Despite the title, I am looking for factual answers.
When I studied the real number system axiomatically, 45 years ago, we began with the Peano Postulates and worked our way up to 1-variable calculus.
Having retired, and with more time than sense, I thought I’d try it again to see if it came out the same. Looking around for an online text or syllabus, however, I found most of these projects beginning with the field axioms instead.
Since associative, distributive, and commutative properties can be derived from the Peano postulates, what is the advantage in beginning with the field axioms? Is it Peano’s reliance on induction? Are the field axioms more firmly grounded in set theory?
Again, I know that each will have its advantages and they will have their costs. I just thought I’d better find out what they are before getting started. Thanks for any help or interest.
Peano axioms are for natural numbers.
pretty sure the guts of Peano is all sorts of properties of integers.
(that an integer + 1 is an integer , and so on )
Fields are for real numbers What is a Field? says that natural numbers are not a field, because the inverse of a natural number is not a natural number (except for the trivial case of ‘1’ ..)
So not sure there is any point to get to algebra from Peano .
Any work you did with Peano would be limited to all values that are integers.
You must switch to fields for real number algebra…
Thanks, Iselder. As I recall, the reals eventually emerge from the Peano system via Cauchy sequences, but like I say, it’s been 45 years. Thank you for the link. I have tried math forums with this, but it seems to fall between two chairs: too useless for homework helpers, too primitive for analysts.
Peano’s axioms and arithmetic are for the natural numbers (1,2,3,…) which are not a field. The real numbers follow the axioms of the (Cauchy) complete ordered (Archmidean) field.
Many maths books will start off with Peano’s axioms to construct the natural numbers and from there construct the integers as equivalence classes of ordered pairs of natural numbers, and then construct the rational numbers as equivalence classes of ordered pairs of integers, and finally construct the real numbers as equivalence classes of Cauchy sequences of rational numbers (or alternatively as Dedekind cuts of rational numbers). Obviously though it’s far more simpler and economical to view the reals as “the complete ordered field” than as “equivalence classes of Cauchy sequences of equivalence classes of ordered pairs of equivalence classes of ordered pairs of natural numbers”
First, there is no opposition between them. Peano axioms are for the natural numbers 0,1,2,… In fact the … is a lot of what the Peano axioms are about. You can start there and define + and * and then embed the natural numbers into the integers (adding negatives), then add multiplicative inverses to get the rational numbers. You could go on and add the Dedekind cuts to get the real numbers (or you can use Cauchy sequences, although without the axiom of choice they are not provably equivalent). (In fact there are models of set theory without AC in which they are different.) There are other ways of getting the real numbers. Then you add a square root of -1 to get the complex numbers.
But why? If you are an aspiring mathematician, it is an interesting, if somewhat boring, exercise in starting with a minimal set of axioms and seeing how far you get. If you merely want to develop calculus, then go for a complete archimedean ordered field. (Complete means every bounded set has an upper bound, archimedean means no infinitesimals.)
I haven’t the slightest clue as to what this thread is about–I checked in just for yucks–but this statement is wonderful.
True for many situations in GQ, and for life in general, in which, one may safely assume, GQ is a part.
Excessive commas and structuring to foster them borrowed from PG Wodehouse.
In addition to what’s been said, there are fields which are not the real numbers. The complex numbers and rational numbers are the most obvious example, but you can also construct finite fields by looking at integer arithmetic modulo some prime p. The simplest case is where you take p = 2, which gives you the following rules of arithmetic:
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 1
0 * 0 = 0
0 * 1 = 0
1 * 0 = 0
1 * 1 = 1
Showing that this system obeys the field axioms is a standard exercise.
Just in case the implications of the answers already given aren’t clear: You would start with the Peano axioms if you want to show how to construct the real numbers from a minimal set of assumptions. On the other hand, if you just want to get down to business, you would assume the reals exist and give the field axioms (and completeness) as the most important properties they have.
Yes, That is what I got out of it. I think I’ll start with Peano, just to make sure the reals exist. At this stage in my life, I want to avoid getting down to business, and boredom will make my last days last longer.
Leo made a good point: as a newbie I’m struck by how well questions normal people don’t ask are answered.
And come to think of it, isn’t the truly simplest field the single-element one, where a+a = a and a*a = a? Or does that violate one of the field axioms?
The only axiom it violates is the implicit axiom 1≠0.
The reason that the trivial ring is excluded from being a field is simply because there are theorems that apply to fields that don’t apply to the trivial ring.
Depends on who’s asked. Most undergraduate textbooks that I’ve used require a field to have an underlying set with at least two distinct elements 0 and 1, but I have seen books that allow a one-element field. Of course, these texts have to specifically exclude this field for almost every single theorem.
Any field axioms I have seen assume that 0 and 1 are distinct.
I will mention in passing that for any prime p and any exponent e, there is a finite field with p^e elements and any two are isomorphic. For example, the field of four elements has two more elements w and w^2 that are cube roots of 1. More generally, the field of p^e elements is generated by a (p^e - 1)th root of 1.