Update comment.

_data/comments/on-complete-ordered-fields/comment-1568399977792.yml

@@ -2,5 +2,5 @@ _id: d6b59760-d655-11e9-b777-a1f56fd4ee18
 name: Andrej
 email: 59d57d95bc7c45ced5f1969279cec06b
 url: 'http://www.andrej.com/'
-message: "As far as I can tell, Toby's argument requires cotransitivity of $<$, above I explicitly noted  this in the proof which gets you excluded middle from a complete order: the supremum $s$ of the set $S$ is either above $a$ or below $b$.\r\n\r\nThe idea to have $\\lnot\\lnot$-bounded Dedekind cuts is good! Let's see, in $Sh(X)$ for a reasonable $X$ the Dedekind cuts are the sheaf of real-valued continuous maps. The unbounded cuts correspond to maps into $[-\\infty,\\infty]$. The $\\lnot\\lnot$-bounded ones would be the maps into $[-\\infty, \\infty]$ which take on values $-\\infty$ and $\\infty$ on a set with empty interior? Indeed, those will fail to be locally bounded (which is what the archimedean axiom amounts to)."
+message: "As far as I can tell, Toby's argument requires cotransitivity of $<$, above I explicitly noted  this in the proof which gets you excluded middle from a complete order: the supremum $s$ of the set $S$ is either above $a$ or below $b$.\r\n\r\nThe idea to have $\\lnot\\lnot$-bounded Dedekind cuts is good! Let's see, in $Sh(X)$ for a reasonable $X$ the Dedekind cuts are the sheaf of real-valued continuous maps. The unbounded cuts correspond to maps into $[-\\infty,\\infty]$. The $\\lnot\\lnot$-bounded ones would be the maps into $[-\\infty, \\infty]$ which take on values $-\\infty$ and $\\infty$ on a set with empty interior? Indeed, those will fail to be locally bounded (which is what the archimedean axiom amounts to). \r\n\r\n But wait, such a sheaf does not form a ring in the topos. Hmmm."
 date: 1568399977