Was there ever an axiom rendered a theorem?












10












$begingroup$


In the history of mathematics, are there notable examples of theorems which have been first considered axioms?



Alternatively, was there any statement first considered an axiom that later have been shown to be derived from other axiom(s), therefore rendering the statement a theorem?






logic math-history axioms







share|cite|improve this question









New contributor




Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$








  • 2




    $begingroup$
    All axioms are theorems, math.stackexchange.com/questions/1242021/… also of interest might be math.stackexchange.com/questions/258346/… and math.stackexchange.com/questions/1383457/… and math.stackexchange.com/questions/1131748/… might also be relevant.
    $endgroup$
    – Asaf Karagila♦
    12 hours ago






  • 1




    $begingroup$
    Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
    $endgroup$
    – Asaf Karagila♦
    10 hours ago






  • 4




    $begingroup$
    And indeed the Axiom of Choice is taken as an axiom and is reduced to a Theorem when assuming ZF+ZL, or or even to a false statement when assuming ZF+AD.
    $endgroup$
    – Asaf Karagila♦
    10 hours ago






  • 2




    $begingroup$
    I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
    $endgroup$
    – Bram28
    9 hours ago






  • 1




    $begingroup$
    @CliveNewstead: The axiom of infinity is ill-defined if you have not yet defined the empty set (because the empty set symbol appears in the axiom of infinity).
    $endgroup$
    – Kevin
    8 hours ago
















10












$begingroup$


In the history of mathematics, are there notable examples of theorems which have been first considered axioms?



Alternatively, was there any statement first considered an axiom that later have been shown to be derived from other axiom(s), therefore rendering the statement a theorem?






logic math-history axioms







share|cite|improve this question









New contributor




Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$








  • 2




    $begingroup$
    All axioms are theorems, math.stackexchange.com/questions/1242021/… also of interest might be math.stackexchange.com/questions/258346/… and math.stackexchange.com/questions/1383457/… and math.stackexchange.com/questions/1131748/… might also be relevant.
    $endgroup$
    – Asaf Karagila♦
    12 hours ago






  • 1




    $begingroup$
    Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
    $endgroup$
    – Asaf Karagila♦
    10 hours ago






  • 4




    $begingroup$
    And indeed the Axiom of Choice is taken as an axiom and is reduced to a Theorem when assuming ZF+ZL, or or even to a false statement when assuming ZF+AD.
    $endgroup$
    – Asaf Karagila♦
    10 hours ago






  • 2




    $begingroup$
    I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
    $endgroup$
    – Bram28
    9 hours ago






  • 1




    $begingroup$
    @CliveNewstead: The axiom of infinity is ill-defined if you have not yet defined the empty set (because the empty set symbol appears in the axiom of infinity).
    $endgroup$
    – Kevin
    8 hours ago














10












10








10


1



$begingroup$


In the history of mathematics, are there notable examples of theorems which have been first considered axioms?



Alternatively, was there any statement first considered an axiom that later have been shown to be derived from other axiom(s), therefore rendering the statement a theorem?






logic math-history axioms







share|cite|improve this question









New contributor




Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




In the history of mathematics, are there notable examples of theorems which have been first considered axioms?



Alternatively, was there any statement first considered an axiom that later have been shown to be derived from other axiom(s), therefore rendering the statement a theorem?






logic math-history axioms




logic math-history axioms






share|cite|improve this question









New contributor




Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









New contributor




Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question








edited 11 hours ago







Eyal Roth













New contributor




Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked 12 hours ago









Eyal RothEyal Roth

1595




1595




New contributor




Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Eyal Roth is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








  • 2




    $begingroup$
    All axioms are theorems, math.stackexchange.com/questions/1242021/… also of interest might be math.stackexchange.com/questions/258346/… and math.stackexchange.com/questions/1383457/… and math.stackexchange.com/questions/1131748/… might also be relevant.
    $endgroup$
    – Asaf Karagila♦
    12 hours ago






  • 1




    $begingroup$
    Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
    $endgroup$
    – Asaf Karagila♦
    10 hours ago






  • 4




    $begingroup$
    And indeed the Axiom of Choice is taken as an axiom and is reduced to a Theorem when assuming ZF+ZL, or or even to a false statement when assuming ZF+AD.
    $endgroup$
    – Asaf Karagila♦
    10 hours ago






