Michael Spivak: basic properties of numbers

[shavuklia]

Hi there,

I'm reading from this book:
http://cale.yi.org/share/Michael Spivak - Calculus.pdf

At the top of page 6, the author writes (rightly after having mentioned the first six basic properties of numbers):

"Moreover, 1 /= 0."

I don't see why it was deemed necessary to add this restriction? The author himself explains:

"The assertion that 1 /= 0 may seem a strange fact to list, but we have to list it, because there is no way it could possibly be proved on the basis of the other properties listed - these properties would all hold if there were only one number, namely, 0."

I hope someone could help me out!

-Sha

 

[pka]

I'm reading from this book:
http://cale.yi.org/share/Michael Spivak - Calculus.pdf
At the top of page 6, the author writes (rightly after having mentioned the first six basic properties of numbers):
"Moreover, 1 /= 0." I don't see why it was deemed necessary to add this restriction? The author himself explains:
"The assertion that 1 /= 0 may seem a strange fact to list, but we have to list it, because there is no way it could possibly be proved on the basis of the other properties listed - these properties would all hold if there were only one number, namely, 0."

In this case, I think you are need to accept Spivak's own explanation. There are several different sets of axioms that can be used to develop the real number system. Although it is not common, nevertheless Spivak chose to use a set of axioms that can be modelled by a set that contains only one element, 0.