Defining square function in $mathbb N$ with only $+$ and the predicate “is a square”












1












$begingroup$


I've got some problems with the following:



Let $C(x)$ be the unary predicate "is a square" (so, for example, $C(4)$, $C(9)$ are true). I want to prove that the function $q(x)=x^2$ is definable in $(mathbb{N}, +, C)$, i.e. there is a formula $phi(x,y)$ such that $phi(a,b) leftrightarrow b=a^2$.



I was thinking about something like $C(x) land z=f(x) $, where $f(x)$ is some expression containing only the + and without any form of recursion.



I can't think of anything, because I can find way to write $x^2$ only using multiplication or using $(x-1)^2$, which I cannot write using only $C$ and $+$. Have you got some suggestions?






logic first-order-logic square-numbers







share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    I've got some problems with the following:



    Let $C(x)$ be the unary predicate "is a square" (so, for example, $C(4)$, $C(9)$ are true). I want to prove that the function $q(x)=x^2$ is definable in $(mathbb{N}, +, C)$, i.e. there is a formula $phi(x,y)$ such that $phi(a,b) leftrightarrow b=a^2$.



    I was thinking about something like $C(x) land z=f(x) $, where $f(x)$ is some expression containing only the + and without any form of recursion.



    I can't think of anything, because I can find way to write $x^2$ only using multiplication or using $(x-1)^2$, which I cannot write using only $C$ and $+$. Have you got some suggestions?






    logic first-order-logic square-numbers







    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      I've got some problems with the following:



      Let $C(x)$ be the unary predicate "is a square" (so, for example, $C(4)$, $C(9)$ are true). I want to prove that the function $q(x)=x^2$ is definable in $(mathbb{N}, +, C)$, i.e. there is a formula $phi(x,y)$ such that $phi(a,b) leftrightarrow b=a^2$.



      I was thinking about something like $C(x) land z=f(x) $, where $f(x)$ is some expression containing only the + and without any form of recursion.



      I can't think of anything, because I can find way to write $x^2$ only using multiplication or using $(x-1)^2$, which I cannot write using only $C$ and $+$. Have you got some suggestions?






      logic first-order-logic square-numbers







      share|cite|improve this question











      $endgroup$




      I've got some problems with the following:



      Let $C(x)$ be the unary predicate "is a square" (so, for example, $C(4)$, $C(9)$ are true). I want to prove that the function $q(x)=x^2$ is definable in $(mathbb{N}, +, C)$, i.e. there is a formula $phi(x,y)$ such that $phi(a,b) leftrightarrow b=a^2$.



      I was thinking about something like $C(x) land z=f(x) $, where $f(x)$ is some expression containing only the + and without any form of recursion.



      I can't think of anything, because I can find way to write $x^2$ only using multiplication or using $(x-1)^2$, which I cannot write using only $C$ and $+$. Have you got some suggestions?






      logic first-order-logic square-numbers




      logic first-order-logic square-numbers






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 17 at 19:42









      Asaf Karagila♦

      308k33441775




      308k33441775










      asked Jan 17 at 19:33









      MarcoMMarcoM

      204




      204






















          1 Answer
          1






          active

          oldest

          votes


















          3












          $begingroup$

          Addition is more expressive than you might think:




          • The ordering $<$ on $mathbb{N}$ is definable from $+$: $a<biffexists c(a+c=b)$.


          • With $<$ in hand, we can talk about "next" - given a number $x$ and a formula $varphi$, the next $y$ after $x$ satisfying $varphi$ is the unique $y$ such that $$varphi(y)wedgeforall z(x<z<yimplies negvarphi(z)).$$


          • Now, can you see a way to tell whether a square $b$ happens to be the square of a given number $a$, based on comparing $a$ with the distance between $b$ and the next square after $b$?







          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            Ah, of course ...
            $endgroup$
            – Henning Makholm
            Jan 17 at 19:57












          Your Answer








          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "69"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });














          draft saved

          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3077414%2fdefining-square-function-in-mathbb-n-with-only-and-the-predicate-is-a-sq%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          3












          $begingroup$

          Addition is more expressive than you might think:




          • The ordering $<$ on $mathbb{N}$ is definable from $+$: $a<biffexists c(a+c=b)$.


          • With $<$ in hand, we can talk about "next" - given a number $x$ and a formula $varphi$, the next $y$ after $x$ satisfying $varphi$ is the unique $y$ such that $$varphi(y)wedgeforall z(x<z<yimplies negvarphi(z)).$$


          • Now, can you see a way to tell whether a square $b$ happens to be the square of a given number $a$, based on comparing $a$ with the distance between $b$ and the next square after $b$?







          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            Ah, of course ...
            $endgroup$
            – Henning Makholm
            Jan 17 at 19:57
















          3












          $begingroup$

          Addition is more expressive than you might think:




          • The ordering $<$ on $mathbb{N}$ is definable from $+$: $a<biffexists c(a+c=b)$.