  • 2




    $begingroup$
    I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
    $endgroup$
    – Bram28
    9 hours ago






  • 1




    $begingroup$
    @CliveNewstead: The axiom of infinity is ill-defined if you have not yet defined the empty set (because the empty set symbol appears in the axiom of infinity).
    $endgroup$
    – Kevin
    8 hours ago














  • 2




    $begingroup$
    All axioms are theorems, math.stackexchange.com/questions/1242021/… also of interest might be math.stackexchange.com/questions/258346/… and math.stackexchange.com/questions/1383457/… and math.stackexchange.com/questions/1131748/… might also be relevant.
    $endgroup$
    – Asaf Karagila♦
    12 hours ago






  • 1




    $begingroup$
    Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
    $endgroup$
    – Asaf Karagila♦
    10 hours ago






  • 4




    $begingroup$
    And indeed the Axiom of Choice is taken as an axiom and is reduced to a Theorem when assuming ZF+ZL, or or even to a false statement when assuming ZF+AD.
    $endgroup$
    – Asaf Karagila♦
    10 hours ago






  • 2




    $begingroup$
    I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
    $endgroup$
    – Bram28
    9 hours ago






  • 1




    $begingroup$
    @CliveNewstead: The axiom of infinity is ill-defined if you have not yet defined the empty set (because the empty set symbol appears in the axiom of infinity).
    $endgroup$
    – Kevin
    8 hours ago








2




2




$begingroup$
All axioms are theorems, math.stackexchange.com/questions/1242021/… also of interest might be math.stackexchange.com/questions/258346/… and math.stackexchange.com/questions/1383457/… and math.stackexchange.com/questions/1131748/… might also be relevant.
$endgroup$
– Asaf Karagila♦
12 hours ago




$begingroup$
All axioms are theorems, math.stackexchange.com/questions/1242021/… also of interest might be math.stackexchange.com/questions/258346/… and math.stackexchange.com/questions/1383457/… and math.stackexchange.com/questions/1131748/… might also be relevant.
$endgroup$
– Asaf Karagila♦
12 hours ago




1




1




$begingroup$
Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
$endgroup$
– Asaf Karagila♦
10 hours ago




$begingroup$
Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
$endgroup$
– Asaf Karagila♦
10 hours ago




4




4




$begingroup$
And indeed the Axiom of Choice is taken as an axiom and is reduced to a Theorem when assuming ZF+ZL, or or even to a false statement when assuming ZF+AD.
$endgroup$
– Asaf Karagila♦
10 hours ago




$begingroup$
And indeed the Axiom of Choice is taken as an axiom and is reduced to a Theorem when assuming ZF+ZL, or or even to a false statement when assuming ZF+AD.
$endgroup$
– Asaf Karagila♦
10 hours ago




2




2




$begingroup$
I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
$endgroup$
– Bram28
9 hours ago




$begingroup$
I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
$endgroup$
– Bram28
9 hours ago




1




1




$begingroup$
@CliveNewstead: The axiom of infinity is ill-defined if you have not yet defined the empty set (because the empty set symbol appears in the axiom of infinity).
$endgroup$
– Kevin
8 hours ago




$begingroup$
@CliveNewstead: The axiom of infinity is ill-defined if you have not yet defined the empty set (because the empty set symbol appears in the axiom of infinity).
$endgroup$
– Kevin
8 hours ago










3 Answers
3






active

oldest

votes


















8












$begingroup$

Fraenkel introduced the axiom schema of replacement to set theory. This implied the axiom schema of comprehension, and allowed the empty set and unordered pair axioms to follow from the axiom of infinity. (Note Zermelo set theory includes the axiom of choice whereas ZF does not, so Zermelo+replacement is ZFC.) The "deleted" axioms are typically listed when describing ZF(C), partly so people realise they're in Zermelo set theory, partly for easier comparisons with other set theories of interest.






share|cite|improve this answer









$endgroup$





















    7












    $begingroup$

    The most famous example I know is that of Hilbert's axiom II.4 for the linear ordering of points on a line, for Euclidean geometry, proven to be superfluous by E.H. Moore. See this wikipedia article, especially "Hilbert's discarded axiom". https://en.wikipedia.org/wiki/Hilbert%27s_axioms



    In the article of Moore linked there, it is stated that also axiom I.4 is superfluous.



    http://www.ams.org/journals/tran/1902-003-01/S0002-9947-1902-1500592-8/S0002-9947-1902-1500592-8.pdf






    share|cite|improve this answer









    $endgroup$





















      6












      $begingroup$

      Yes, everywhere. What is an axiom from one theory can be a theorem in another.



