By the axiom of powers, there exists the power set $\powerset \N$, where $\N$ is the set of natural numbers. By Power Set of Natural Numbers is not Countable, this set is uncountable. Consider the set of ordered pairs: :$\AA = \left\{ { \left({A,\prec}\right) : A \in \powerset \N}\right\}$ where: :the first coordinate is a (possibly empty) subset of $\N$ :the second coordinate is a strict well-ordering on $A$. There is at least one pair of this form for each $A \subseteq \N$, taking $\prec$ to be the usual (strict) ordering of the natural numbers. The usual ordering is a well-ordering, from the Well-Ordering Principle. Define the relation: :$(A, \prec) \sim (A',\prec')$ {{iff}}: :$(A, \prec)$ is order isomorphic to  $(A',\prec')$. By Order Isomorphism is Equivalence Relation, $\sim$ is an equivalence relation. Let $E$ be the set of all equivalence classes $\left[\!\left[{\left({A,\prec}\right)}\right]\!\right]$ defined by $\sim$ imposed on $\powerset \N$  Define the relation: :$\left[\!\left[{\left({A,\prec_A}\right)}\right]\!\right] \ll \left[\!\left[{\left({B,\prec_B}\right)}\right]\!\right]$ {{iff}} :$(A, \prec_A)$ is order isomorphic to an initial segment $S_b$ of  $(B,\prec_B)$. We claim that $\left({E,\ll}\right) = \Omega$. To see that $\ll$ is well-defined, we use a commutative diagram:  $\xymatrix { \tuple {A, \prec_A} \ar[d]^f \ar[r]^{g_1} & \map {S_b} B \ar@{-->}[d]^{g_2 \circ f}\\ \tuple {C, \prec_C} \ar[r]^{g_2} & \map {S_d} B}$ where $f$ is a mapping defining $\sim$, and $g_1,g_2$ are mappings defining $\ll$. From No Isomorphism from Woset to Initial Segment, no equivalence class $\left[\!\left[{\left({A,\prec_A}\right)}\right]\!\right]$ can bear $\ll$ to itself. From Wosets are Isomorphic to Each Other or Initial Segments, $\ll$ is connected. From the definition of an order isomorphism, $\ll$ is an ordering. Thus $\ll$ is a strict total ordering. The empty set $\O$ contains no elements that could define an initial segment in it. Consider the empty mapping: $\nu: \O \to \O$. Such a mapping $\nu$ is bijective, by Empty Mapping to Empty Set is Bijective. But $\varnothing$ is itself an initial segment of any non-empty set, from Initial Segment Determined by Smallest Element is Empty. Thus there is an order-isomorphism from $\varnothing$ to an initial segment in any non empty $A$, as $\nu$ is vacuously order-preserving. Thus $\left[\!\left[{\left({\varnothing,\varnothing}\right)}\right]\!\right]$ is the smallest element of $E$. Let $\alpha = \left[\!\left[{\left({A,\prec}\right)}\right]\!\right]$ be an element of $E$. We claim that $\left({A,\prec}\right)$ is order isomorphic to $S_\alpha(E)$, the initial segment of $E$ determined by $\alpha$. To see this, define the mapping: :$f_\alpha: A \to E$: :$f_\alpha(x) = \left[\!\left[{\left({S_x(A),{\prec \restriction_{S_x(A)}}}\right)}\right]\!\right]$ where ${\restriction}$ denotes restriction. This is a strict ordering for each $\prec \restriction_{S_x(A)}$, from Restriction of Strict Well-Ordering is Strict Well-Ordering. Suppose $x \prec y$ in $A$. Then $S_x$ is an initial segment of $S_y$. From No Order Isomorphism Between Distinct Initial Segments of Woset, $S_x \ne S_y$. Thus $f_\alpha(x) \ll f_\alpha(y)$. Conclude that $f_\alpha$ is strictly increasing. From Strictly Monotone Mapping with Totally Ordered Domain is Injective, $f_\alpha$ is therefore an injection. For all $x \in A$, $f_\alpha(x)$ is an element of $E$. Thus it is an element of $S_\alpha(E)$, by the construction of $f_\alpha$. By the definition of an image set: :$f[A] = S_\alpha(E)$ From Injection to Image is Bijection, there is a bijection from $A$ to $f[A]$. That is, from $A$ to $S_\alpha(E)$. As $f_\alpha$ was shown to be (strictly) increasing, there is an order isomorphism between $\left({A,\prec}\right)$ and $S_\alpha(E)$. Recall that $E$ has a smallest element $\left[\!\left[{\left({\varnothing,\varnothing}\right)}\right]\!\right]$. Let $F$ be any non-empty proper subset of $E$ As it is non-empty, $F$ has some element in it. Call the element $a$. If such an $a$ precedes all elements in $F$, then $a$ is the smallest element of $F$. Suppose $a$ is not the smallest element in $F$. Then $F$ has at least two elements. Then $a \in S_a(E) \cap F$. From Inverse of Order Isomorphism is Order Isomorphism, $f^{-1}$ is also an isomorphism. That means that $a$ is isomorphic to some $(A,\prec) \in \AA$ But $A$ has a minimal element, by the definition of $\AA$. From Order Isomorphism on Well-Ordered Set preserves Well-Ordering, $S_a(E) \cap F$ is a well-ordered set. Thus it has a smallest element. The smallest element in $S_a(E)$ is the smallest element of $E$, by the definition of initial segment. We assumed that $a$ is not the smallest element in $F$, so the smallest element of $S_a(E) \cap F$ is the smallest element of $F$. As $F$ was arbitrary, we can conclude that every non-empty subset of $E$ has a smallest element. So $E$ is well ordered. It remains to be proven that $E$ is uncountable and every initial segment of $E$ is countable. {{AimForCont}} that $E$ is countable. Then there is a bijection: :$h: E \leftrightarrow \N$. From Order Isomorphism on Well-Ordered Set preserves Well-Ordering, we can define the strict ordering: :$f(x) \prec f(y)$ in $\N$ {{iff}} :$x \ll y$ in $E$. From Composite of Order Isomorphisms is Order Isomorphism, we can consider the isomorphism $f_\alpha \circ h$, where $f_\alpha$ is the above isomorphism to some $S_\alpha(E)$ in $E$, which is isomorphic to some $(A,\prec) \in \AA$. Then $h^{-1}[\N]$ is order isomorphic to all of $\AA$. Then there is a bijection between $\powerset \N$ and $\N$, contradicting Power Set of Natural Numbers is not Countable. From this contradiction, $E$ cannot be countable. Consider any initial segment $S_\alpha(E)$  We have shown that $S_\alpha(E)$ is order isomorphic to some $\tuple {A, \prec} \in \AA$. But $A$ is a subset of the natural numbers. From Subset of Countably Infinite Set is Countable, $A$ is countable. We have shown that $\left({E,\ll}\right)$ is an uncountable well-ordered set, every initial segment of which is countable. {{qed}}