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Most mathematicians don't spend their time considering extensions of ZFC, but mathematicians are always adding axioms in a different sense. Add a commutativity

by 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
    - nmrm2 11y ago
      This is a silly argument. 1. Assume false. 2. Anything follows. For what odd-ball definition of "far reaching" is "anything" not "far reaching"? An assumption doesn't have to be an Axiom to have "far reaching" consequences. Period. So, you're wrong -- assuming non-foundational axioms can have far reaching consequences. Assumptions are assumptions, and flaws are flaws. Assuming "false" is just as bad as working with an inconsistent foundations. It's just a fact. That said, the actual difference you're trying to elucidate is sociological, not technical -- you suppose that people are less likely to accept non-foundational assumptions, and so that'll check back something from a sociological viewpoint. You assume people won't build results on top of a system with an unproven assumption thrown in. That argument would be a lot more compelling if Mathematicians formalized their arguments in terms of an underlying foundations, because then we would always know that all assumptions are checked. But Mathematicians don't work like that. And errors creep in. And if people build on top of those errors, it can be pretty catastrophic. In short, what matters is that people accept the incorrect fact and move forward from it, NOT whether that incorrectness comes from foundations or from something built on top of them. The impact of either depends entirely upon how many people trust the result and build on top of it.

by xamuel 11y ago

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
  - nmrm2 11y ago
    This is a silly argument. 1. Assume false. 2. Anything follows. For what odd-ball definition of "far reaching" is "anything" not "far reaching"? An assumption doesn't have to be an Axiom to have "far reaching" consequences. Period. So, you're wrong -- assuming non-foundational axioms can have far reaching consequences. Assumptions are assumptions, and flaws are flaws. Assuming "false" is just as bad as working with an inconsistent foundations. It's just a fact. That said, the actual difference you're trying to elucidate is sociological, not technical -- you suppose that people are less likely to accept non-foundational assumptions, and so that'll check back something from a sociological viewpoint. You assume people won't build results on top of a system with an unproven assumption thrown in. That argument would be a lot more compelling if Mathematicians formalized their arguments in terms of an underlying foundations, because then we would always know that all assumptions are checked. But Mathematicians don't work like that. And errors creep in. And if people build on top of those errors, it can be pretty catastrophic. In short, what matters is that people accept the incorrect fact and move forward from it, NOT whether that incorrectness comes from foundations or from something built on top of them. The impact of either depends entirely upon how many people trust the result and build on top of it.