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I was going through Stillwell's otherwise nice book The Real Numbers when I noticed that his definition of the Dedekind product of reals seemed wrong.

His definition of the product is quite simple: if $L$ and $L'$ are lower Dedekind cuts, then $$LL' = \{rr':\ r \in L \text{ and } r' \in L'\}$$

This seems wrong, for otherwise, wouldn't $LL'$ be unbounded above? For instance, as $L,L' \neq \emptyset$, there are rationals $l,l' \in L,L'$ respectively. Now, for any rational $r > 1$, the negative rational $\rho = \min(- r, l)$ is in $L$ by downward closure. Similarly, the negative rational $\rho' = \min(- r, l')$ is in $L'$. So, according to his definition, $\rho\rho' \in LL'$. But since $1 < r < r^2 \leq (- \rho)(- \rho') = \rho\rho'$, this means that no rational $r$ can bound $LL'$ above.

Even thought, I am quite sure about my reasoning, this book, sadly, does not have any errata listings to confirm my thoughts. I just want to make sure that I am not missing anything obvious before reporting this issue to the author.

balddraz
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    You're certainly correct, special care is needed to define the multiplication to avoid the problem you're discussing. The wikipedia page on construction of the real numbers by Dedekind cuts does a good job of explaining the correct method. – CyclotomicField Jan 13 '18 at 17:32
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    Yeah, this doesn't work. The correct definition can be found, for example, here: https://www.math.brown.edu/~res/INF/handout3.pdf towards the end – Ashwin Iyengar Jan 13 '18 at 17:38
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    I too encountered problems with the book. Here is one: https://math.stackexchange.com/questions/2358700/if-inaccessible-sets-exist-their-existence-is-not-provable-in-zf. –  Jan 13 '18 at 17:45
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    Hmm. I guess I will report this problem to the author. It's a shame. Now about 3 entire sections of the book are unreadable. – balddraz Jan 13 '18 at 17:59

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