Is an axiom a proof?

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From this comments discussion on Philosophy.SE:




"Check out formal logic resources - I'm not going to dig them out for you. Alternatively ask on Math.SE. An 'axiom is a proof' is a definition in formal logic - and not an axiom. In philosophical logic, you can dispute this - but then there you can dispute what counts as proof."




This comment did not align with the definitions I have always used in my head. I have always treated an axiom as a statement that is assumed true without a proof, and a proof is a structure with which one proves the truth of a conclusion given the assumption that its premises are true.



As one descends deeper into formal logic, is there a school of thought where "an axiom is a proof" is actually a definition? I haven't been able to find anything to support this in my searches on the internet, but the original poster of the comment is quite confident that searching will reveal said defintion.










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  • it’s a bit obtuse, and sounds more like Philosophy than Mathematics.
    – Thomas Andrews
    2 days ago










  • @ThomasAndrews It definitely did feel obtuse, but from my own forays into formal logic, some of the things which one feels need to define or assume boggle my mind, so I couldn't immediately discount the possibility. And, from experience, things that are definitions in mathematics do indeed behave differently from other statements that are assumed true.
    – Cort Ammon
    2 days ago










  • "but then there you can dispute what counts as proof" -- did they just imply that there cannot be disputes over what counts as proof in mathematics?
    – Theoretical Economist
    2 days ago















up vote
4
down vote

favorite












From this comments discussion on Philosophy.SE:




"Check out formal logic resources - I'm not going to dig them out for you. Alternatively ask on Math.SE. An 'axiom is a proof' is a definition in formal logic - and not an axiom. In philosophical logic, you can dispute this - but then there you can dispute what counts as proof."




This comment did not align with the definitions I have always used in my head. I have always treated an axiom as a statement that is assumed true without a proof, and a proof is a structure with which one proves the truth of a conclusion given the assumption that its premises are true.



As one descends deeper into formal logic, is there a school of thought where "an axiom is a proof" is actually a definition? I haven't been able to find anything to support this in my searches on the internet, but the original poster of the comment is quite confident that searching will reveal said defintion.










share|cite|improve this question





















  • it’s a bit obtuse, and sounds more like Philosophy than Mathematics.
    – Thomas Andrews
    2 days ago










  • @ThomasAndrews It definitely did feel obtuse, but from my own forays into formal logic, some of the things which one feels need to define or assume boggle my mind, so I couldn't immediately discount the possibility. And, from experience, things that are definitions in mathematics do indeed behave differently from other statements that are assumed true.
    – Cort Ammon
    2 days ago










  • "but then there you can dispute what counts as proof" -- did they just imply that there cannot be disputes over what counts as proof in mathematics?
    – Theoretical Economist
    2 days ago













up vote
4
down vote

favorite









up vote
4
down vote

favorite











From this comments discussion on Philosophy.SE:




"Check out formal logic resources - I'm not going to dig them out for you. Alternatively ask on Math.SE. An 'axiom is a proof' is a definition in formal logic - and not an axiom. In philosophical logic, you can dispute this - but then there you can dispute what counts as proof."




This comment did not align with the definitions I have always used in my head. I have always treated an axiom as a statement that is assumed true without a proof, and a proof is a structure with which one proves the truth of a conclusion given the assumption that its premises are true.



As one descends deeper into formal logic, is there a school of thought where "an axiom is a proof" is actually a definition? I haven't been able to find anything to support this in my searches on the internet, but the original poster of the comment is quite confident that searching will reveal said defintion.










share|cite|improve this question













From this comments discussion on Philosophy.SE:




"Check out formal logic resources - I'm not going to dig them out for you. Alternatively ask on Math.SE. An 'axiom is a proof' is a definition in formal logic - and not an axiom. In philosophical logic, you can dispute this - but then there you can dispute what counts as proof."




This comment did not align with the definitions I have always used in my head. I have always treated an axiom as a statement that is assumed true without a proof, and a proof is a structure with which one proves the truth of a conclusion given the assumption that its premises are true.



