# Is this Correct?

• Jan 30th 2014, 08:41 AM
davidciprut
Is this Correct?
Statement: if m and n are natural numbers this condition can't happen: n<m<n+1

Proof: So I said that let's assume that n<m<n+1 is true.
So from the first part we get n<m and from here we get that m-n is a natural number.
So there exist k in N and we can express m-n=k+1 (I depended on a statement that I proved before)
and from here we get m-k=n+1
So from the second part of the inequality we get m<n+1=m-k and from here we get that k<0 and this a contradiction to the fact that we can express m-n as k+1 and therefore this condition can't happen.

Happy if someone can give feedback thanks!
• Jan 30th 2014, 08:48 AM
romsek
Re: Is this Correct?
can't you just subtract n from both sides to obtain

0 < m < 1

which is clearly a contradiction to m being a natural number?

does the concept of subtraction exist at this point in your algebra yet? (seriously, it looks like you are still building the natural numbers)
• Jan 30th 2014, 09:02 AM
davidciprut
Re: Is this Correct?
Yes now that I realized, all the things that I did was kinda redundant...
Look I am confused too, the way we built the natural numbers was with properties of inductive sets, definition of inductive he defined it as follows
There is and inductive set I
1)1 is in I
2)n is in I then n+1 is in I

But he didn't even defined any arithmetic operation on N. He defined Fields previously but N is not a field so I guess it's not valid for N.
And then he defined N as the intersection of all inductive sets... So this is what I know..
I would be more than happy if you could suggest a good analysis book for me..
• Jan 30th 2014, 09:34 AM
HallsofIvy
Re: Is this Correct?
Quote:

Originally Posted by romsek
can't you just subtract n from both sides to obtain

0 < m < 1

You mean 0< m- n< 1 which is still a contradiction since, with m and n natural numbers, m-n is also a natural number.

Quote:

which is clearly a contradiction to m being a natural number?

does the concept of subtraction exist at this point in your algebra yet? (seriously, it looks like you are still building the natural numbers)
• Jan 30th 2014, 10:42 AM
Hartlw
Re: Is this Correct?
A def of the natural numbers which satisfies any other definition is:
1=1
2=1+2
3=1+3
.
.
There is no natural number between n and n+1, by def.
• Jan 30th 2014, 01:46 PM
Plato
Re: Is this Correct?
Quote:

Originally Posted by Hartlw
A def of the natural numbers which satisfies any other definition is:
1=1
2=1+2
3=1+3

.
.
There is no natural number between n and n+1, by def.

There is almost nothing in the above quote that is in the least correct.
Have see many, many different ways to develop the natural numbers. Nothing in the above could possibly be correct.
I hope the OP will not be confused by useless and jumbled reply.
• Jan 30th 2014, 10:15 PM
MINOANMAN
Re: Is this Correct?
Hey Boy.....
(There is almost nothing in the above quote that is in the least correct.
Have see many, many different ways to develop the natural numbers. Nothing in the above could possibly be correct.
I hope the OP will not be confused by useless and jumbled reply. )

it is time I believe to stop your vicious rhetoric and study more.... some algebra definitions....I sugest first to start with Peano's axioms...here
Peano axioms - Wikipedia, the free encyclopedia
• Jan 31st 2014, 03:39 AM
Plato
Re: Is this Correct?
Quote:

Originally Posted by MINOANMAN;809594it is time I believe to stop your vicious rhetoric and study more.... some algebra definitions....I sugest first to start with Peano's axioms...here [url=http://en.wikipedia.org/wiki/Peano_axioms
Peano axioms - Wikipedia, the free encyclopedia[/url]

Actually you should take you own advice to heart. You clearly have no idea about the history of Peano axioms.
• Jan 31st 2014, 07:42 AM
Hartlw
Re: Is this Correct?
My definition is perfectly consistent with the Peano Postulates:

1) 1 is a natural number.
2) Each number x has a successor, x’
3) x’≠1
4) If x’=y’ then x=y
5) axiom of induction

Ref: Landau, Foundations of Analysis.
• Jan 31st 2014, 08:04 AM
MINOANMAN
Re: Is this Correct?
Quote:

Originally Posted by Hartlw
My definition is perfectly consistent with the Peano Postulates:

1) 1 is a natural number.
2) Each number x has a successor, x’
3) x’≠1
4) If x’=y’ then x=y
5) axiom of induction

Ref: Landau, Foundations of Analysis.

Mr Hartlw :You are absolutely right....your comments as well as those of Mr HallsofIvy are in comply with Peano's Axioms... disregard vicius comments...
• Jan 31st 2014, 10:05 AM
Hartlw
Re: Is this Correct?
I note that the definition:
1=1, 2=1+1, 3=1+2…
came before the Peano Axioms.
Peano didn’t create the natural numbers, he axiomatized them.

My personal preference is to go to the source, rather than its abstraction, whenever it is convenient and simpler.