Talk:Real Analysis/Dedekind's construction

Multiplication error
Dedekind multiplication is tricky. The definition for multiplication as stated currently doesn't work if both numbers are negative. Let $$\alpha_1$$ and $$\alpha_2$$ both be the cut for the real number -2, which we'll assume has been constructed by other methods for now :). -2 * -2 = 4. Let's consider some number in our two Dedekind cuts. I like -10. $$-10 \in \alpha_1$$ and $$-10 \in \alpha_2$$, so by your definition $$100 \in \alpha_1 \cdot \alpha_2$$. However, 100 is firmly not in the Dedekind cut for 4. --Shadytrees (talk) 02:47, 22 June 2008 (UTC)

Your of course correct. I have made some change that attempts to be closer to the truth, but technically the multiplication as I defined it is still incorrect because this page defines cuts as open. It may be easier to allow them to be open or closed, and define an equivalence relation. I shall have to think on this a bit. Thenub314 (talk) 10:31, 9 January 2009 (UTC)

Naming
It seems like a bad idea to call the set of cuts $$\mathbb R$$ before we prove that it satisfies the axioms we put forth for $$\mathbb R$$.