As one descends deeper into formal logic, is there a school of thought where "an axiom is a proof" is actually a definition? I haven't been able to find anything to support this in my searches on the internet, but the original poster of the comment is quite confident that searching will reveal said defintion.







formal-proofs






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asked 2 days ago









Cort Ammon

2,185613




2,185613











  • it’s a bit obtuse, and sounds more like Philosophy than Mathematics.
    – Thomas Andrews
    2 days ago










  • @ThomasAndrews It definitely did feel obtuse, but from my own forays into formal logic, some of the things which one feels need to define or assume boggle my mind, so I couldn't immediately discount the possibility. And, from experience, things that are definitions in mathematics do indeed behave differently from other statements that are assumed true.
    – Cort Ammon
    2 days ago










  • "but then there you can dispute what counts as proof" -- did they just imply that there cannot be disputes over what counts as proof in mathematics?
    – Theoretical Economist
    2 days ago

















  • it’s a bit obtuse, and sounds more like Philosophy than Mathematics.
    – Thomas Andrews
    2 days ago










  • @ThomasAndrews It definitely did feel obtuse, but from my own forays into formal logic, some of the things which one feels need to define or assume boggle my mind, so I couldn't immediately discount the possibility. And, from experience, things that are definitions in mathematics do indeed behave differently from other statements that are assumed true.
    – Cort Ammon
    2 days ago










  • "but then there you can dispute what counts as proof" -- did they just imply that there cannot be disputes over what counts as proof in mathematics?
    – Theoretical Economist
    2 days ago
















it’s a bit obtuse, and sounds more like Philosophy than Mathematics.
– Thomas Andrews
2 days ago




it’s a bit obtuse, and sounds more like Philosophy than Mathematics.
– Thomas Andrews
2 days ago












@ThomasAndrews It definitely did feel obtuse, but from my own forays into formal logic, some of the things which one feels need to define or assume boggle my mind, so I couldn't immediately discount the possibility. And, from experience, things that are definitions in mathematics do indeed behave differently from other statements that are assumed true.
– Cort Ammon
2 days ago




@ThomasAndrews It definitely did feel obtuse, but from my own forays into formal logic, some of the things which one feels need to define or assume boggle my mind, so I couldn't immediately discount the possibility. And, from experience, things that are definitions in mathematics do indeed behave differently from other statements that are assumed true.
– Cort Ammon
2 days ago












"but then there you can dispute what counts as proof" -- did they just imply that there cannot be disputes over what counts as proof in mathematics?
– Theoretical Economist
2 days ago





"but then there you can dispute what counts as proof" -- did they just imply that there cannot be disputes over what counts as proof in mathematics?
– Theoretical Economist
2 days ago











3 Answers
3






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up vote
13
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One can say that a proof of a statement $X$ is a finite sequence of statements where a) each statement in the sequence is either an axiom or follows from some previous staements via one of a handful of rules of inference and b) the last statement in the sequence is $X$.
- So if the statement $X$ is already an axiom, the one-term sequence with only term $X$ constitutes a proof of statement $X$. Hence if you do not distinguish between $X$ and the one-term sequence with only term $X$, tha claim is correct and an axiom fulfills the definition of a proof.






share|cite|improve this answer
















  • 1




    Well, a proof is a sequence of statements, and an axiom is a statement. So technically, it is not true that an axiom os a proof. :) It does mean that an axiom has a proof.
    – Thomas Andrews
    2 days ago











  • IMO, the question is not about the statement "an axiom is a proof", but about the meta-statement ' "an axiom is a proof" is a definition'.
    – Yves Daoust
    2 days ago







  • 5




    @ThomasAndrews A single statement is a sequence of length one.
    – David Richerby
    2 days ago










  • Nope, it really is not. There is a relationship, but they are not the same thing. Any more than $x$ and $x$ are the same thing. (We are, by nature, being pedantic here.)
    – Thomas Andrews
    2 days ago






