|
|
| Line 1: |
Line 1: |
| In [[mathematical logic]], '''Tarski's high school algebra problem''' was a question posed by [[Alfred Tarski]]. It asks whether there are [[Identity (mathematics)|identities]] involving [[addition]], [[multiplication]], and [[exponentiation]] over the positive integers that cannot be proved using eleven [[axioms]] about these operations that are taught in high school-level mathematics. The question was solved in 1980 by [[Alex Wilkie]] who showed that such unprovable identities do exist.
| | If an existing Word - Press code is found vulnerable, Word - Press will immediately issue an update for that. Also, you may want to opt for a more professioanl theme if you are planning on showing your site off to a high volume of potential customers each day. In case you have any kind of questions regarding wherever and also tips on how to use [http://deurl.de/backup_plugin_581128 wordpress backup plugin], you'll be able to email us from our own web-page. * A community forum for debate of the product together with some other customers in the comments spot. 2- Ask for the designs and graphics that will be provided along with the Word - Press theme. The number of options offered here is overwhelming, but once I took the time to begin to review the video training, I was amazed at how easy it was to create a squeeze page and a membership site. <br><br> |
|
| |
|
| ==Statement of the problem==
| | Creating a website from scratch can be such a pain. Infertility can cause a major setback to the couples due to the inability to conceive. This plugin is a must have for anyone who is serious about using Word - Press. You can up your site's rank with the search engines by simply taking a bit of time with your site. The biggest advantage of using a coupon or deal plugin is that it gives your readers the coupons and deals within minutes of them becoming available. <br><br>It is also popular because willing surrogates,as well as egg and sperm donors,are plentiful. Word - Press has different exciting features including a plug-in architecture with a templating system. For a much deeper understanding of simple wordpress themes", check out Upon browsing such, you'll be able to know valuable facts. Every single Theme might be unique, providing several alternatives for webpage owners to reap the benefits of in an effort to instantaneously adjust their web page appear. Converting HTML to Word - Press theme for your website can allow you to enjoy the varied Word - Press features that aid in consistent growth your online business. <br><br>It is the convenient service through which professionals either improve the position or keep the ranking intact. Quttera - Quttera describes itself as a 'Saa - S [Software as a Service] web-malware monitoring and alerting solution for websites of any size and complexity. When we talk about functional suitability, Word - Press proves itself as one of the strongest contestant among its other rivals. If you are looking for Hire Wordpress Developer then just get in touch with him. Look for experience: When you are searching for a Word - Press developer you should always look at their experience level. <br><br>As a open source platform Wordpress offers distinctive ready to use themes for free along with custom theme support and easy customization. Visit our website to learn more about how you can benefit. Just download it from the website and start using the same. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. Likewise, professional publishers with a multi author and editor setup often find that Word - Press lack basic user and role management capabilities. |
| | |
| Tarski considered the following eleven axioms about addition ('+'), multiplication ('·'), and exponentiation to be standard axioms taught in high school:
| |
| # ''x'' + ''y'' = ''y'' + ''x''
| |
| # (''x'' + ''y'') + ''z'' = ''x'' + (''y'' + ''z'')
| |
| # ''x'' · 1 = ''x''
| |
| # ''x'' · ''y'' = ''y'' · ''x''
| |
| # (''x'' · ''y'') · ''z'' = ''x'' · (''y'' · ''z'')
| |
| # ''x'' · (''y'' + ''z'') = ''x'' · ''y'' + ''x'' ·''z''
| |
| # 1<sup>''x''</sup> = 1
| |
| # ''x''<sup>1</sup> = ''x''
| |
| # ''x''<sup>''y'' + ''z''</sup> = ''x''<sup>''y''</sup> · ''x''<sup>''z''</sup>
| |
| # (''x'' · ''y'')<sup>''z''</sup> = ''x''<sup>''z''</sup> · ''y''<sup>''z''</sup>
| |
| # (''x''<sup>''y''</sup>)<sup>''z''</sup> = ''x''<sup>''y'' · ''z''</sup>.
| |
| | |
| These eleven axioms, sometimes called the high school identities,<ref name="BurrisLee">Stanley Burris, Simon Lee, ''Tarski's high school identities'', [[American Mathematical Monthly]], '''100''', (1993), no.3, pp.231–236.</ref> are related to the axioms of an [[Exponential field|exponential ring]].<ref>Strictly speaking an exponential ring has an exponential function ''E'' that takes each element ''x'' to something that acts like ''a''<sup>''x''</sup> for a fixed number ''a''. But a slight generalisation gives the axioms listed here. The lack of axioms about additive inverses means the axioms actually describe an exponential [[Semiring|commutative semiring]].</ref> Tarski's problem then becomes: are there identities involving only addition, multiplication, and exponentiation, that are true for all positive integers, but that cannot be proved using only the axioms 1–11?