      Euclid's fifth postulate can be replaced by the statement that the angles on the inside of each triangle add up to $pi$ radians.



      Another notable example is the axiom of choice, which is equivalent in some axiomatic systems to Zorn's Lemma.



      Also, watch this Feynman clip.






      share|cite|improve this answer











      $endgroup$













      • $begingroup$
        That is an interesting clip (and I love the accent). If I understand correctly, Feynman discusses axioms which have bi-directional relations; i.e, one can be deduced from the other and vice-versa; or perhaps, any two of three axioms can imply the third. I'm rather interested in cases of uni-directional axioms which have been discovered to be implied from another axiom or set of axioms.
        $endgroup$
        – Eyal Roth
        11 hours ago






      • 1




        $begingroup$
        These cases are considered to be alternative statements of the same axiom. You choose whichever one you want as an axiom and prove the other. If you have an axiom you suspect is redundant you might find a way to prove one of the statements, or you might find a statement that is obvious enough that people accept it as an axiom. In both of these cases, the axiom has been proven to be independent of the others. I think OP wants a case where a statement was thought to be independent of the other axioms of a subject and was shown to be a consequence of them.
        $endgroup$
        – Ross Millikan
        5 hours ago












      Your Answer





      StackExchange.ifUsing("editor", function () {
      return StackExchange.using("mathjaxEditing", function () {
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      });
      });
      }, "mathjax-editing");

      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
      });


      }
      });






      Eyal Roth is a new contributor. Be nice, and check out our Code of Conduct.










      draft saved

      draft discarded


















      StackExchange.ready(
      function () {
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3179606%2fwas-there-ever-an-axiom-rendered-a-theorem%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      8












      $begingroup$

      Fraenkel introduced the axiom schema of replacement to set theory. This implied the axiom schema of comprehension, and allowed the empty set and unordered pair axioms to follow from the axiom of infinity. (Note Zermelo set theory includes the axiom of choice whereas ZF does not, so Zermelo+replacement is ZFC.) The "deleted" axioms are typically listed when describing ZF(C), partly so people realise they're in Zermelo set theory, partly for easier comparisons with other set theories of interest.






      share|cite|improve this answer









      $endgroup$


















        8












        $begingroup$

        Fraenkel introduced the axiom schema of replacement to set theory. This implied the axiom schema of comprehension, and allowed the empty set and unordered pair axioms to follow from the axiom of infinity. (Note Zermelo set theory includes the axiom of choice whereas ZF does not, so Zermelo+replacement is ZFC.) The "deleted" axioms are typically listed when describing ZF(C), partly so people realise they're in Zermelo set theory, partly for easier comparisons with other set theories of interest.






        share|cite|improve this answer









        $endgroup$
















          8












          8








          8





          $begingroup$

          Fraenkel introduced the axiom schema of replacement to set theory. This implied the axiom schema of comprehension, and allowed the empty set and unordered pair axioms to follow from the axiom of infinity. (Note Zermelo set theory includes the axiom of choice whereas ZF does not, so Zermelo+replacement is ZFC.) The "deleted" axioms are typically listed when describing ZF(C), partly so people realise they're in Zermelo set theory, partly for easier comparisons with other set theories of interest.






          share|cite|improve this answer









          $endgroup$



          Fraenkel introduced the axiom schema of replacement to set theory. This implied the axiom schema of comprehension, and allowed the empty set and unordered pair axioms to follow from the axiom of infinity. (Note Zermelo set theory includes the axiom of choice whereas ZF does not, so Zermelo+replacement is ZFC.) The "deleted" axioms are typically listed when describing ZF(C), partly so people realise they're in Zermelo set theory, partly for easier comparisons with other set theories of interest.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 10 hours ago









          J.G.J.G.

