According to Michael Spivak's book on calculus, in his initial discussion of the basic properties of numbers, he talks about 1 not being equal to 0 in regards to the existence of a multiplicative identity property. He does attempt a brief explanation as to why this needs to be stated, however, his meaning is still very unclear to me. Can someone please clarify this for me?
He is a link to the book in google books. It is P6 on pp. 5-6.
Calculus - Google Books
Thank you for your quick response. I am not very familiar with fields so this may simply be beyond my abilities at this time, unless there is a more intuitive way to understand this idea. Basically though, is "one is not equal to zero" an axiom that must be stated in order for a working proof of the multiplicative identity property?
Because "2" is meaningless. It is kind of confusing to think of them as which is why I wrote them with subscripts. A field need not have "numbers" in them in the traditional sense. Really, all " " means is that anything "added" to it doesn't change. I could have equally called it , , or .
I'm definitely going to have to ponder this a bit more, but I'm starting to get the line of thinking. This is very interesting. So would it be correct to think of "x" as an element that has both additive and multiplicative identity properties? And that the only way such an element could exist is if were the only element in such a field? And therefore could not exist in a field with multiple elements, such as the field of real numbers?