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Suppose that $A$ is a Riesz space (vector lattice). Recall the following terminology:

  • A is Archimedean if for any $x,y\geq 0$ such that $n x\leq y$ for $n=1,2,\ldots$ is follows that $x=0$;

  • A is Dedekind complete if every upper bounded subset of $A$ has a supremum.

As the title of my post suggest, my question is: Does Dedekind completeness imply the Archimeadean property?

I have the following proof, which I would like to check if it is right:

Suppose that $x,y\geq 0$ satisfied that $n x\leq y$ for $n=1,2,\ldots$ in a Dedekind complete Riesz space $A$. To reach a contradiction, suppose that $x\neq 0$. Then the set $$B:=\{ n x\colon n=1,2,\ldots\} $$ has an upper bound; namely $y$. Since $A$ is Dedekind complete, we can take $z:=\sup B$. Since $x> 0$ (i.e. $x\geq 0$ and $x\neq 0$), it follows that $z>z-x$. Hence $z-x$ is not an upper bound of $B$, hence there exists $m\in\mathbb{N}$ such that $z-x\ngeq m x$. But this means that $z\ngeq m x + x=(m+1)x$, a contradiction.

John Paul Littlefield
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1 Answers1

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Yes, your argument is valid. In particular, were a supremum of the natural numbers (embedded in some fashion) to exist within a Dedekind-complete system, that system will either imply the existence of a biggest natural, as you've essentially observed, and thus contradict itself, or the supremum will have to behave badly and one or more algebraic properties will break - typically, addition fails to be injective, meaning the supremum "absorbs" things added to it (i.e. $z \pm x = z$), and subtraction is no longer always possible. What this means is you get "stuck" on it, so that when you do the subtraction of $x$ you remain there at the supremum, instead of getting back into the naturals to the "left" of you, as though it were made of a sort of magic glue that sticks fast and cakes hard on your feet and worse, requires infinite strength to break, at which point there is no telling where you will land. The first part of this is the non-injectivity of addition, and the second corresponds to subtracting the infinite supremum from itself: it is a kind of "irresistible force meets immovable object" paradox, or as the Chinese would call it, the Mao Dun - Spear-Shield. The result of this operation is undefined.

The_Sympathizer
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