          33.1k23251




          33.1k23251























              7












              $begingroup$

              The most famous example I know is that of Hilbert's axiom II.4 for the linear ordering of points on a line, for Euclidean geometry, proven to be superfluous by E.H. Moore. See this wikipedia article, especially "Hilbert's discarded axiom". https://en.wikipedia.org/wiki/Hilbert%27s_axioms



              In the article of Moore linked there, it is stated that also axiom I.4 is superfluous.



              http://www.ams.org/journals/tran/1902-003-01/S0002-9947-1902-1500592-8/S0002-9947-1902-1500592-8.pdf






              share|cite|improve this answer









              $endgroup$


















                7












                $begingroup$

                The most famous example I know is that of Hilbert's axiom II.4 for the linear ordering of points on a line, for Euclidean geometry, proven to be superfluous by E.H. Moore. See this wikipedia article, especially "Hilbert's discarded axiom". https://en.wikipedia.org/wiki/Hilbert%27s_axioms



                In the article of Moore linked there, it is stated that also axiom I.4 is superfluous.



                http://www.ams.org/journals/tran/1902-003-01/S0002-9947-1902-1500592-8/S0002-9947-1902-1500592-8.pdf






                share|cite|improve this answer









                $endgroup$
















                  7












                  7








                  7





                  $begingroup$

                  The most famous example I know is that of Hilbert's axiom II.4 for the linear ordering of points on a line, for Euclidean geometry, proven to be superfluous by E.H. Moore. See this wikipedia article, especially "Hilbert's discarded axiom". https://en.wikipedia.org/wiki/Hilbert%27s_axioms



                  In the article of Moore linked there, it is stated that also axiom I.4 is superfluous.



                  http://www.ams.org/journals/tran/1902-003-01/S0002-9947-1902-1500592-8/S0002-9947-1902-1500592-8.pdf






                  share|cite|improve this answer









                  $endgroup$



                  The most famous example I know is that of Hilbert's axiom II.4 for the linear ordering of points on a line, for Euclidean geometry, proven to be superfluous by E.H. Moore. See this wikipedia article, especially "Hilbert's discarded axiom". https://en.wikipedia.org/wiki/Hilbert%27s_axioms



                  In the article of Moore linked there, it is stated that also axiom I.4 is superfluous.



                  http://www.ams.org/journals/tran/1902-003-01/S0002-9947-1902-1500592-8/S0002-9947-1902-1500592-8.pdf







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 7 hours ago









                  roy smithroy smith

                  999711




                  999711























                      6












                      $begingroup$

                      Yes, everywhere. What is an axiom from one theory can be a theorem in another.



                      Euclid's fifth postulate can be replaced by the statement that the angles on the inside of each triangle add up to $pi$ radians.



                      Another notable example is the axiom of choice, which is equivalent in some axiomatic systems to Zorn's Lemma.



                      Also, watch this Feynman clip.






                      share|cite|improve this answer











                      $endgroup$













                      • $begingroup$
                        That is an interesting clip (and I love the accent). If I understand correctly, Feynman discusses axioms which have bi-directional relations; i.e, one can be deduced from the other and vice-versa; or perhaps, any two of three axioms can imply the third. I'm rather interested in cases of uni-directional axioms which have been discovered to be implied from another axiom or set of axioms.
                        $endgroup$
                        – Eyal Roth
                        11 hours ago






                      • 1




                        $begingroup$
                        These cases are considered to be alternative statements of the same axiom. You choose whichever one you want as an axiom and prove the other. If you have an axiom you suspect is redundant you might find a way to prove one of the statements, or you might find a statement that is obvious enough that people accept it as an axiom. In both of these cases, the axiom has been proven to be independent of the others. I think OP wants a case where a statement was thought to be independent of the other axioms of a subject and was shown to be a consequence of them.
                        $endgroup$
                        – Ross Millikan
                        5 hours ago
















                      6












                      $begingroup$

                      Yes, everywhere. What is an axiom from one theory can be a theorem in another.



                      Euclid's fifth postulate can be replaced by the statement that the angles on the inside of each triangle add up to $pi$ radians.



