Existence of Minimal Uncountable Well-Ordered Set/Proof Without Using Choice

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Theorem

There exists a minimal uncountable well-ordered set.

That is, there exists an uncountable well-ordered set $\Omega$ with the property that every initial segment in $\Omega$ is countable.


Proof

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:

$\AA = \set {\struct {A, \prec}: A \in \powerset \N}$

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:

$\struct {A, \prec} \sim \struct {A', \prec'}$

if and only if:

$\struct {A, \prec}$ is order isomorphic to $\struct {A', \prec'}$.

By Order Isomorphism is Equivalence Relation, $\sim$ is an equivalence relation.

Let $E$ be the set of all equivalence classes $\eqclass {\struct {A, \prec} } {}$ defined by $\sim$ imposed on $\powerset \N$


Define the relation:

$\eqclass {\struct {A, \prec_A} } {} \ll \eqclass {\struct {B, \prec_B} } {}$

if and only if:

$\struct {A, \prec_A}$ is order isomorphic to an initial segment $S_b$ of $\struct {B, \prec_B}$.


We claim that $\struct {E, \ll} = \Omega$.

To see that $\ll$ is well-defined, we use a commutative diagram:

$\xymatrix { \struct {A, \prec_A} \ar[d]^f \ar[r]^{g_1} & \map {S_b} B \[email protected]{-->}[d]^{g_2 \circ f} \\ \struct {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 $\eqclass {\struct {A, \prec_A} } {}$ 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 $\O$ 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 $\O$ to an initial segment in any non-empty $A$, as $\nu$ is vacuously order-preserving.

Thus $\eqclass {\struct {\O, \O} } {}$ is the smallest element of $E$.


Let $\alpha = \eqclass {\struct {A, \prec} } {}$ be an element of $E$.

We claim that $\struct {A, \prec}$ is order isomorphic to $\map {S_\alpha} E$, the initial segment of $E$ determined by $\alpha$.

To see this, define the mapping:

$f_\alpha: A \to E$:
$\map {f_\alpha} x = \eqclass {\struct {\map {S_x} A, {\prec \restriction_{\map {S_x} A} } } } {}$

where $\restriction$ denotes restriction.

This is a strict ordering for each $\prec \restriction_{\map {S_x} A}$, 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 $\map {f_\alpha} x \ll \map {f_\alpha} y$.

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 $x \in A$, $\map {f_\alpha} x$ is an element of $E$.

Thus it is an element of $\map {S_\alpha} E$, by the construction of $f_\alpha$.

By the definition of image set:

$f \sqbrk A = \map {S_\alpha} E$

From Injection to Image is Bijection, there is a bijection from $A$ to $f \sqbrk A$.

That is, from $A$ to $\map {S_\alpha} E$.

As $f_\alpha$ was shown to be (strictly) increasing, there is an order isomorphism between $\struct {A, \prec}$ and $\map {S_\alpha} E$.

Recall that $E$ has a smallest element $\eqclass {\struct {\O, \O} } {}$.

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 $a \in \map {S_a} E \cap F$.

From Inverse of Order Isomorphism is Order Isomorphism, $f^{-1}$ is also an isomorphism.

That means that $a$ is isomorphic to some $\struct {A, \prec} \in \AA$


But $A$ has a minimal element, by the definition of $\AA$.

From Order Isomorphism on Well-Ordered Set preserves Well-Ordering, $\map {S_a} E \cap F$ is a well-ordered set.

Thus it has a smallest element.

The smallest element in $\map {S_a} E$ 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 $\map {S_a} E \cap F$ 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.

Aiming for a contradiction, suppose 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:

$\map f x \prec \map f y$ in $\N$

if and only if:

$x \ll y$ 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 $\map {S_\alpha} E$ in $E$, which is isomorphic to some $\struct {A, \prec} \in \AA$.

Then $h^{-1} \sqbrk \N$ 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 $\map {S_\alpha} E$

We have shown that $\map {S_\alpha} E$ 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 $\struct {E, \ll}$ is an uncountable well-ordered set, every initial segment of which is countable.

$\blacksquare$

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