          • With $<$ in hand, we can talk about "next" - given a number $x$ and a formula $varphi$, the next $y$ after $x$ satisfying $varphi$ is the unique $y$ such that $$varphi(y)wedgeforall z(x<z<yimplies negvarphi(z)).$$


          • Now, can you see a way to tell whether a square $b$ happens to be the square of a given number $a$, based on comparing $a$ with the distance between $b$ and the next square after $b$?







          share|cite|improve this answer









          $endgroup$









          • 1




            $begingroup$
            Ah, of course ...
            $endgroup$
            – Henning Makholm
            Jan 17 at 19:57














          3












          3








          3





          $begingroup$

          Addition is more expressive than you might think:




          • The ordering $<$ on $mathbb{N}$ is definable from $+$: $a<biffexists c(a+c=b)$.


          • With $<$ in hand, we can talk about "next" - given a number $x$ and a formula $varphi$, the next $y$ after $x$ satisfying $varphi$ is the unique $y$ such that $$varphi(y)wedgeforall z(x<z<yimplies negvarphi(z)).$$


          • Now, can you see a way to tell whether a square $b$ happens to be the square of a given number $a$, based on comparing $a$ with the distance between $b$ and the next square after $b$?







          share|cite|improve this answer









          $endgroup$



          Addition is more expressive than you might think:




          • The ordering $<$ on $mathbb{N}$ is definable from $+$: $a<biffexists c(a+c=b)$.


          • With $<$ in hand, we can talk about "next" - given a number $x$ and a formula $varphi$, the next $y$ after $x$ satisfying $varphi$ is the unique $y$ such that $$varphi(y)wedgeforall z(x<z<yimplies negvarphi(z)).$$


          • Now, can you see a way to tell whether a square $b$ happens to be the square of a given number $a$, based on comparing $a$ with the distance between $b$ and the next square after $b$?








          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 17 at 19:56









          Noah SchweberNoah Schweber

          129k10152294




          129k10152294








          • 1




            $begingroup$
            Ah, of course ...
            $endgroup$
            – Henning Makholm
            Jan 17 at 19:57














          • 1




            $begingroup$
            Ah, of course ...
            $endgroup$
            – Henning Makholm
            Jan 17 at 19:57








          1




          1




          $begingroup$
          Ah, of course ...
          $endgroup$
          – Henning Makholm
          Jan 17 at 19:57




          $begingroup$
          Ah, of course ...
          $endgroup$
          – Henning Makholm
          Jan 17 at 19:57


















          draft saved

          draft discarded




















































          Thanks for contributing an answer to Mathematics Stack Exchange!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid …



          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.


          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3077414%2fdefining-square-function-in-mathbb-n-with-only-and-the-predicate-is-a-sq%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          -

          Popular posts from this blog

          Questions related to Moebius Transform of Characteristic Function of the Primes

          Image
          2 $begingroup$ Consider the function defined in (1) below related to the fundamental prime counting function $pi(x)$ . Note that A143519(n) is not multiplicative. (1) $quad f(x)=sumlimits_{n=1}^{x}A143519(n)$ https://oeis.org/A143519 The following plot illustrates $f(x)$ defined in formula (1) above. Figure (1) : Illustration of $f(x)$ defined in formula (1) The integer zeros of $f(x)$ for $xle 10,000$ are listed in (2) below. (2) $quad$ {1,6,9,12,19,30,79,80,81,116,193,201,287,288,291,668,673,679,680,685,686,1109} The zero crossings of $f(x)$ for $xle 10,000$ where $f(x)$ doesn't settle at zero are listed in (3) below. (3) $quad$ {14,21,33,114,115,118,195,286,290,295,442,445,665,667,670,671,678,682} Question (1) : Does $f(x)$ have a finite number of integer zeros, and if so ...
          Read more

          List of scandals in India

          Image
          List of scandals in India From Wikipedia, the free encyclopedia Jump to navigation Jump to search This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by expanding it with reliably sourced entries. This article has multiple issues. Please help improve it or discuss these issues on the talk page . (Learn how and when to remove these template messages) This article has an unclear citation style . The references used may be made clearer with a different or consistent style of citation and footnoting. ( January 2014 ) (Learn how and when to remove this template message) This list is in reverse order. As a consequence it appears to be slanted towards recent events. Please try to keep recent events in historical perspective and add more content related to non-recent events. ( May 2013 ) (Learn how and when to remove this template message) ...
          Read more

          Can not write log (Is /dev/pts mounted?) - openpty in Ubuntu-on-Windows?

          Image
          0 1 What exactly does the error message beginning with E: mean? I assume it has to do with the file structure on Ubuntu-on-Windows, but what exactly? The following package was automatically installed and is no longer required: os-prober Use 'apt-get autoremove' to remove it. The following extra packages will be installed: libxslt1.1 The following NEW packages will be installed: libxslt1.1 xmlstarlet 0 upgraded, 2 newly installed, 0 to remove and 0 not upgraded. Need to get 435 kB of archives. After this operation, 1023 kB of additional disk space will be used. Get:1 http://archive.ubuntu.com/ubuntu/ trusty/main libxslt1.1 amd64 1.1.28-2build1 [145 kB] Get:2 http://archive.ubuntu.com/ubuntu/ trusty/universe xmlstarlet amd64 1.5.0-1 [290 kB] ...
          Read more