                      Another notable example is the axiom of choice, which is equivalent in some axiomatic systems to Zorn's Lemma.



                      Also, watch this Feynman clip.






                      share|cite|improve this answer











                      $endgroup$













                      • $begingroup$
                        That is an interesting clip (and I love the accent). If I understand correctly, Feynman discusses axioms which have bi-directional relations; i.e, one can be deduced from the other and vice-versa; or perhaps, any two of three axioms can imply the third. I'm rather interested in cases of uni-directional axioms which have been discovered to be implied from another axiom or set of axioms.
                        $endgroup$
                        – Eyal Roth
                        11 hours ago






                      • 1




                        $begingroup$
                        These cases are considered to be alternative statements of the same axiom. You choose whichever one you want as an axiom and prove the other. If you have an axiom you suspect is redundant you might find a way to prove one of the statements, or you might find a statement that is obvious enough that people accept it as an axiom. In both of these cases, the axiom has been proven to be independent of the others. I think OP wants a case where a statement was thought to be independent of the other axioms of a subject and was shown to be a consequence of them.
                        $endgroup$
                        – Ross Millikan
                        5 hours ago














                      6












                      6








                      6





                      $begingroup$

                      Yes, everywhere. What is an axiom from one theory can be a theorem in another.



                      Euclid's fifth postulate can be replaced by the statement that the angles on the inside of each triangle add up to $pi$ radians.



                      Another notable example is the axiom of choice, which is equivalent in some axiomatic systems to Zorn's Lemma.



                      Also, watch this Feynman clip.






                      share|cite|improve this answer











                      $endgroup$



                      Yes, everywhere. What is an axiom from one theory can be a theorem in another.



                      Euclid's fifth postulate can be replaced by the statement that the angles on the inside of each triangle add up to $pi$ radians.



                      Another notable example is the axiom of choice, which is equivalent in some axiomatic systems to Zorn's Lemma.



                      Also, watch this Feynman clip.







                      share|cite|improve this answer














                      share|cite|improve this answer



                      share|cite|improve this answer








                      edited 12 hours ago


























                      community wiki





                      2 revs
                      Shaun













                      • $begingroup$
                        That is an interesting clip (and I love the accent). If I understand correctly, Feynman discusses axioms which have bi-directional relations; i.e, one can be deduced from the other and vice-versa; or perhaps, any two of three axioms can imply the third. I'm rather interested in cases of uni-directional axioms which have been discovered to be implied from another axiom or set of axioms.
                        $endgroup$
                        – Eyal Roth
                        11 hours ago






                      • 1




                        $begingroup$
                        These cases are considered to be alternative statements of the same axiom. You choose whichever one you want as an axiom and prove the other. If you have an axiom you suspect is redundant you might find a way to prove one of the statements, or you might find a statement that is obvious enough that people accept it as an axiom. In both of these cases, the axiom has been proven to be independent of the others. I think OP wants a case where a statement was thought to be independent of the other axioms of a subject and was shown to be a consequence of them.
                        $endgroup$
                        – Ross Millikan
                        5 hours ago


















                      • $begingroup$
                        That is an interesting clip (and I love the accent). If I understand correctly, Feynman discusses axioms which have bi-directional relations; i.e, one can be deduced from the other and vice-versa; or perhaps, any two of three axioms can imply the third. I'm rather interested in cases of uni-directional axioms which have been discovered to be implied from another axiom or set of axioms.
                        $endgroup$
                        – Eyal Roth
                        11 hours ago






                      • 1




                        $begingroup$
                        These cases are considered to be alternative statements of the same axiom. You choose whichever one you want as an axiom and prove the other. If you have an axiom you suspect is redundant you might find a way to prove one of the statements, or you might find a statement that is obvious enough that people accept it as an axiom. In both of these cases, the axiom has been proven to be independent of the others. I think OP wants a case where a statement was thought to be independent of the other axioms of a subject and was shown to be a consequence of them.
                        $endgroup$
                        – Ross Millikan
                        5 hours ago
















                      $begingroup$
                      That is an interesting clip (and I love the accent). If I understand correctly, Feynman discusses axioms which have bi-directional relations; i.e, one can be deduced from the other and vice-versa; or perhaps, any two of three axioms can imply the third. I'm rather interested in cases of uni-directional axioms which have been discovered to be implied from another axiom or set of axioms.
                      $endgroup$
                      – Eyal Roth
                      11 hours ago




