This comment is intended as a service to those students who will stumble on pile of trash called sci.math.
I discuss the so-called Peano "axioms". But before I do, I must define what the words axiom and postulate mean because most idiots are unable to differentiate between the two.
Postulate: an *assertion* assumed to be true, as a basis of inference. Axiom: a self-evident fact, known to be true, and used as a basis of inference.
The five Peano "axioms" are stated as follows:
1. Zero is a natural number.
I was a teenager when I first read that. My first response was a good chuckle. How could this useless information be written in an Encyclopedia with the reputation of Britannica?
Our bonobo mathematician Peano introduces two terms, the subject (zero) and a qualified object (natural number). You are supposed to know what these are, except the only problem, is that the imbeciles (professors of math and mathematicians) don't know what is a magnitude, never mind a number. As for a natural number, our Simian friends in universities worldwide, have no idea how much thought and effort went into the construction of the natural numbers, that was made possible by ratios of equal magnitudes.
So, the first so-called "axiom" contains two undefined and unqualified terms. You shall see how rigorous and sound John Gabriel's axioms are at the end of this comment.
Without any proof and no justification Peano 1 tells us that zero is a *number*, but not just any rational number, it is a *natural number*. That you don't know what is a natural number, is your problem.
You can ask: What is zero? What is a number? What is a "natural number"? And the idiot Peano simply stares at you blankly.
2. Every natural number has a successor in the natural numbers.
Peano 2 gets more interesting. After introducing the natural number, the next bombshell is that there is *more than one natural number*! :-) But if this were not shocking enough, we see that these numbers have successors (whatever the fuck that means). So, since we are not told what a successor means, we simply assume that there exists some kind of order such that one number follows the number before it. Spaghetti brain Peano seemed to think these were sound concepts. His fellow primate Bertrand Russell once said that Peano had a sharpness of mind. From this, can we infer the Russell make have lost a substantial amount of brain cells to tobacco smoking? Hmmm.
3. Zero is not the successor of any natural number.
Peano 3 tells us that the in the imagined ordering, zero appears first. Never mind that a standard set does not care about the order of elements. Does this mean that if each set is like a brown bag, then if I look inside, the first object I shall see is zero? :-) All natural numbers can be written as S(x), but not 0 according to the imbecile Peano. That's what Peano 3 is saying.
4. If the successor of two natural numbers is the same, then the two original numbers are the same.
Peano 4 tells us S(x)=S(y) => x=y. What a profound statement! :-) This introduces the vague notion of difference.
5. If a set contains zero and the successor of every number is in the set, then the set contains the natural numbers.
There are so many assumptions in Peano 5, that it's hard to even think of where to begin addressing the bullshit.
It infuriates me that the amoebas on this forum DARED to compare my axioms with this fucking rot.
To subscribe to such rot exposes your lack of intelligence.
And now, for the sound construction of numbers from scratch and the new axioms:
Construction of rational numbers:
1. A magnitude is the idea of size of extent. We can either tell that two magnitudes are equal or not. If we can tell they are not equal, then we know which is smaller or bigger, but we can't tell how much bigger or smaller. This is called qualitative measurement (without numbers).
2. We can form ratios of magnitudes. AB : CD where AB and CD are line segments. The expression AB : CD means the comparison of magnitudes AB and CD.
3. A ratio of equal magnitudes, say AB : AB or CD : CD allows us to use either as the standard of measurement, that is, the unit. The unit is a ratio of equal magnitudes.
4. The unit enables us now to compare AB and CD if both are exact multiples of the unit that measures both. We can now perform quantitative measurement, because we can tell how much greater AB is than CD or how much less AB is than CD.
5. Finally, if a magnitude is only part of a unit, then we arrive at a ratio of numbers, say AB : CD where AB and CD are multiples of the unit. AB : CD now means the comparison of numbers AB and CD. When we write AB/CD, it is called a fraction.
So, in five steps I have derived the concept of number for you. There is one thing left - what happens when you can't measure a magnitude that is not a multiple of a unit and can't be expressed exactly using any part of a unit? This is called an incommensurable magnitude and the best you can do is provide an approximation such as 3.14159... or 1.414..., etc.
Euclid's Elements: Definition of magnitude: Bk V. My definition of magnitude is better than Euclid's because it is not circular. Definition of number: Bk. VII
The Axioms of Arithmetic:
1. The difference (or subtraction) of two numbers is that number which describes how much the larger exceeds the smaller.
Explanation: We start with the primitive operator of subtraction (difference) which is '-'. We cannot do monkey things like 1-3 because 1 is smaller than 3. The smaller is subtracted from the bigger, like this: 3 - 1 = 2.
Besides, you *can't even begin* to do subtraction with the Peano rot axioms!!!
There is NO WAY you can say 1 - 3 in Peano's fartioms. :-) You have to use '+' because it is the operator of the successor function.
Using my axioms, '-' is used in *every operation*.
2. The difference of equal numbers is zero.
Explanation: k - k = 0
3. The sum (or addition) of two numbers is that number whose difference with either of the two numbers is either of the two numbers.
Explanation: m - n = d where m > n. So, n + d = m.
4. The quotient (or division) of two numbers is that magnitude that measures either number in terms of the other.
Explanation: 2/3 measures 3. How? 3 - (2/3 + 2/3 + 2/3)=0 or 2/3 + 2/3 + 2/3 = 3 since we defined addition in (3).
1/3 measures 2. How? 2 - (1/3+1/3+1/3+1/3+1/3+1/3) = 0. In this case, 2 is measured by the reciprocal of 3. The axioms says "in terms of the other".
5. If a unit is divided by a number into *equal* parts, then each of these parts of a unit, is called the reciprocal of that number.
Explanation: 1/k + 1/k + +1/k (k times) = 1.
6. Division by zero is undefined.
Explanation: Zero does not measure ANY other number except itself.
7. The product (or multiplication) of two numbers is the quotient of either number with the reciprocal of the other.
Explanation: m x n = m / (1/n) or n / (1/m)
8. The difference of any number and zero is the number.
Observe that all the basic arithmetic operations are defined in terms of difference.