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?Sufficient Conditions for proving $V$ is a vector spaceCan a sequence whose final term is an axiom, be considered a formal proof?Why is Zorn's Lemma called a lemma?Axiom Systems and Formal SystemsWhen the mathematical community consider the inclusion of a new axiom?.Why is the axiom of choice not taught from the start to mathematics undergraduates?Why 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

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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?Sufficient Conditions for proving $V$ is a vector spaceCan a sequence whose final term is an axiom, be considered a formal proof?Why is Zorn's Lemma called a lemma?Axiom Systems and Formal SystemsWhen the mathematical community consider the inclusion of a new axiom?.Why is the axiom of choice not taught from the start to mathematics undergraduates?Why 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













16












$begingroup$


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
    Apr 8 at 13:16






  • 2




    $begingroup$
    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
    Apr 8 at 13:26






  • 2




    $begingroup$
    Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
    $endgroup$
    – Asaf Karagila
    Apr 8 at 14:38






  • 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
    Apr 8 at 14:41






  • 2




    $begingroup$
    I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
    $endgroup$
    – Bram28
    Apr 8 at 16:05















16












$begingroup$


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
    Apr 8 at 13:16






  • 2




    $begingroup$
    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
    Apr 8 at 13:26






  • 2




    $begingroup$
    Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
    $endgroup$
    – Asaf Karagila
    Apr 8 at 14:38






  • 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
    Apr 8 at 14:41






  • 2




    $begingroup$
    I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
    $endgroup$
    – Bram28
    Apr 8 at 16:05













16












16








16


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 soft-question math-history axioms






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Apr 13 at 19:09


























community wiki





4 revs, 3 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
    Apr 8 at 13:16






  • 2




    $begingroup$
    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
    Apr 8 at 13:26






  • 2




    $begingroup$
    Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
    $endgroup$
    – Asaf Karagila
    Apr 8 at 14:38






  • 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
    Apr 8 at 14:41






  • 2




    $begingroup$
    I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
    $endgroup$
    – Bram28
    Apr 8 at 16:05












  • 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
    Apr 8 at 13:16






  • 2




    $begingroup$
    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
    Apr 8 at 13:26






  • 2




    $begingroup$
    Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
    $endgroup$
    – Asaf Karagila
    Apr 8 at 14:38






  • 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
    Apr 8 at 14:41






  • 2




    $begingroup$
    I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
    $endgroup$
    – Bram28
    Apr 8 at 16:05







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
Apr 8 at 13:16




$begingroup$
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
Apr 8 at 13:16




2




2




$begingroup$
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
Apr 8 at 13:26




$begingroup$
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
Apr 8 at 13:26




2




2




$begingroup$
Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
$endgroup$
– Asaf Karagila
Apr 8 at 14:38




$begingroup$
Eyal, the main point here is that "axiom" is a social agreement, rather than a mathematical definition.
$endgroup$
– Asaf Karagila
Apr 8 at 14:38




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
Apr 8 at 14:41




$begingroup$
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
Apr 8 at 14:41




2




2




$begingroup$
I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
$endgroup$
– Bram28
Apr 8 at 16:05




$begingroup$
I believe all the the axioms from Peano Arithmetic (PA) can be derived from ZF?
$endgroup$
– Bram28
Apr 8 at 16:05










3 Answers
3






active

oldest

votes


















26












$begingroup$

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











$endgroup$




















    17












    $begingroup$

    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
        Apr 8 at 13:57






      • 8




        $begingroup$
        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
        Apr 8 at 20:03











      Your Answer








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      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      26












      $begingroup$

      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











      $endgroup$

















        26












        $begingroup$

        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











        $endgroup$















          26












          26








          26





          $begingroup$

          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 Apr 8 at 18:18


























          community wiki





          roy smith






















              17












              $begingroup$

              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











              $endgroup$

















                17












                $begingroup$

                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











                $endgroup$















                  17












                  17








                  17





                  $begingroup$

                  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 Apr 8 at 14:26


























                  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
                        Apr 8 at 13:57






                      • 8




                        $begingroup$
                        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
                        Apr 8 at 20:03















                      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
                        Apr 8 at 13:57






                      • 8




                        $begingroup$
                        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
                        Apr 8 at 20:03













                      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 Apr 8 at 13:07


























                      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
                        Apr 8 at 13:57






                      • 8




                        $begingroup$
                        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
                        Apr 8 at 20:03
















                      • $begingroup$
                        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
                        Apr 8 at 13:57






                      • 8




                        $begingroup$
                        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
                        Apr 8 at 20:03















                      $begingroup$
                      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
                      Apr 8 at 13:57




                      $begingroup$
                      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
                      Apr 8 at 13:57




                      8




                      8




                      $begingroup$
                      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
                      Apr 8 at 20:03




                      $begingroup$
                      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
                      Apr 8 at 20:03

















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