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Was there ever an axiom rendered a theorem?


How can the axiom of choice be called “axiom” if it is false in Cohen's model?What is the difference between an axiom and a postulate?Why is Zorn's Lemma called a lemma?Can a sequence whose final term is an axiom, be considered a formal proof?Axiom Systems and Formal SystemsWhen the mathematical community consider the inclusion of a new axiom?.A Concept Which Has Been 'Specialized' In the Course of HistoryWhy is the Generalization Axiom considered a Pure Axiom?Euclid's Elements missing axiom of M. Pasch examplesZermelo-Fraenkel set theory and Hilbert's axioms for geometryWhich is the first theorem in Euclid's Elements which uses Pasch's Axiom?Axiom of Choice — Why is it an axiom and not a theorem?Is consistency an axiom of mathematics?Redunduncy of Pasch's Axiom of Hilbert's Foundations of Geometry













11












$begingroup$


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?










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.
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  • 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
    17 hours ago






  • 1




    $begingroup$
    Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
    $endgroup$
    – Asaf Karagila
    16 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
    16 hours ago






  • 2




    $begingroup$
    I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
    $endgroup$
    – Bram28
    14 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
    14 hours ago
















11












$begingroup$


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?










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
    17 hours ago






  • 1




    $begingroup$
    Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
    $endgroup$
    – Asaf Karagila
    16 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
    16 hours ago






  • 2




    $begingroup$
    I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
    $endgroup$
    – Bram28
    14 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
    14 hours ago














11












11








11


2



$begingroup$


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?










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






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 16 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 17 hours ago









Eyal RothEyal Roth

1646




1646




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
    17 hours ago






  • 1




    $begingroup$
    Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
    $endgroup$
    – Asaf Karagila
    16 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
    16 hours ago






  • 2




    $begingroup$
    I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
    $endgroup$
    – Bram28
    14 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
    14 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
    17 hours ago






  • 1




    $begingroup$
    Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
    $endgroup$
    – Asaf Karagila
    16 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
    16 hours ago






  • 2




    $begingroup$
    I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
    $endgroup$
    – Bram28
    14 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
    14 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
17 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
17 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
16 hours ago




$begingroup$
Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
$endgroup$
– Asaf Karagila
16 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
16 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
16 hours ago




2




2




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




$begingroup$
I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
$endgroup$
– Bram28
14 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
14 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
14 hours ago










3 Answers
3






active

oldest

votes


















8












$begingroup$

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
    16 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
    10 hours ago



















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









    $endgroup$














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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      8












      $begingroup$

      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
        16 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
        10 hours ago
















      8












      $begingroup$

      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
        16 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
        10 hours ago














      8












      8








      8





      $begingroup$

      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 17 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
        16 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
        10 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
        16 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
        10 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
      16 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
      16 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
      10 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
      10 hours ago











      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 16 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 12 hours ago









                  roy smithroy smith

                  999711




                  999711






















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