I think we’re in agreement with each other. The integers are not well-ordered by ‘<’ as ‘<’ is traditionally interpreted; they are well-ordered by other, different relations (that can be formalized in logic); see the Wikipedia article on well-ordering.
The reason that we don’t start out with “There are no chains that do not start with zero.” is, I speculate, at least two-fold:
In Peano’s original axioms, he used the other formulation. So there is a precedent.
Peano’s formulation can be expressed easily in first-order Peano arithmetic. Peano’s formulation describes what is going on within the system; whereas “there are no chains that do not start with zero” is discussing the structure of the system from the outside. They do come out equivalent (I think) in second-order logic, but Peano’s formulation is the one that is easily expressed in first-order logic.
“All natural numbers can be generated by iterating the successor function on zero.”
“The smallest set k which includes zero and the successor of every member of itself is the set of natural numbers.”
I think that both of those formulations can be phrased in first-order logic...
Properties are sets of numbers, so without getting into technicalities, you need second-order logic to talk about the smallest set such that whatever (since you need to quantify over all candidate sets).
Similarly, to say that you can get x by iterating the successor function on zero requires second-order logic. First-order logic isn’t even sufficient to define addition without adding axioms for what addition does.
If you use set theory, then yes. Usually, however, mathematicians don’t want to have to worry about things like the axiom of regularity when all they wanted to talk about in the first place was the natural numbers!
You can’t talk about what the natural numbers are and are not without some form of set theory.
But you can talk about some of the properties they have, and quite often that is all we care about.
Also, the stronger your system is, the more likely it is that your formulation is inconsistent (and if the system is inconsistent, you’re definitely not describing anything meaningful). I’m much more confident that first-order Peano arithmetic is consistent than I am that first-order ZFC set theory is consistent.
Enjoy A Problem Course in Mathematical Logic. Read Definition 6.4, Definition 6.5, and Definition 6.6 (Edit: They are on PDF pages 47-50, book pages 35-38.). It means that, within each model of the axioms, it is the case that every object in the model has the specified property. The natural numbers happen to be a model of first-order Peano arithmetic.
Let me ask you what “every x” means in first-order ZFC set theory. Answer carefully—it has a countable model.
I think we’re in agreement with each other. The integers are not well-ordered by ‘<’ as ‘<’ is traditionally interpreted; they are well-ordered by other, different relations (that can be formalized in logic); see the Wikipedia article on well-ordering.
The reason that we don’t start out with “There are no chains that do not start with zero.” is, I speculate, at least two-fold:
In Peano’s original axioms, he used the other formulation. So there is a precedent.
Peano’s formulation can be expressed easily in first-order Peano arithmetic. Peano’s formulation describes what is going on within the system; whereas “there are no chains that do not start with zero” is discussing the structure of the system from the outside. They do come out equivalent (I think) in second-order logic, but Peano’s formulation is the one that is easily expressed in first-order logic.
“All natural numbers can be generated by iterating the successor function on zero.” “The smallest set k which includes zero and the successor of every member of itself is the set of natural numbers.”
I think that both of those formulations can be phrased in first-order logic...
Properties are sets of numbers, so without getting into technicalities, you need second-order logic to talk about the smallest set such that whatever (since you need to quantify over all candidate sets).
Similarly, to say that you can get x by iterating the successor function on zero requires second-order logic. First-order logic isn’t even sufficient to define addition without adding axioms for what addition does.
If you use set theory, then yes. Usually, however, mathematicians don’t want to have to worry about things like the axiom of regularity when all they wanted to talk about in the first place was the natural numbers!
You can’t talk about what the natural numbers are and are not without some form of set theory.
“0 is the only number which is not the successor of any number” requires set theory to be meaningful.
But you can talk about some of the properties they have, and quite often that is all we care about.
Also, the stronger your system is, the more likely it is that your formulation is inconsistent (and if the system is inconsistent, you’re definitely not describing anything meaningful). I’m much more confident that first-order Peano arithmetic is consistent than I am that first-order ZFC set theory is consistent.
No. You can rephrase that as: “Every natural number is either 0 or the successor of some number”.
What does “Every x” mean in the absence of set theory?
Enjoy A Problem Course in Mathematical Logic. Read Definition 6.4, Definition 6.5, and Definition 6.6 (Edit: They are on PDF pages 47-50, book pages 35-38.). It means that, within each model of the axioms, it is the case that every object in the model has the specified property. The natural numbers happen to be a model of first-order Peano arithmetic.
Let me ask you what “every x” means in first-order ZFC set theory. Answer carefully—it has a countable model.