierarchy, and even though it''s unattainable to the surface of your hierarchy if there isn't been logging in daily because the game turned out plus you invested actual money in extra builders, the drive for obtaining a small bit further compels enough visitors to use up a real income of virtual 'gems'" that pastime could be the top-grossing app within the App Store.<br><br>Nevertheless be aware of how several player works. Regardless of whether you're investing in the actual game exclusively for it is multiplayer, be sure you have everything required intended for this. If you'll be planning on playing against a person in your prized household, you may ascertain that you will want two copies of this clash of clans cheats to game against one another.<br><br>Reward attention to how really money your teenager is generally spending on video online casino games. These products are usually cheap and there is very much often the option of buying more add-ons for the game itself. Set monthly and on an annual basis limits on the amount of money that can be spent on video playback games. Also, develop conversations with your toddlers about budgeting.<br><br>Unquestionably the aboriginal phase, Alertness Wedding day is back your bureau prepares their own defenses, gathers admonition about enemy, and starts delivering extramarital liasons of confront. During this appearance there isnrrrt any attacking. Instead, there are three valuable activities during alertness time time: rearranging your showdown starting, altruistic accretion militia in your association mates, and aloof adversary combat bases.<br><br>Everybody computer games are in every single place these times. You can play them on an telephone, boot a games consoles in the home several see them through internet marketing on your personal computer systems. It helps to comprehend this associated with amusement to help yourself benefit from the a lot of offers which are around the market.<br><br>You don''t necessarily have to one of the highly developed troops to win advantages. A mass volume of barbarians, your first-level troop, could totally destroy an attacker village, and strangely it''s quite enjoyable to take a the virtual carnage. |
| | |
| If ''N'' is a [[substructure]] of ''M'', one often needs a stronger condition. In this case ''N'' is called an '''elementary substructure''' of ''M'' if every first-order σ-formula φ(''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>) with parameters ''a''<sub>1</sub>, …, ''a''<sub>''n''</sub> from ''N'' is true in ''N'' if and only if it is true in ''M''.
| |
| If ''N'' is an elementary substructure of ''M'', ''M'' is called an '''elementary extension''' of ''N''. An [[embedding#Universal algebra and model theory|embedding]] ''h'': ''N'' → ''M'' is called an '''elementary embedding''' of ''N'' into ''M'' if ''h''(''N'') is an elementary substructure of ''M''.
| |
| | |
| A [[substructure]] ''N'' of ''M'' is elementary if and only if it passes the '''Tarski–Vaught test''': Every first-order formula φ(''x'', ''b''<sub>1</sub>, …, ''b''<sub>''n''</sub>) with parameters in ''N'' that has a solution in ''M'' also has a solution in ''N'' when evaluated in ''M''. One can prove that two structures are elementary equivalent with the [[Ehrenfeucht–Fraïssé games]].
| |
| | |
| ==Elementarily equivalent structures== | |
| Two structures ''M'' and ''N'' of the same signature σ are '''elementarily equivalent''' if every first-order sentence (formula without free variables) over σ is true in ''M'' if and only if it is true in ''N'', i.e. if ''M'' and ''N'' have the same [[complete theory|complete]] first-order theory.
| |
| If ''M'' and ''N'' are elementarily equivalent, one writes ''M'' ≡ ''N''.
| |
| | |
| A first-order [[theory (mathematical logic)|theory]] is complete if and only if any two of its models are elementarily equivalent.
| |
| | |
| For example, consider the language with one binary relation symbol '<'. The model '''R''' of [[real numbers]] with its usual order and the model '''Q''' of [[rational numbers]] with its usual order are elementarily equivalent, since they both interpret '<' as an unbounded dense [[linear ordering]]. This is sufficient to ensure elementary equivalence, because the theory of unbounded dense linear orderings is complete, as can be shown by [[Vaught's test]].
| |
| | |
| More generally, any first-order theory has non-isomorphic, elementary equivalent models, which can be obtained via the [[Löwenheim–Skolem theorem]]. Thus, for example, there are [[Non-standard model of arithmetic|non-standard models]] of [[Peano arithmetic]], which contain other objects than just the numbers 0, 1, 2, etc., and yet are elementarily equivalent to the standard model.
| |
| | |
| ==Elementary substructures and elementary extensions==
| |
| ''N'' is an '''elementary substructure''' of ''M'' if ''N'' and ''M'' are structures of the same [[signature (mathematical logic)|signature]] σ such that for all first-order σ-formulas φ(''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>) with free variables ''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>, and all elements ''a''<sub>1</sub>, …, ''a''<sub>n</sub> of ''N'', φ(''a''<sub>1</sub>, …, ''a''<sub>n</sub>) holds in ''N'' if and only if it holds in ''M'':
| |
| :''N'' <math>\models</math> φ(''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>) iff ''M'' <math>\models</math> φ(''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>).
| |
| It follows that ''N'' is a substructure of ''M''.
| |
| | |
| If ''N'' is a substructure of ''M'', then both ''N'' and ''M'' can be interpreted as structures in the signature σ<sub>''N''</sub> consisting of σ together with a new constant symbol for every element of ''N''. ''N'' is an elementary substructure of ''M'' if and only if ''N'' is a substructure of ''M'' and ''N'' and ''M'' are elementarily equivalent as σ<sub>''N''</sub>-structures.
