Proving reflexive, symmetric, transitive properties

[issacnewton]

Hello

I was reading Spivak's calculus. It starts with discussing the familiar axioms of the real numbers. He calls them properties. At some another forum, I came across the reference to Landau's "Foundation of Analysis" as a background for analysis. So I referred to that book. On the very first page , he says that reflexive, symmetric, and transitive properties of the natural numbers are taken for granted on logical grounds. So Landau is taking reflexive, symmetric and transitive properties as axioms of natural numbers. I was wondering if we can prove reflexive, symmetric and transitive properties from the field axioms given in Spivak's calculus book.

thanks

 

[micromass]

So first of all, the approach of Landau and the approach of Spivak are very different. What Spivak does is introduce the real number axiomatically and then construct the natural numbers as a subset of the reals. What Landau does is to take the natural numbers axiomatically, and then construct the real numbers from then. So the two approaches are kind of inverses to each other.

Now, what you are referring to are the axioms of equality. These are axioms of basic logic, they cannot be proven in the approach of Spivak or Landau. They have to be taken for granted because they define what equality actually is. Other axioms of equality are surely possible, but you need to start from something.

 

[issacnewton]

Thanks micromass. Book like Spivak's Calculus should mention what you are saying, just for completeness. I was trying to prove something in Spivak, and I wanted to
use transitive property of real numbers. Since Spivak supposedly starts from scratch, I started wondering where does transitive property come from.