  • 3




    There is rarely any loss of clarity or understanding if one identifies one-element sequences with their single element.
    – David Richerby
    2 days ago

















up vote
2
down vote













In the usual formulation of a proof system, a formula $phi$ is provable if either



  • $phi$ is an axiom of the proof system, or

  • $phi$ can be concluded from formulae $phi_1,...,phi_n$ that are provable in the proof system, using one of the proof rules of the system

Therefore we cannot say that "an axiom is a proof", since we have to say that a proof is a proof of something. But we can say that if $phi$ is an axiom in our proof system, then the axiom $phi$ constitutes a proof of $phi$.






share|cite|improve this answer




















  • “Is provable” is different from “is a proof.” A proof is a sequence of statements, as usually defined. But a statement is not a sequence of statements (though it can trivially be made to be one.) If this is what the philosophy people mean - that an axiom can be proved from an axiom - then that really is obtuse.
    – Thomas Andrews
    2 days ago










  • Please reread my comment, I’m not sure what you are responding to. An axiom is provable. A statement is not a sequence of statements, and a proof is a sequence uence of (one) statement.
    – Thomas Andrews
    2 days ago










  • @ThomasAndrews: but the proof of the axiom (within that logic that assumes the relevant axiom) is just the statement of the axiom.
    – nomen
    2 days ago










  • No, a proof is a sequence of axioms. A sequence with one element is not the same thing as the element. “What is the length of the proof of (axiom)?” “One line.” What is the length of (axiom)? Depends on what you mean by the length of an axiom, but nobody would say “one line.” That’s because an axiom isn’t a proof. It is a statement. A proof is a sequence of statements.
    – Thomas Andrews
    2 days ago

















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I guess both can hold, though this discussion has little to do with formal logics.



"An axiom is a proof" could be seen as a definition in a (meta)theory where you previously defined a proof. But without more restrictions, this merely creates an alias and is usually not done.



But in a theory where you defined both an axiom and a proof, with different meanings, you could adopt an axiom saying "an axiom is a proof".




In common mathematics, an axiom is a proposition accepted without a proof, whereas a proof is the logical deduction of a proposition from the available axioms. (The axioms are independent, you can't derive one from the others.)






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    3 Answers
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    3 Answers
    3






    active

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    up vote
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    One can say that a proof of a statement $X$ is a finite sequence of statements where a) each statement in the sequence is either an axiom or follows from some previous staements via one of a handful of rules of inference and b) the last statement in the sequence is $X$.
    - So if the statement $X$ is already an axiom, the one-term sequence with only term $X$ constitutes a proof of statement $X$. Hence if you do not distinguish between $X$ and the one-term sequence with only term $X$, tha claim is correct and an axiom fulfills the definition of a proof.






    share|cite|improve this answer
















    • 1




      Well, a proof is a sequence of statements, and an axiom is a statement. So technically, it is not true that an axiom os a proof. :) It does mean that an axiom has a proof.
      – Thomas Andrews
      2 days ago











    • IMO, the question is not about the statement "an axiom is a proof", but about the meta-statement ' "an axiom is a proof" is a definition'.
      – Yves Daoust
      2 days ago







    • 5




      @ThomasAndrews A single statement is a sequence of length one.
      – David Richerby
      2 days ago










    • Nope, it really is not. There is a relationship, but they are not the same thing. Any more than $x$ and $x$ are the same thing. (We are, by nature, being pedantic here.)
      – Thomas Andrews
      2 days ago






    • 3




      There is rarely any loss of clarity or understanding if one identifies one-element sequences with their single element.
      – David Richerby
      2 days ago














    up vote
    13
    down vote













    One can say that a proof of a statement $X$ is a finite sequence of statements where a) each statement in the sequence is either an axiom or follows from some previous staements via one of a handful of rules of inference and b) the last statement in the sequence is $X$.
    - So if the statement $X$ is already an axiom, the one-term sequence with only term $X$ constitutes a proof of statement $X$. Hence if you do not distinguish between $X$ and the one-term sequence with only term $X$, tha claim is correct and an axiom fulfills the definition of a proof.