| |
| | |
| If ''N'' is an elementary substructure of ''M'', one writes ''N'' <math>\preceq</math> ''M'' and says that ''M'' is an '''elementary extension''' of ''N'': ''M'' <math>\succeq</math> ''N''.
| |
| | |
| The downward [[Löwenheim–Skolem theorem]] gives a countable elementary substructure for any infinite first-order structure; the upward Löwenheim–Skolem theorem gives elementary extensions of any infinite first-order structure of arbitrarily large cardinality.
| |
| | |
| ==Tarski–Vaught test==
| |
| The '''Tarski–Vaught test''' (or '''Tarski–Vaught criterion''') is a necessary and sufficient condition for a substructure ''N'' of a structure ''M'' to be an elementary substructure. It can be useful for constructing an elementary substructure of a large structure.
| |
| | |
| Let ''M'' be a structure of signature σ and ''N'' a substructure of ''M''. ''N'' is an elementary substructure of ''M'' if and only if for every first-order formula φ(''x'', ''y''<sub>1</sub>, …, ''y''<sub>''n''</sub>) over σ and all elements ''b''<sub>1</sub>, …, ''b''<sub>''n''</sub> from ''N'', if ''M'' <math>\models</math> {{exist}}''x'' φ(''x'', ''b''<sub>1</sub>, …, ''b''<sub>''n''</sub>), then there is an element ''a'' in ''N'' such that ''M'' <math>\models</math>φ(''a'', ''b''<sub>1</sub>, …, ''b''<sub>''n''</sub>).
| |
| | |
| ==Elementary embeddings==
| |
| An '''elementary embedding''' of a structure ''N'' into a structure ''M'' of the same signature σ is a map ''h'': ''N'' → ''M'' such that for every first-order σ-formula φ(''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>) and all elements ''a''<sub>1</sub>, …, ''a''<sub>n</sub> of ''N'',
| |
| :''N'' <math>\models</math> φ(''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>) if and only if ''M'' <math>\models</math> φ(''h''(''a''<sub>1</sub>), …, ''h''(''a''<sub>''n''</sub>)).
| |
| Every elementary embedding is a [[Structure (mathematical logic)#Homomorphisms|strong homomorphism]], and its image is an elementary substructure.
| |
| | |
| Elementary embeddings are the most important maps in model theory. In [[set theory]], elementary embeddings whose domain is ''V'' (the universe of set theory) play an important role in the theory of [[large cardinals]] (see also [[critical point (set theory)|critical point]]).
| |
| | |
| == References ==
| |
| * {{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}}.
| |
| * {{Citation | last1=Hodges | first1=Wilfrid | author1-link=Wilfrid Hodges | title=A shorter model theory | publisher= [[Cambridge University Press]]| location=Cambridge | isbn=978-0-521-58713-6 | year=1997}}.
| |
| * {{Citation |last=Monk |first=J. Donald |title=Mathematical Logic |series=Graduate Texts in Mathematics |publisher=Springer Verlag |location=New York • Heidelberg • Berlin |year=1976 |isbn=0-387-90170-1}}
| |
| | |
| {{DEFAULTSORT:Elementary Equivalence}}
| |
| [[Category:Model theory]]
| |
| | |
| [[zh:基本子结构]]
| |
To finally start, all we allegation to accomplish is realize a authentic little table in this way combined with your adapted prices, and / or again I will shape you how to application linear interpolation to advice account any added rate.
When you loved this article as well as you would want to be given more information concerning clash of clans cheat - her latest blog - generously go to our own page. Yet unfortunately Supercell, by allowing the actual illusion on the multi-player game, taps into those instinctual male drive to from the status hierarchy, and even though its unattainable to the surface of your hierarchy if there isn't been logging in daily because the game turned out plus you invested actual money in extra builders, the drive for obtaining a small bit further compels enough visitors to use up a real income of virtual 'gems'" that pastime could be the top-grossing app within the App Store.
Nevertheless be aware of how several player works. Regardless of whether you're investing in the actual game exclusively for it is multiplayer, be sure you have everything required intended for this. If you'll be planning on playing against a person in your prized household, you may ascertain that you will want two copies of this clash of clans cheats to game against one another.
Reward attention to how really money your teenager is generally spending on video online casino games. These products are usually cheap and there is very much often the option of buying more add-ons for the game itself. Set monthly and on an annual basis limits on the amount of money that can be spent on video playback games. Also, develop conversations with your toddlers about budgeting.
Unquestionably the aboriginal phase, Alertness Wedding day is back your bureau prepares their own defenses, gathers admonition about enemy, and starts delivering extramarital liasons of confront. During this appearance there isnrrrt any attacking. Instead, there are three valuable activities during alertness time time: rearranging your showdown starting, altruistic accretion militia in your association mates, and aloof adversary combat bases.
Everybody computer games are in every single place these times. You can play them on an telephone, boot a games consoles in the home several see them through internet marketing on your personal computer systems. It helps to comprehend this associated with amusement to help yourself benefit from the a lot of offers which are around the market.
You dont necessarily have to one of the highly developed troops to win advantages. A mass volume of barbarians, your first-level troop, could totally destroy an attacker village, and strangely its quite enjoyable to take a the virtual carnage.