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Old 2005-07-17, 00:49   #1
Numbers
 
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Default Peano Postulates

Dr. Silverman will no doubt be delighted to know that this melon-head actually got himself a book on number theory. And I got all the way to page 8 before I have a question.

In my book the first Peano postulate is: 1) 1 is a natural number.

The third postulate says: 3) 1 is not the successor of any natural number.

On Mathworld, the first Peano postulate is: 1) Zero is a number.

In other words, my book specifically says that zero is not a number, while Mathworld says it is.

So my question is: Is William J LeVeque (University of Michigan) – the author of my book - a crank, or has Mathworld got it wrong?
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Old 2005-07-17, 01:02   #2
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Most mathematicians consider {0,1,2,3,...} to be the natural numbers. However, some like to exclude 0. So, in about every math paper the author has to tell you which one he means.
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Old 2005-07-17, 01:40   #3
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Look at this link for more info:

http://mathworld.wolfram.com/WholeNumber.html

Zero is considered a whole number, thus 1 is the first natural number.
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Old 2005-07-17, 15:56   #4
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Thumbs down Peano postulates


According to Mathworld zero is both a positive and a negative integer. This is confusing to say the least.
I will have to go back to Brahmagupta about (598-665) the renowned Indian math'ician who is credited for first systematically using -ve numbers and zero for the first time.
If I may be permitted to use my own axioms I can formulate a logical answer but time does not permit me right now.
Mally
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Old 2005-07-17, 17:10   #5
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Zeta Flux & Rogue,

From the fact that you two are obviously in different camps on this, and yet manage to survive in peaceful harmony, respecting the other’s right to his point of view, I am going to assume that this difference is not a schismatic issue, and move on. Thank you for bringing it to my attention, though, as I dare say it will at some point come back to bite me.

Mally,
I think that if you go back and read Mathworld again you will find that it says that zero is both a non-positive and a non-negative number. But I am non-positive about this.
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Old 2005-07-17, 17:31   #6
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Quote:
Originally Posted by mfgoode

According to Mathworld zero is both a positive and a negative integer. This is confusing to say the least.
As I understand it, zero has no sign, but that doesn't mean that I understand it correctly.
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Old 2005-07-17, 19:07   #7
R.D. Silverman
 
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Quote:
Originally Posted by Numbers
Dr. Silverman will no doubt be delighted to know that this melon-head actually got himself a book on number theory. And I got all the way to page 8 before I have a question.

In my book the first Peano postulate is: 1) 1 is a natural number.

The third postulate says: 3) 1 is not the successor of any natural number.

On Mathworld, the first Peano postulate is: 1) Zero is a number.

In other words, my book specifically says that zero is not a number, while Mathworld says it is.

So my question is: Is William J LeVeque (University of Michigan) – the author of my book - a crank, or has Mathworld got it wrong?
Neither. A universal agreement on the definition of N, the natural numbers,
does not exist. Sometimes it is defined as {0,1,2,...}, sometimes as {1,2,3...}

It is a matter of definition. Both conventions are used.

While you are on this subject, may I suggest that you also investigate the
concept of "well-ordered", and its relation to the axiom of induction? Both
are tied into the construction of N via Peano arithmetic.....
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Old 2005-07-19, 09:07   #8
mfgoode
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Cool

Quote:
Originally Posted by Numbers
Zeta Flux & Rogue,

From the fact that you two are obviously in different camps on this, and yet manage to survive in peaceful harmony, respecting the other’s right to his point of view, I am going to assume that this difference is not a schismatic issue, and move on. Thank you for bringing it to my attention, though, as I dare say it will at some point come back to bite me.

Mally,
I think that if you go back and read Mathworld again you will find that it says that zero is both a non-positive and a non-negative number. But I am non-positive about this.

Good thinking Numbers. Keep it up!
Whats the difference in the meaning of non -negative but positive and non
positive but negative? If not either, then zero has no sign as rogue states.
If so IMHO it has no business to be on the number line of integers.
We even have a term for + ve infinity and -ve infinity. If infinity has signs why shouldn't zero be the exception?
May I boldly state that Zero is a positive integer! I am not as yet qualifying this statement with logical proof and axioms due lack of time.
I am going to the interior of India to night, where computers are scarce,
But it will give me time to cook up my theory which is at present all in my mind
Mally
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Old 2005-07-19, 11:51   #9
R.D. Silverman
 
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Quote:
Originally Posted by mfgoode

Good thinking Numbers. Keep it up!
Whats the difference in the meaning of non -negative but positive and non
positive but negative? If not either, then zero has no sign as rogue states.
If so IMHO it has no business to be on the number line of integers.
We even have a term for + ve infinity and -ve infinity. If infinity has signs why shouldn't zero be the exception?
May I boldly state that Zero is a positive integer! I am not as yet qualifying this statement with logical proof and axioms due lack of time.
I am going to the interior of India to night, where computers are scarce,
But it will give me time to cook up my theory which is at present all in my mind
Mally
Mathematics is not deemed to be correct or incorrect based
upon one's opinion.

As for the meaning of non-negative and non-positive, these terms are
well defined. The non-negative numbers is the set of reals that is the
complement of the negative numbers. Negative means less than 0.
Thus, non-negative means {x | x >= 0}. Similarly, non-positive means
{x | x <= 0}.

As for zero being unsigned, the DEFINITION of positive means greater than 0.
The DEFINITION of negative means less than 0. 0 is the unique real number
x such that x = -x. 0 is neither positive nor negative.

Why people turn this into a mystery is beyond my understanding. It is really
quite simple.

What do you mean by your statement "shouldn't 0 be the exception"?

You may NOT state that 0 is a positive integer. It is trivially not true.

You, of course, are free to dream up your own fantasy version of mathematics.

As for your claim "I am not as yet qualifying this statement with logical proof and axioms due lack of time.", it is not lack of time, but the fact that you
will not get a consistent set of axioms.
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Old 2005-07-19, 11:57   #10
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And maybe 0 isn't positive or negative as it equals nothing? I mean that you write down 0 to tell that you have none, so how can nothing be positive or negative? My 2 sents...
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Old 2005-07-19, 15:19   #11
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Quote:
Originally Posted by olmari
And maybe 0 isn't positive or negative as it equals nothing? I mean that you write down 0 to tell that you have none, so how can nothing be positive or negative? My 2 sents...
Zero equals Zero.

"Nothing" is not a mathematical concept. You are applying the common
English meaning of the word "nothing" in a context where it is not meaningful.

0 is a number. It is not "nothing"
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