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Topic: John Gabriel's Thread on Mathematics.
Replies: 231   Last Post: Mar 22, 2014 9:23 PM

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 Wizard-Of-Oz Posts: 404 Registered: 12/28/13
Re: Construction of rational numbers and arithmetic axioms.
Posted: Mar 1, 2014 7:58 PM

John Gabriel <thenewcalculus@gmail.com> wrote in

Subject: Re: Infinity: The Story So Far
From: "Wizard-Of-Oz" <somewhere@over-the-rainbow.com>
Newsgroups: sci.math

John Gabriel <thenewcalculus@gmail.com> wrote in

> 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.
>
> (difference) which is '-'. We cannot do monkey things like 1-3 because
> 1 is smaller than 3.

That is not ruled out by your axiom. 1 - 3 is not forbidden by your
axiom which simply refers to two number.

1 and 3 are two number. Subtraction is how much the larger exceed the
smaller, so for 1 - 3, 3 is the larger, 1 is the smaller, 2 is how much
the larger exceed the smaller so 1 - 3 = 2

> 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!!!

You have to define subtraction, which is done based upon the axioms and
other operations which you define.

> 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.

No .. its not. Though once addition is defined from peano you can
operations can be derived from the successor.

> Using my axioms, '-' is used in *every operation*.

Not really a benefit.

> 2. The difference of equal numbers is zero.
>
> Explanation: k - k = 0

That is fine

> 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.

So, lets take then numbers 2 and 3. We need to the look at either one
or the other of those numbers. Lets take 3. We have that 6 - 3 = 3 and
3 is either of the numbers. So we get 2 + 3 = 6

> 4. The quotient (or division) of two numbers is that magnitude that
> measures either number in terms of the other.

Lets take 2 / 6. So we want the magnitude that measures either in terms
of the other. 3 measure 6 in terms of 2. so 2 / 6 = 3

> Explanation: 2/3 measures 3. How? 3 - (2/3 + 2/3 + 2/3)=0
> or 2/3 +
> 2/3 + 2/3 = 3

BAHAHAH .. 3 = 2/3 + 2/3 + 2/3

You're still a moron

> 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.

You need to thank me for getting you to add the word "equal". I expect
an acknowledgment for my contribution to improving this axiom

Now .. one can see how to divide a unit into 3 equal parts, but how do
you divide a unit into 1/3 of an equal part?

Your definition only works for the reciprocal by a whole 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.
>

Observe you updated your axioms with my correction, but they still are
not correct.

Observe your axions give 2 - 3 = 1, 2 + 3 - 6, 2 / 6 = 3 and cannot
handle reciprocal of (1/3), which means you can only multiply by
fractions with a unit numerator.

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