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> And just like mathematics, adding an another "axiom" to the law has far, far-reaching consequences. Actually mathematicians virtually never do this. Almost a

> And just like mathematics, adding an another "axiom" to the law has far, far-reaching consequences. Actually mathematicians virtually never do this. Almost all of mathematics (from arithmetic to calculus to category theory) operates in the confines of ZFC[1] and adds no further axioms. All of these fields may add definitions, but these are just shorthand; the are conservative and have no actual consequences. It is more like fixing Newton's laws and then experimenting with all the machines you can build with them. [1]: https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t...

xamuel 11y ago
Most mathematicians don't spend their time considering extensions of ZFC, but mathematicians are always adding axioms in a different sense. Add a commutativity axiom to group theory and you get commutative group theory. Add an axiom that "the space has total measure 1" to measure theory and you get probability theory. And so on.
- johncolanduoni 11y ago
  These are generally called axioms, but these are totally distinct from axioms in the sense of ZFC's axioms. They are simply part of the definition of a group, commutative group, etc. Since they are just part of the definition of the structure, choosing the "wrong axioms" will only result in a structure which has no actual instances, not in inconsistency of the underlying theory. As such, they most certainly don't have "far reaching consequences", except that some axioms may produce uninteresting structures.
  - ihm 11y ago
    > These are generally called axioms, but these are totally distinct from axioms in the sense of ZFC's axioms. I wouldn't really say they're "totally distinct". For example, one could say that general group theory is studying the theory (in the logic sense) of a group (usually the second order theory rather than first order, but you get idea). Then adding assumptions about the structure of the group under consideration is like adding axioms and looking at the new theory.
  - deleted 11y ago
    [deleted]
  - nmrm2 11y ago
    This is a really hair-splitting distinction that only makes sense if mathematicians wrote proofs (e.g., about commutative group theory) while keeping ZFC's axioms in mind. But even then, it's not really an appropriate distinction -- accidentally assuming inconsistency and moving forward with it is exactly as far-reaching as assuming inconsistency at the foundations. And, doing some obscure work on an inconsistent foundations that no body uses is exactly as harmless as doing some obscure work on a structure without elements.
    - johncolanduoni 11y ago
      > This is a really hair-splitting distinction that only makes sense if mathematicians wrote proofs (e.g., about commutative group theory) while keeping ZFC's axioms in mind. They may not do so explicitly, but it is easy to see where the axioms become important when dealing with mathematical structures. For example, if you are considering the group of integers with addition, the existence of this group is predicated on the existence of the integers, which is predicated on the existence of the natural numbers as a set, which is an axiom of ZFC (axiom of infinity[1]). The case for other foundations is similar. > But even then, it's not really an appropriate distinction -- accidentally assuming inconsistency and moving forward with it is exactly as far-reaching as assuming inconsistency at the foundations. No, because when studying a mathematical structure we always consider concrete instances of it, which are usually in fact the motivation for the definition of the structure. So mathematics based on possibly (relatively) inconsistent structures (other than foundational systems) is not really a thing; group theory was not initiated prior to the discovery of structures satisfying the groups axioms, and neither was the study of rings, fields, modules, topological spaces etc. On the other hand, we cannot consider models of ZFC unless we build it within the assumptions of a foundational system (possibly ZFC itself). So at some point we have to pick a set of axioms to be the bottom of our ladder of turtles, and the axioms of that system are very special indeed. Another difference is that the axioms of a group (or a field, or a topological space) cannot stand on their own; they clearly reference sets and functions. ZFC's axioms only reference sets, all of the fundamental properties of which are enumerated in the axioms themselves. If you gave an (english speaking) alien the definition of a group, he would need to ask you what the definition of sets and functions were. So even if we wanted the group axioms to be our foundations, this would not be possible. [1] : https://en.wikipedia.org/wiki/Axiom_of_infinity https://en.wikipedia.org/wiki/Axiom_of_infinity
eli_gottlieb 11y ago
>Actually mathematicians virtually never do this. Almost all of mathematics (from arithmetic to calculus to category theory) operates in the confines of ZFC[1] and adds no further axioms. Except for all the non-ZFC foundations, like type theory or category-theoretic foundations.
- cwzwarich 11y ago
  Relatively speaking, there are very few mathematicians using those systems to discover new mathematics as opposed to studying them for their foundational interest.
- johncolanduoni 11y ago
  They (and those who study ZFC extensions) are the reason I said "virtually"; they are most certainly a small minority among mathematicians. I myself have been studying homotopy type theory as of late.