    share|cite|improve this answer
















    • 1




      Well, a proof is a sequence of statements, and an axiom is a statement. So technically, it is not true that an axiom os a proof. :) It does mean that an axiom has a proof.
      – Thomas Andrews
      2 days ago











    • IMO, the question is not about the statement "an axiom is a proof", but about the meta-statement ' "an axiom is a proof" is a definition'.
      – Yves Daoust
      2 days ago







    • 5




      @ThomasAndrews A single statement is a sequence of length one.
      – David Richerby
      2 days ago










    • Nope, it really is not. There is a relationship, but they are not the same thing. Any more than $x$ and $x$ are the same thing. (We are, by nature, being pedantic here.)
      – Thomas Andrews
      2 days ago






    • 3




      There is rarely any loss of clarity or understanding if one identifies one-element sequences with their single element.
      – David Richerby
      2 days ago












    up vote
    13
    down vote










    up vote
    13
    down vote









    One can say that a proof of a statement $X$ is a finite sequence of statements where a) each statement in the sequence is either an axiom or follows from some previous staements via one of a handful of rules of inference and b) the last statement in the sequence is $X$.
    - So if the statement $X$ is already an axiom, the one-term sequence with only term $X$ constitutes a proof of statement $X$. Hence if you do not distinguish between $X$ and the one-term sequence with only term $X$, tha claim is correct and an axiom fulfills the definition of a proof.






    share|cite|improve this answer












    One can say that a proof of a statement $X$ is a finite sequence of statements where a) each statement in the sequence is either an axiom or follows from some previous staements via one of a handful of rules of inference and b) the last statement in the sequence is $X$.
    - So if the statement $X$ is already an axiom, the one-term sequence with only term $X$ constitutes a proof of statement $X$. Hence if you do not distinguish between $X$ and the one-term sequence with only term $X$, tha claim is correct and an axiom fulfills the definition of a proof.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered 2 days ago









    Hagen von Eitzen

    267k21259481




    267k21259481







    • 1




      Well, a proof is a sequence of statements, and an axiom is a statement. So technically, it is not true that an axiom os a proof. :) It does mean that an axiom has a proof.
      – Thomas Andrews
      2 days ago











    • IMO, the question is not about the statement "an axiom is a proof", but about the meta-statement ' "an axiom is a proof" is a definition'.
      – Yves Daoust
      2 days ago







    • 5




      @ThomasAndrews A single statement is a sequence of length one.
      – David Richerby
      2 days ago










    • Nope, it really is not. There is a relationship, but they are not the same thing. Any more than $x$ and $x$ are the same thing. (We are, by nature, being pedantic here.)
      – Thomas Andrews
      2 days ago






    • 3




      There is rarely any loss of clarity or understanding if one identifies one-element sequences with their single element.
      – David Richerby
      2 days ago












    • 1




      Well, a proof is a sequence of statements, and an axiom is a statement. So technically, it is not true that an axiom os a proof. :) It does mean that an axiom has a proof.
      – Thomas Andrews
      2 days ago











    • IMO, the question is not about the statement "an axiom is a proof", but about the meta-statement ' "an axiom is a proof" is a definition'.
      – Yves Daoust
      2 days ago







    • 5




      @ThomasAndrews A single statement is a sequence of length one.
      – David Richerby
      2 days ago










    • Nope, it really is not. There is a relationship, but they are not the same thing. Any more than $x$ and $x$ are the same thing. (We are, by nature, being pedantic here.)
      – Thomas Andrews
      2 days ago






    • 3




      There is rarely any loss of clarity or understanding if one identifies one-element sequences with their single element.
      – David Richerby
      2 days ago







    1




    1




    Well, a proof is a sequence of statements, and an axiom is a statement. So technically, it is not true that an axiom os a proof. :) It does mean that an axiom has a proof.
    – Thomas Andrews
    2 days ago





    Well, a proof is a sequence of statements, and an axiom is a statement. So technically, it is not true that an axiom os a proof. :) It does mean that an axiom has a proof.
    – Thomas Andrews
    2 days ago