                      $begingroup$
                      That is an interesting clip (and I love the accent). If I understand correctly, Feynman discusses axioms which have bi-directional relations; i.e, one can be deduced from the other and vice-versa; or perhaps, any two of three axioms can imply the third. I'm rather interested in cases of uni-directional axioms which have been discovered to be implied from another axiom or set of axioms.
                      $endgroup$
                      – Eyal Roth
                      11 hours ago




                      1




                      1




                      $begingroup$
                      These cases are considered to be alternative statements of the same axiom. You choose whichever one you want as an axiom and prove the other. If you have an axiom you suspect is redundant you might find a way to prove one of the statements, or you might find a statement that is obvious enough that people accept it as an axiom. In both of these cases, the axiom has been proven to be independent of the others. I think OP wants a case where a statement was thought to be independent of the other axioms of a subject and was shown to be a consequence of them.
                      $endgroup$
                      – Ross Millikan
                      5 hours ago




                      $begingroup$
                      These cases are considered to be alternative statements of the same axiom. You choose whichever one you want as an axiom and prove the other. If you have an axiom you suspect is redundant you might find a way to prove one of the statements, or you might find a statement that is obvious enough that people accept it as an axiom. In both of these cases, the axiom has been proven to be independent of the others. I think OP wants a case where a statement was thought to be independent of the other axioms of a subject and was shown to be a consequence of them.
                      $endgroup$
                      – Ross Millikan
                      5 hours ago










                      Eyal Roth is a new contributor. Be nice, and check out our Code of Conduct.










                      draft saved

                      draft discarded


















                      Eyal Roth is a new contributor. Be nice, and check out our Code of Conduct.













                      Eyal Roth is a new contributor. Be nice, and check out our Code of Conduct.












                      Eyal Roth is a new contributor. Be nice, and check out our Code of Conduct.
















                      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%2f3179606%2fwas-there-ever-an-axiom-rendered-a-theorem%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

                      A CLEAN and SIMPLE way to add appendices to Table of Contents and bookmarks

                      Image
                      0 I found a lot of questions abount appendices and ToC. Many users want appendices to be grouped in an Appendix part, however some problems arise with ToC, hyperref, PDF viewer bookmarks, and so on. There are different solutions which require extra packages, command patching and other extra code, however none of them satisfies me. I almost found an easy way to accomplish a good result, where appendices are added to bookmarks in the right way and hyperref links point to the right page. However, the number of the "Appendix" part page is wrong (it's the number of appendix A). Is there any EASY way to fix that? This is a MWE: documentclass{book} usepackage[nottoc,notlot,notlof]{tocbibind} usepackage{hyperref} begin{document} frontmatter tableofcontents mainmatter part{First} chapter{...
                      Read more

                      Calculate evaluation metrics using cross_val_predict sklearn

                      Image
                      1 In the sklearn.model_selection.cross_val_predict page it is stated: Generate cross-validated estimates for each input data point. It is not appropriate to pass these predictions into an evaluation metric. Can someone explain what does it mean? If this gives estimate of Y (y prediction) for every Y (true Y), why can't I calculate metrics such as RMSE or coefficient of determination using these results? python scikit-learn cross-validation share | improve this question edited Nov 28 '18 at 17:52 desertnaut 20.3k 7 43 79 ...
                      Read more

                      Insert data from modal to MySQL (multiple modal on website)

                      Image
                      0 I have problem with inserting data to MySQL from modal. My modal: <a href="#" class="badge badge-pill badge-success">6 komentarzy</a> <a href="#" class="badge badge-pill badge-danger">brak komentarzy</a> <a data-toggle="modal" href="#add_desk_comm_{$desk_['desk_id']}" data-target="#add_desk_comm_{$desk_['desk_id']}" class="ediiit">(dodaj)</a> <div class="modal fade" id="add_desk_comm_{$desk_['desk_id']}" tabindex="-1" role="dialog" aria-labelledby="edit_printer" aria-hidden="true"> <div class="modal-dialog" role="document"> ...
                      Read more