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So the title is cryptic, but I was hoping someone could help me understand the following

This assumes a knowledge of Dedekind cuts.

Let us define $0^*$to be the following cut

$0^* = \lbrace p: p<0 \rbrace$

and suppose $\alpha \in \mathbb{R}$, i.e. $\alpha$ is a cut.

Now, presumably, $\alpha \cdot 0^* = 0^* \cdot \alpha = 0^*$

My question is the following: Is this by definition, and if so why are we allowed to simply define it? Or is this something that can be proved, and if so how is it proved?

I'm just not sure how to go about doing this one, and the reference I'm using (Rudin) says virtually nothing about it.

Oh, the other pertinent detail is the way multiplication is defined for Dedekind cuts, which (up the point I've studied) only has done so for positive reals, i.e. give $\alpha, \beta \in \mathbb{R}$, then

$\alpha \beta = \lbrace p: p\leq rs, r\in \alpha, s\in\beta, r>0, s>0 \rbrace$

This is also sometimes defined

$\alpha \beta = \lbrace \lbrace p: p \leq 0\rbrace \cup \lbrace rs : r\in \alpha, s \in \beta, r>0, s>0\rbrace \rbrace$

I.e, all the positive products from alpha and beta and everything below (and including) zero.

Since $0^*$ contains exclusively negative rationals, the above definition for real multiplication can't even be used to discuss the product $\alpha \cdot 0^*$

Thanks.

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    What book of Rudin do you use and which chapter? There are several ways to resolve these problems, I think, the simplest is probably to declare $0\cdot a=a\cdot 0=0$ by definition and for non-zero $\alpha$ and $\beta$ multiplication is $\pm|\alpha|\cdot|\beta|$ (for positive numbers it is defined in the question) with the negative sign iff exactly one of $\alpha,\beta$ is negative. – 2017-02-07
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    Principles of mathematical analysis, Appendix to chapter 1 believe (construction of the reals). Is there really no proof so to speak? It just seems so arbitrary. Dedekind cuts make little enough sense to me as it is, without just declaring things to be true with virtually zero justification. Anyway, pg 29: https://notendur.hi.is/vae11/%C3%9Eekking/principles_of_mathematical_analysis_walter_rudin.pdf – 2017-02-07
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    Yes, Rudin defines this as I said. "We confine ourselves first to $\mathbb R^+$, so everything in step $6$ is about positive numbers only. In step 7 he extends multiplication to all $\mathbb R$. Most proofs are omitted there (and in most books) because they are straitforward and tedious. I didn't got about virtually zero justification. There are several constructions of reals, one that is principally different and appeals to me more is via Cauchy sequences, see https://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Construction_from_Cauchy_sequences Dedekind cuts also varies. – 2017-02-07
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    But the problem is that all constructions lead to some pretty unpleasant proofs. You don't lose much, if you don't understand them now fully though, the constructions of reals require some experience in math from you for not appearing to be black magic. – 2017-02-07
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    Well so far I've proven everything regarding Dedekind cuts up to multiplication. I want to know the details about the constructions, in case it can answer questions down the road. Cauchy sequences are next. – 2017-02-07
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    @Wolfram One of the problems I have is the following: when multiplication is extended to $\mathbb{R}$ in step 7, do all the field axioms for multiplication have to be reproved, or do they essentially follow semi-trivially from definitions previously made? Basically, do we have to prove the field axioms for multiplication twice: once for $\mathbb{R}^+$, then again for $\mathbb{R}$, or are the field axioms implied for $\mathbb{R}$ from having proved them for $\mathbb{R}^+$? – 2017-02-07
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    Not all field axioms hold for $\mathbb R^+$. If you mean the properties of multiplication, then stricly speaking we need to prove them again for $\mathbb R$, for we can just consider several cases of the signs and use already known properties of $\mathbb R^+$. But of course they are not obvious from definitions, some work should still be done. – 2017-02-08

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