    IMO, the question is not about the statement "an axiom is a proof", but about the meta-statement ' "an axiom is a proof" is a definition'.
    – Yves Daoust
    2 days ago





    IMO, the question is not about the statement "an axiom is a proof", but about the meta-statement ' "an axiom is a proof" is a definition'.
    – Yves Daoust
    2 days ago





    5




    5




    @ThomasAndrews A single statement is a sequence of length one.
    – David Richerby
    2 days ago




    @ThomasAndrews A single statement is a sequence of length one.
    – David Richerby
    2 days ago












    Nope, it really is not. There is a relationship, but they are not the same thing. Any more than $x$ and $x$ are the same thing. (We are, by nature, being pedantic here.)
    – Thomas Andrews
    2 days ago




    Nope, it really is not. There is a relationship, but they are not the same thing. Any more than $x$ and $x$ are the same thing. (We are, by nature, being pedantic here.)
    – Thomas Andrews
    2 days ago




    3




    3




    There is rarely any loss of clarity or understanding if one identifies one-element sequences with their single element.
    – David Richerby
    2 days ago




    There is rarely any loss of clarity or understanding if one identifies one-element sequences with their single element.
    – David Richerby
    2 days ago










    up vote
    2
    down vote













    In the usual formulation of a proof system, a formula $phi$ is provable if either



    • $phi$ is an axiom of the proof system, or

    • $phi$ can be concluded from formulae $phi_1,...,phi_n$ that are provable in the proof system, using one of the proof rules of the system

    Therefore we cannot say that "an axiom is a proof", since we have to say that a proof is a proof of something. But we can say that if $phi$ is an axiom in our proof system, then the axiom $phi$ constitutes a proof of $phi$.






    share|cite|improve this answer




















    • “Is provable” is different from “is a proof.” A proof is a sequence of statements, as usually defined. But a statement is not a sequence of statements (though it can trivially be made to be one.) If this is what the philosophy people mean - that an axiom can be proved from an axiom - then that really is obtuse.
      – Thomas Andrews
      2 days ago










    • Please reread my comment, I’m not sure what you are responding to. An axiom is provable. A statement is not a sequence of statements, and a proof is a sequence uence of (one) statement.
      – Thomas Andrews
      2 days ago










    • @ThomasAndrews: but the proof of the axiom (within that logic that assumes the relevant axiom) is just the statement of the axiom.
      – nomen
      2 days ago










    • No, a proof is a sequence of axioms. A sequence with one element is not the same thing as the element. “What is the length of the proof of (axiom)?” “One line.” What is the length of (axiom)? Depends on what you mean by the length of an axiom, but nobody would say “one line.” That’s because an axiom isn’t a proof. It is a statement. A proof is a sequence of statements.
      – Thomas Andrews
      2 days ago














    up vote
    2
    down vote













    In the usual formulation of a proof system, a formula $phi$ is provable if either



    • $phi$ is an axiom of the proof system, or

    • $phi$ can be concluded from formulae $phi_1,...,phi_n$ that are provable in the proof system, using one of the proof rules of the system

    Therefore we cannot say that "an axiom is a proof", since we have to say that a proof is a proof of something. But we can say that if $phi$ is an axiom in our proof system, then the axiom $phi$ constitutes a proof of $phi$.






    share|cite|improve this answer




















    • “Is provable” is different from “is a proof.” A proof is a sequence of statements, as usually defined. But a statement is not a sequence of statements (though it can trivially be made to be one.) If this is what the philosophy people mean - that an axiom can be proved from an axiom - then that really is obtuse.
      – Thomas Andrews
      2 days ago










    • Please reread my comment, I’m not sure what you are responding to. An axiom is provable. A statement is not a sequence of statements, and a proof is a sequence uence of (one) statement.
      – Thomas Andrews
      2 days ago










    • @ThomasAndrews: but the proof of the axiom (within that logic that assumes the relevant axiom) is just the statement of the axiom.
      – nomen
      2 days ago










