__I. Formalities__

A set S is *uncountable* if there is no injection from S to a subset of the natural numbers N = {0,1,2,...}

The *Cartesian product* A x B is the set of all pairs (a,b) such that a is in A and b is in B. In the case where A=B=R, the set of real numbers, the Cartesian product is the familiar Cartesian coordinate system.

A *relation* R on S is a subset of S x S; if (x,y) is in R, write xRy.

A *partial order* on S is a relation R on S that satisfies three criteria:

(1) *R is reflexive*: For all x in S, xRx

(2) __R is symmetric__: For all x,y in S, if xRy then yRx

(3) __R is transitive__: For all x,y,z in S, if xRy and yRz, then xRz

Examples include the "less than or equal to" relation on the real numbers, the "subset" relation on the power set of S, etc.

A *paritally-ordered* set is a pair (S,*) where S is a set and * is a partial order on S; where the context is clear, this formalism will be dropped.

A *total order* on S is a partial order that satisfies:

(4) For all x, y in S, either xRy or yRx

A *totally-ordered* set is a partially-ordered set whose partial order is a total order.

A partially-ordered set (S, <=) is *locally finite* if:

(5) For all x,y in S, the *interval* [x,y] = {z in S | x <= z and z <= y} is finite

**II. Problem Statement**

Does there exist an uncountable totally-ordered locally finite set?

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