| |
| | |
| ==Example of a provable identity==
| |
| | |
| Since the axioms seem to list all the basic facts about the operations in question it is not immediately obvious that there should be anything one can state using only the three operations that is not provably true. However, proving seemingly innocuous statements can require long proofs using only the above eleven axioms. Consider the following proof that (''x'' + 1)<sup>2</sup> = ''x''<sup>2</sup> + 2 · ''x'' + 1:
| |
| :(''x'' + 1)<sup>2</sup>
| |
| := (''x'' + 1)<sup>1 + 1</sup>
| |
| := (''x'' + 1)<sup>1</sup> · (''x'' + 1)<sup>1</sup> by 9.
| |
| := (''x'' + 1) · (''x'' + 1) by 8.
| |
| := (''x'' + 1) · ''x'' + (''x'' + 1) · 1 by 6.
| |
| := ''x'' · (''x'' + 1) + ''x'' + 1 by 4. and 3.
| |
| := ''x'' · ''x'' + ''x'' · 1 + ''x'' · 1 + 1 by 6. and 3.
| |
| := ''x''<sup>1</sup> · ''x''<sup>1</sup> + ''x'' · (1 + 1) + 1 by 8. and 6.
| |
| := ''x''<sup>1 + 1</sup> + ''x'' · 2 + 1 by 9.
| |
| := ''x''<sup>2</sup> + 2 · ''x'' + 1 by 4.
| |
| | |
| Here brackets are omitted when axiom 2. tells us that there is no confusion about grouping.
| |
| | |
| The length of proofs is not an issue; proofs of similar identities to that above for things like (''x'' + ''y'')<sup>100</sup> would take a lot of lines, but would really involve little more than the above proof.
| |
| | |
| ==History of the problem==
| |
| | |
| The list of eleven axioms can be found explicitly written down in the works of [[Richard Dedekind]],<ref>Richard Dedekind, ''Was sind und was sollen die Zahlen?'', 8te unveränderte Aufl. Friedr. Vieweg & Sohn, Braunschweig (1960). English translation: ''What are numbers and what should they be?'' Revised, edited, and translated from the German by [[H. A. Pogorzelski]], W.
| |
| Ryan, and W. Snyder, RIM Monographs in Mathematics, Research Institute for Mathematics, (1995).</ref> although they were obviously known and used by mathematicians long before then. Dedekind was the first, though, who seemed to be asking if these axioms were somehow sufficient to tell us everything we could want to know about the integers. The question was put on a firm footing as a problem in logic and [[model theory]] sometime in the 1960s by Alfred Tarski,<ref name="BurrisLee"/><ref name="Gurevic">R. Gurevič, ''Equational theory of positive numbers with exponentiation'', Proc. Amer. Math. Soc. '''94''' no.1, (1985), pp.135–141.</ref> and by the 1980s it had become known as Tarski's high school algebra problem.
| |
| | |
| ==Solution==
| |
| | |
| In 1980 Alex Wilkie proved that not every identity in question can be proved using the axioms above.<ref>A.J. Wilkie, ''On exponentiation – a solution to Tarski's high school algebra problem'', Connections between model theory and algebraic and analytic geometry, Quad. Mat., '''6''', Dept. Math., Seconda Univ. Napoli, Caserta, (2000), pp.107–129.</ref> He did this by explicitly finding such an identity. By introducing new function symbols corresponding to polynomials that map positive numbers to positive numbers he proved this identity, and showed that these functions together with the eleven axioms above were both sufficient and necessary to prove it. The identity in question is
| |
| :<math>\begin{align}&\left((1+x)^y+(1+x+x^2)^y\right)^x\cdot\left((1+x^3)^x+(1+x^2+x^4)^x\right)^y\\&\quad=\left((1+x)^x+(1+x+x^2)^x\right)^y\cdot\left((1+x^3)^y+(1+x^2+x^4)^y\right)^x.\end{align}</math> | |
| This identity is usually denoted ''W''(''x'',''y'') and is true for all positive integers ''x'' and ''y'', as can be seen by factoring <math>(1-x+x^2)^{xy}</math> out of the second terms; yet it cannot be proved true using the eleven high school axioms.