    • No, a proof is a sequence of axioms. A sequence with one element is not the same thing as the element. “What is the length of the proof of (axiom)?” “One line.” What is the length of (axiom)? Depends on what you mean by the length of an axiom, but nobody would say “one line.” That’s because an axiom isn’t a proof. It is a statement. A proof is a sequence of statements.
      – Thomas Andrews
      2 days ago












    up vote
    2
    down vote










    up vote
    2
    down vote









    In the usual formulation of a proof system, a formula $phi$ is provable if either



    • $phi$ is an axiom of the proof system, or

    • $phi$ can be concluded from formulae $phi_1,...,phi_n$ that are provable in the proof system, using one of the proof rules of the system

    Therefore we cannot say that "an axiom is a proof", since we have to say that a proof is a proof of something. But we can say that if $phi$ is an axiom in our proof system, then the axiom $phi$ constitutes a proof of $phi$.






    share|cite|improve this answer












    In the usual formulation of a proof system, a formula $phi$ is provable if either



    • $phi$ is an axiom of the proof system, or

    • $phi$ can be concluded from formulae $phi_1,...,phi_n$ that are provable in the proof system, using one of the proof rules of the system

    Therefore we cannot say that "an axiom is a proof", since we have to say that a proof is a proof of something. But we can say that if $phi$ is an axiom in our proof system, then the axiom $phi$ constitutes a proof of $phi$.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered 2 days ago









    Hans Hüttel

    3,0242819




    3,0242819











    • “Is provable” is different from “is a proof.” A proof is a sequence of statements, as usually defined. But a statement is not a sequence of statements (though it can trivially be made to be one.) If this is what the philosophy people mean - that an axiom can be proved from an axiom - then that really is obtuse.
      – Thomas Andrews
      2 days ago










    • Please reread my comment, I’m not sure what you are responding to. An axiom is provable. A statement is not a sequence of statements, and a proof is a sequence uence of (one) statement.
      – Thomas Andrews
      2 days ago










    • @ThomasAndrews: but the proof of the axiom (within that logic that assumes the relevant axiom) is just the statement of the axiom.
      – nomen
      2 days ago










    • No, a proof is a sequence of axioms. A sequence with one element is not the same thing as the element. “What is the length of the proof of (axiom)?” “One line.” What is the length of (axiom)? Depends on what you mean by the length of an axiom, but nobody would say “one line.” That’s because an axiom isn’t a proof. It is a statement. A proof is a sequence of statements.
      – Thomas Andrews
      2 days ago
















    • “Is provable” is different from “is a proof.” A proof is a sequence of statements, as usually defined. But a statement is not a sequence of statements (though it can trivially be made to be one.) If this is what the philosophy people mean - that an axiom can be proved from an axiom - then that really is obtuse.
      – Thomas Andrews
      2 days ago










    • Please reread my comment, I’m not sure what you are responding to. An axiom is provable. A statement is not a sequence of statements, and a proof is a sequence uence of (one) statement.
      – Thomas Andrews
      2 days ago










    • @ThomasAndrews: but the proof of the axiom (within that logic that assumes the relevant axiom) is just the statement of the axiom.
      – nomen
      2 days ago










    • No, a proof is a sequence of axioms. A sequence with one element is not the same thing as the element. “What is the length of the proof of (axiom)?” “One line.” What is the length of (axiom)? Depends on what you mean by the length of an axiom, but nobody would say “one line.” That’s because an axiom isn’t a proof. It is a statement. A proof is a sequence of statements.
      – Thomas Andrews
      2 days ago















    “Is provable” is different from “is a proof.” A proof is a sequence of statements, as usually defined. But a statement is not a sequence of statements (though it can trivially be made to be one.) If this is what the philosophy people mean - that an axiom can be proved from an axiom - then that really is obtuse.
    – Thomas Andrews
    2 days ago




