The Well-Ordering Number

Intuition: The well-ordering number can be seen as a measure of the compactness properties of a logic $\mathcal{L},$ in that the larger $w(\mathcal{L})$ is, then more fragments of compactness are lost

Pinning Down Ordinals: A set of sentences $\Phi$ of a logic $\mathcal{L}$, $\Phi \subseteq \mathcal{L}[\tau ]$, where $<\in \tau ,$ pins down an ordinal $\alpha$ iff i) For all $\mathfrak{A}\models \Phi ,{<}^{\mathfrak{A}}$ is a well-ordering of its field (domain) and ii) There is $\mathfrak{B}\models \Phi$ such that ${<}^{\mathfrak{B}}$ is of order type $\alpha$

$w_{\kappa }(\mathcal{L})=$ sup{$\alpha |\Phi$ pins down $\alpha$ and $|\Phi |\le \kappa$} sup${}_{\kappa \in ORD}{w}_{\kappa }(\mathcal{L})$ is defined as the supremum of all ordinals that are pinned down by an arbitrary $\Phi \subseteq \mathcal{L}[\tau ]$. Equivalent to the supremum of all $w_{\kappa }(\mathcal{L})$ for arbitrary $\kappa$.

The well-ordering number of $\mathcal{L}$ is the case where $\kappa =1$, i.e., $w(\mathcal{L})=w_1(\mathcal{L})$ $w(\mathcal{L})=$ sup{$\alpha |\phi$ pins down $\alpha$ and $\phi$ is an $\mathcal{L}$-sentence}

If a logic $\mathcal{L}$ does not have a well-ordering number, then $w(\mathcal{L})=\infty$

Note: $w_1(\mathcal{L})=w(\mathcal{L})=w_{\kappa }(\mathcal{L})$

E.g., Consider ${\mathcal{L}}_{\omega ,\omega }$ first-order logic The well-ordering number $w({\mathcal{L}}_{\omega ,\omega })$ is $\omega$ and in fact for any $\kappa \in ORD$, $w_k({\mathcal{L}}_{\omega ,\omega })=\omega$. $⟹$ Consider the least uncountable ordinal ${\omega }_1$, for some uncountable subset $S\subseteq {\omega }_1$, then we will not be able to define $S$ Note: ‘Definability’ means that there is a WFF describing a set. i.e., a definable set $S$ satisfies a WFF $\phi$. $\to$ For a model $\mathfrak{A}\models \phi ,{<}^{\mathfrak{A}}$ is a well-ordering of order type $\alpha <w_{\kappa }(\mathcal{L})$ $\to$ For $\beta <\alpha ,\beta$ is also characterized by ${<}^{\mathfrak{A}}$

Note: If $<\notin \tau$, for $\tau$ is some vocabulary, then nothing can be said about a logic based on $\tau ,\mathcal{L}(\tau )$, as the notion of well-ordering cannot be expressed w/in the language.

Boundedness:

• We say a logic $\mathcal{L}$ is bounded if it has a well-ordering number.
• A logic $\mathcal{L}$ is unbounded it has no well-ordering number, i.e., $w(\mathcal{L})=\infty$

Strong Logics: A logic $\mathcal{L}$ is said to be strong if there exists an $\mathcal{L}$-sentence $\varphi$ which pins down arbitrarily large ordinals, that is, $\varphi$ characterizes the entire class of well-ordering

Relation to The Well-Ordering Number: It can be observed that is $\mathcal{L}$ is strong, then $\mathcal{L}$ is unbounded by implication (i.e., $\mathcal{L}$ is strong $⟹w(\mathcal{L})=\infty$) This is because if a single sentence can pin down arbitrarily large ordinals than the supremum of all ordinals pinned down by a set of $\mathcal{L}$-sentences does not exist (and no upper bounds as well!). This converse is not necessarily true i.e., $w(\mathcal{L})=\infty /⟹\mathcal{L}$ is strong.

Strong vs $w(\mathcal{L})=\infty$ $w(\mathcal{L})=w_1(\mathcal{L})=\infty$ means that all ordinals can be pinned down by some (potentially different) $\mathcal{L}$-sentence $\phi$ (i.e., consider an ordered sequence of ordinals being pinned down, if the logic is proper-class sized, there can be a sentence for every pinned down ordinal, there doesn’t exist a supremum and not a single sentence pins down all sentences). While a logic is strong if there exists a single $\mathcal{L}$-sentence $\phi$ which does pin down all ordinals and thus, characterizes the class of well-orderings (i.e., a definition of well-ordering, and where the class of well-orderings is elementary in $\mathcal{L}$).