| |
| | |
| Intuitively, the identity cannot be proved because the high school axioms can't be used to discuss the polynomial <math>1-x+x^2</math>. Reasoning about that polynomial and the subterm <math>-x</math> requires a concept of negation or subtraction, and these are not present in the high school axioms. Lacking this, it is then impossible to use the axioms to manipulate the polynomial and prove true properties about it. Wilkie's results from his paper show, in more formal language, that the "only gap" in the high school axioms is the inability to manipulate polynomials with negative coefficients.
| |
| | |
| ==Generalisations==
| |
| | |
| Wilkie proved that there are statements about the positive integers that cannot be proved using the eleven axioms above and showed what extra information is needed before such statements can be proved. Using [[Nevanlinna theory]] it has also been proved that if one restricts the kinds of exponential one takes then the above eleven axioms are sufficient to prove every true statement.<ref>C. Ward Henson, Lee A. Rubel, ''Some applications of Nevanlinna theory to mathematical logic: Identities of exponential functions'', [[Transactions of the American Mathematical Society]], vol.282 '''1''', (1984), pp.1–32.</ref>
| |
| | |
| Another problem stemming from Wilkie's result that remains open is that which asks what the smallest [[Algebra (ring theory)|algebra]] is for which ''W''(''x'', ''y'') is not true but the eleven axioms above are. In 1985 an algebra with 59 elements was found that satisfied the axioms but for which ''W''(''x'', ''y'') was false.<ref name="Gurevic"/> Smaller such algebras have since been found, and it is now known that the smallest such one must have either 11 or 12 elements.<ref>Jian Zhang, ''Computer search for counterexamples to Wilkie's identity'', Automated Deduction – CADE-20, [[Springer Science+Business Media|Springer]] (2005), pp.441–451, {{doi|10.1007/11532231_32}}.</ref>
| |
| | |
| ==Notes==
| |
| {{Reflist}}
| |
| | |
| ==References==
| |
| * Stanley N. Burris, Karen A. Yeats, ''The saga of the high school identities'', [[Algebra Universalis]] '''52''' no.2–3, (2004), pp. 325–342, {{MathSciNet | id = 2161657}}.
| |
| | |
| {{DEFAULTSORT:Tarski's High School Algebra Problem}}
| |
| [[Category:Universal algebra]]
| |
If an existing Word - Press code is found vulnerable, Word - Press will immediately issue an update for that. Also, you may want to opt for a more professioanl theme if you are planning on showing your site off to a high volume of potential customers each day. In case you have any kind of questions regarding wherever and also tips on how to use wordpress backup plugin, you'll be able to email us from our own web-page. * A community forum for debate of the product together with some other customers in the comments spot. 2- Ask for the designs and graphics that will be provided along with the Word - Press theme. The number of options offered here is overwhelming, but once I took the time to begin to review the video training, I was amazed at how easy it was to create a squeeze page and a membership site.
Creating a website from scratch can be such a pain. Infertility can cause a major setback to the couples due to the inability to conceive. This plugin is a must have for anyone who is serious about using Word - Press. You can up your site's rank with the search engines by simply taking a bit of time with your site. The biggest advantage of using a coupon or deal plugin is that it gives your readers the coupons and deals within minutes of them becoming available.
It is also popular because willing surrogates,as well as egg and sperm donors,are plentiful. Word - Press has different exciting features including a plug-in architecture with a templating system. For a much deeper understanding of simple wordpress themes", check out Upon browsing such, you'll be able to know valuable facts. Every single Theme might be unique, providing several alternatives for webpage owners to reap the benefits of in an effort to instantaneously adjust their web page appear. Converting HTML to Word - Press theme for your website can allow you to enjoy the varied Word - Press features that aid in consistent growth your online business.
It is the convenient service through which professionals either improve the position or keep the ranking intact. Quttera - Quttera describes itself as a 'Saa - S [Software as a Service] web-malware monitoring and alerting solution for websites of any size and complexity. When we talk about functional suitability, Word - Press proves itself as one of the strongest contestant among its other rivals. If you are looking for Hire Wordpress Developer then just get in touch with him. Look for experience: When you are searching for a Word - Press developer you should always look at their experience level.
As a open source platform Wordpress offers distinctive ready to use themes for free along with custom theme support and easy customization. Visit our website to learn more about how you can benefit. Just download it from the website and start using the same. This is because of the customization that works as a keystone for a SEO friendly blogging portal website. Likewise, professional publishers with a multi author and editor setup often find that Word - Press lack basic user and role management capabilities.