Was there ever an axiom rendered a theorem?












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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 has been shown to be derived from other axiom(s), therefore rendering the statement a theorem?










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$endgroup$








  • 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
    2 days ago






  • 1




    $begingroup$
    I think the history of $C^*$-algebras is somewhat like that. In the early days a $C^*$-algebra was defined through a whole laundry list of properties. More and more of these where shown to be consequences of some of the others. So today the list of defining properties is quite short and most of the originally defining properties are now theorems.
    $endgroup$
    – quarague
    2 days ago






  • 2




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






  • 2




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
















14












$begingroup$


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 has been shown to be derived from other axiom(s), therefore rendering the statement a theorem?










share|cite|improve this question











$endgroup$








  • 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
    2 days ago






  • 1




    $begingroup$
    I think the history of $C^*$-algebras is somewhat like that. In the early days a $C^*$-algebra was defined through a whole laundry list of properties. More and more of these where shown to be consequences of some of the others. So today the list of defining properties is quite short and most of the originally defining properties are now theorems.
    $endgroup$
    – quarague
    2 days ago






  • 2




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






  • 2




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














14












14








14


3



$begingroup$


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 has been shown to be derived from other axiom(s), therefore rendering the statement a theorem?










share|cite|improve this question











$endgroup$




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 has been shown to be derived from other axiom(s), therefore rendering the statement a theorem?







logic math-history axioms






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 13 hours ago


























community wiki





3 revs, 2 users 57%
Eyal Roth









  • 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
    2 days ago






  • 1




    $begingroup$
    I think the history of $C^*$-algebras is somewhat like that. In the early days a $C^*$-algebra was defined through a whole laundry list of properties. More and more of these where shown to be consequences of some of the others. So today the list of defining properties is quite short and most of the originally defining properties are now theorems.
    $endgroup$
    – quarague
    2 days ago






  • 2




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






  • 2




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






  • 1




    $begingroup$
    I think the history of $C^*$-algebras is somewhat like that. In the early days a $C^*$-algebra was defined through a whole laundry list of properties. More and more of these where shown to be consequences of some of the others. So today the list of defining properties is quite short and most of the originally defining properties are now theorems.
    $endgroup$
    – quarague
    2 days ago






  • 2




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






  • 2




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




1




1




$begingroup$
I think the history of $C^*$-algebras is somewhat like that. In the early days a $C^*$-algebra was defined through a whole laundry list of properties. More and more of these where shown to be consequences of some of the others. So today the list of defining properties is quite short and most of the originally defining properties are now theorems.
$endgroup$
– quarague
2 days ago




$begingroup$
I think the history of $C^*$-algebras is somewhat like that. In the early days a $C^*$-algebra was defined through a whole laundry list of properties. More and more of these where shown to be consequences of some of the others. So today the list of defining properties is quite short and most of the originally defining properties are now theorems.
$endgroup$
– quarague
2 days ago




2




2




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




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




2




2




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




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










3 Answers
3






active

oldest

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22












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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











$endgroup$





















    14












    $begingroup$

    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











    $endgroup$





















      10












      $begingroup$

      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
        2 days ago






      • 6




        $begingroup$
        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
        2 days ago












      Your Answer





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      3 Answers
      3






      active

      oldest

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      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      22












      $begingroup$

      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











      $endgroup$


















        22












        $begingroup$

        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











        $endgroup$
















          22












          22








          22





          $begingroup$

          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 2 days ago


























          community wiki





          roy smith
























              14












              $begingroup$

              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











              $endgroup$


















                14












                $begingroup$

                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











                $endgroup$
















                  14












                  14








                  14





                  $begingroup$

                  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 2 days ago


























                  community wiki





                  J.G.
























                      10












                      $begingroup$

                      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
                        2 days ago






                      • 6




                        $begingroup$
                        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
                        2 days ago
















                      10












                      $begingroup$

                      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
                        2 days ago






                      • 6




                        $begingroup$
                        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
                        2 days ago














                      10












                      10








                      10





                      $begingroup$

                      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 2 days 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
                        2 days ago






                      • 6




                        $begingroup$
                        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
                        2 days 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
                        2 days ago






                      • 6




                        $begingroup$
                        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
                        2 days 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
                      2 days 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
                      2 days ago




                      6




                      6




                      $begingroup$
                      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
                      2 days 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
                      2 days ago


















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