    “Is provable” is different from “is a proof.” A proof is a sequence of statements, as usually defined. But a statement is not a sequence of statements (though it can trivially be made to be one.) If this is what the philosophy people mean - that an axiom can be proved from an axiom - then that really is obtuse.
    – Thomas Andrews
    2 days ago












    Please reread my comment, I’m not sure what you are responding to. An axiom is provable. A statement is not a sequence of statements, and a proof is a sequence uence of (one) statement.
    – Thomas Andrews
    2 days ago




    Please reread my comment, I’m not sure what you are responding to. An axiom is provable. A statement is not a sequence of statements, and a proof is a sequence uence of (one) statement.
    – Thomas Andrews
    2 days ago












    @ThomasAndrews: but the proof of the axiom (within that logic that assumes the relevant axiom) is just the statement of the axiom.
    – nomen
    2 days ago




    @ThomasAndrews: but the proof of the axiom (within that logic that assumes the relevant axiom) is just the statement of the axiom.
    – nomen
    2 days ago












    No, a proof is a sequence of axioms. A sequence with one element is not the same thing as the element. “What is the length of the proof of (axiom)?” “One line.” What is the length of (axiom)? Depends on what you mean by the length of an axiom, but nobody would say “one line.” That’s because an axiom isn’t a proof. It is a statement. A proof is a sequence of statements.
    – Thomas Andrews
    2 days ago




    No, a proof is a sequence of axioms. A sequence with one element is not the same thing as the element. “What is the length of the proof of (axiom)?” “One line.” What is the length of (axiom)? Depends on what you mean by the length of an axiom, but nobody would say “one line.” That’s because an axiom isn’t a proof. It is a statement. A proof is a sequence of statements.
    – Thomas Andrews
    2 days ago










    up vote
    2
    down vote













    I guess both can hold, though this discussion has little to do with formal logics.



    "An axiom is a proof" could be seen as a definition in a (meta)theory where you previously defined a proof. But without more restrictions, this merely creates an alias and is usually not done.



    But in a theory where you defined both an axiom and a proof, with different meanings, you could adopt an axiom saying "an axiom is a proof".




    In common mathematics, an axiom is a proposition accepted without a proof, whereas a proof is the logical deduction of a proposition from the available axioms. (The axioms are independent, you can't derive one from the others.)






    share|cite|improve this answer


























      up vote
      2
      down vote













      I guess both can hold, though this discussion has little to do with formal logics.



      "An axiom is a proof" could be seen as a definition in a (meta)theory where you previously defined a proof. But without more restrictions, this merely creates an alias and is usually not done.



      But in a theory where you defined both an axiom and a proof, with different meanings, you could adopt an axiom saying "an axiom is a proof".




      In common mathematics, an axiom is a proposition accepted without a proof, whereas a proof is the logical deduction of a proposition from the available axioms. (The axioms are independent, you can't derive one from the others.)






      share|cite|improve this answer
























        up vote
        2
        down vote










        up vote
        2
        down vote









        I guess both can hold, though this discussion has little to do with formal logics.



        "An axiom is a proof" could be seen as a definition in a (meta)theory where you previously defined a proof. But without more restrictions, this merely creates an alias and is usually not done.



        But in a theory where you defined both an axiom and a proof, with different meanings, you could adopt an axiom saying "an axiom is a proof".




        In common mathematics, an axiom is a proposition accepted without a proof, whereas a proof is the logical deduction of a proposition from the available axioms. (The axioms are independent, you can't derive one from the others.)






        share|cite|improve this answer














        I guess both can hold, though this discussion has little to do with formal logics.



        "An axiom is a proof" could be seen as a definition in a (meta)theory where you previously defined a proof. But without more restrictions, this merely creates an alias and is usually not done.



        But in a theory where you defined both an axiom and a proof, with different meanings, you could adopt an axiom saying "an axiom is a proof".




        In common mathematics, an axiom is a proposition accepted without a proof, whereas a proof is the logical deduction of a proposition from the available axioms. (The axioms are independent, you can't derive one from the others.)







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited 2 days ago

























        answered 2 days ago









        Yves Daoust

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