E.g., Consider the infinitary logic ${\mathcal{L}}_{\infty ,\omega }$, ${\mathcal{L}}_{\infty ,\omega }$ does not have a well-ordering number (i.e.,$w(\mathcal{L})=\infty$) and so is unbounded. This is because the class of ${\mathcal{L}}_{\infty ,\omega }$-sentences forms a proper-class, even in finite vocabularies ($|\tau |<\omega$), but ${\mathcal{L}}_{\infty ,\omega }$ is not strong. Consider the disjunction of the proper-class of ${\mathcal{L}}_{\infty ,\omega }$ formulae, $\underset{\alpha \in ORD}{\bigvee }{\phi }_{\alpha }$, this pins down every ordinal $\alpha$ and so defines the entire class of well-orderings $WO$. But this is not expressible as a single ${\mathcal{L}}_{\infty ,\omega }$-sentence, and so the logic ${\mathcal{L}}_{\infty ,\omega }$ is not strong. $\to {\mathcal{L}}_{\infty ,\omega }$ not being strong is due to the class $WO$ not forming a projective class ($PC$) in ${\mathcal{L}}_{\infty ,\omega }$. ($w({\mathcal{L}}_{\infty ,\omega })=\infty$ and yet ${\mathcal{L}}_{\infty ,\omega }$ not being strong can be interpreted that a vestige of compactness is still preserved for this logic)

Proof: (Do proof)

E.g., A logic which is strong is Second-Order Logic, ${\mathcal{L}}^2$. This is because the notion of well-ordering can be expressed as a single sentence in ${\mathcal{L}}^2$, that is, it contains a definition of well-ordering and so characterizes all of $WO$ $\to$ and by implication of ${\mathcal{L}}^2$ being strong, $w({\mathcal{L}}^2)=\infty$

We define by recursion: i) ${\beth }_0(\kappa )=\kappa$ ii) ${\beth }_{\alpha +1}(\kappa )=2^{{\beth }_{\alpha }(\kappa )}$ iii) ${\beth }_{\lambda }(\kappa )=$ sup{${\beth }_{\alpha }(\kappa )|\alpha <\lambda$} for $\lambda$ is a limit ordinal $\to$ The size of these Beth numbers can be illustrated via a modification of the Von Neumann universe i) $V_0^{\ast }(A)=A$ ii) $V_{\alpha +1}(A{)}^{\ast }=\mathcal{P}(V_{\alpha }^{\ast }(A))$ iii) $V_{\lambda }^{\ast }(A)=\bigcup _{\alpha <\lambda }{V}_{\alpha }^{\ast }(A)$ for $\lambda$ is a limit ordinal

Note: For all $\alpha$, we have $|V_{\alpha }^{\ast }(A)|={\beth }_{\alpha }(\kappa )$

Assume that $\lambda <h_{\kappa }(\mathcal{L})$ is pinned down by a set $\Phi \subseteq \mathcal{L}[\tau ]$ of power $\le \kappa$. The vocabulary, $\tau$, can be chosen relational with new binary relation symbols $v,\epsilon$ and constants $c_0,c_1$. Let ${\Phi }^{\prime }$ consist of the following: i) $\exists zv{c}_{0}z\wedge \forall z(v{c}_{0}z\vee v{c}_{1}z);$ ii) ${\phi }^{\left\{x|v{c}_{0}z\right\}}$ for $\phi \in \Phi ;$ iii) $\forall x,y(\forall z(z\epsilon x\leftrightarrow z\epsilon y)\to x=y);$ that is, ”$\epsilon$ is s extensίonal”; iv) $\forall z(v{c}_{1}z\leftrightarrow \forall u\epsilon v{c}_{0}z)$ Then for any models $\mathfrak{A}\models {\Phi }^{\prime }$, we have ${\mu }_i=$|{$a\in A|c_i,a\in v^{\mathfrak{A}}$}| such that $|A|\le {\mu }_0+{\mu }_1$, where ${\mu }_0>h_{\kappa }(\mathcal{L})$ and ${\mu }_1\le {\beth }_1({\mu }_0)$. Hence ${\Phi }^{\prime }$ pins down cardinals and $h_{\kappa }(\mathcal{L})>{\beth }_1(\lambda )$

Theorem: Given that each ordinal $\alpha <w_{\kappa }(\mathcal{L})$ can be pinned down by a set ${\Psi }_{\alpha }$ of sentences, $|{\Psi }_{\alpha }|\le \kappa$ having a model $\mathfrak{A}$ of power $<h_k(\mathcal{L})$ where ${<}^{\mathfrak{A}}$ if of order type $\alpha$. Then for every $\lambda <h_{\kappa }(\mathcal{L}),h_{\kappa }(\mathcal{L})\ge {\beth }_{w_{\kappa }(\mathcal{L})}(\lambda )$ $\to$ The Hanf number depends on the well-ordering number

Compactness properties yield small Hanf numbers, while large well-ordering numbers cause compactness properties to ‘fade away’ (i.e., less fragments of compactness), which, as a result, yields large Hanf numbers.

References

The Hanf Number The Compactness Theorem Model Theoretic Logics.pdf gupea_2077_25506_1.pdf