xamuel 11y ago

Most mathematicians don't spend their time considering extensions of ZFC, but mathematicians are always adding axioms in a different sense. Add a commutativity axiom to group theory and you get commutative group theory. Add an axiom that "the space has total measure 1" to measure theory and you get probability theory. And so on.

johncolanduoni 11y ago
These are generally called axioms, but these are totally distinct from axioms in the sense of ZFC's axioms. They are simply part of the definition of a group, commutative group, etc. Since they are just part of the definition of the structure, choosing the "wrong axioms" will only result in a structure which has no actual instances, not in inconsistency of the underlying theory. As such, they most certainly don't have "far reaching consequences", except that some axioms may produce uninteresting structures.
- ihm 11y ago
  > These are generally called axioms, but these are totally distinct from axioms in the sense of ZFC's axioms. I wouldn't really say they're "totally distinct". For example, one could say that general group theory is studying the theory (in the logic sense) of a group (usually the second order theory rather than first order, but you get idea). Then adding assumptions about the structure of the group under consideration is like adding axioms and looking at the new theory.
- deleted 11y ago
  [deleted]
- nmrm2 11y ago
  This is a really hair-splitting distinction that only makes sense if mathematicians wrote proofs (e.g., about commutative group theory) while keeping ZFC's axioms in mind. But even then, it's not really an appropriate distinction -- accidentally assuming inconsistency and moving forward with it is exactly as far-reaching as assuming inconsistency at the foundations. And, doing some obscure work on an inconsistent foundations that no body uses is exactly as harmless as doing some obscure work on a structure without elements.
  - johncolanduoni 11y ago
    > This is a really hair-splitting distinction that only makes sense if mathematicians wrote proofs (e.g., about commutative group theory) while keeping ZFC's axioms in mind. They may not do so explicitly, but it is easy to see where the axioms become important when dealing with mathematical structures. For example, if you are considering the group of integers with addition, the existence of this group is predicated on the existence of the integers, which is predicated on the existence of the natural numbers as a set, which is an axiom of ZFC (axiom of infinity[1]). The case for other foundations is similar. > But even then, it's not really an appropriate distinction -- accidentally assuming inconsistency and moving forward with it is exactly as far-reaching as assuming inconsistency at the foundations. No, because when studying a mathematical structure we always consider concrete instances of it, which are usually in fact the motivation for the definition of the structure. So mathematics based on possibly (relatively) inconsistent structures (other than foundational systems) is not really a thing; group theory was not initiated prior to the discovery of structures satisfying the groups axioms, and neither was the study of rings, fields, modules, topological spaces etc. On the other hand, we cannot consider models of ZFC unless we build it within the assumptions of a foundational system (possibly ZFC itself). So at some point we have to pick a set of axioms to be the bottom of our ladder of turtles, and the axioms of that system are very special indeed. Another difference is that the axioms of a group (or a field, or a topological space) cannot stand on their own; they clearly reference sets and functions. ZFC's axioms only reference sets, all of the fundamental properties of which are enumerated in the axioms themselves. If you gave an (english speaking) alien the definition of a group, he would need to ask you what the definition of sets and functions were. So even if we wanted the group axioms to be our foundations, this would not be possible. [1] : https://en.wikipedia.org/wiki/Axiom_of_infinity https://en.wikipedia.org/wiki/Axiom_of_infinity

eli_gottlieb 11y ago

>Actually mathematicians virtually never do this. Almost all of mathematics (from arithmetic to calculus to category theory) operates in the confines of ZFC[1] and adds no further axioms. Except for all the non-ZFC foundations, like type theory or category-theoretic foundations.

cwzwarich 11y ago
Relatively speaking, there are very few mathematicians using those systems to discover new mathematics as opposed to studying them for their foundational interest.
johncolanduoni 11y ago
They (and those who study ZFC extensions) are the reason I said "virtually"; they are most certainly a small minority among mathematicians. I myself have been studying homotopy type theory as of late.