Indescribable cardinal

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In set theory, a branch of mathematics, a Q-indescribable cardinal is a certain kind of large cardinal number that is hard to axiomatize in some language Q. There are many different types of indescribable cardinals corresponding to different choices of languages Q. They were introduced by Hanf & Scott (1961).

A cardinal number ${\displaystyle \kappa }$ is called ${\displaystyle \Pi _{m}^{n}}$-indescribable if for every ${\displaystyle \Pi _{m}}$ proposition ${\displaystyle \phi }$, and set ${\displaystyle A\subseteq V_{\kappa }}$ with ${\displaystyle (V_{\kappa +n},\in ,A)\vDash \phi }$ there exists an ${\displaystyle \alpha <\kappa }$ with ${\displaystyle (V_{\alpha +n},\in ,A\cap V_{\alpha })\vDash \phi }$.^[1] Following Lévy's hierarchy, here one looks at formulas with m-1 alternations of quantifiers with the outermost quantifier being universal. ${\displaystyle \Sigma _{m}^{n}}$-indescribable cardinals are defined in a similar way, but with an outermost existential quantifier. Prior to defining the structure ${\displaystyle (V_{\kappa +n},\in ,A)}$, one new predicate symbol is added to the language of set theory, which is interpreted as ${\displaystyle A}$.^[2] The idea is that ${\displaystyle \kappa }$ cannot be distinguished (looking from below) from smaller cardinals by any formula of n+1-th order logic with m-1 alternations of quantifiers even with the advantage of an extra unary predicate symbol (for A). This implies that it is large because it means that there must be many smaller cardinals with similar properties. ^{[citation needed]}

The cardinal number ${\displaystyle \kappa }$ is called totally indescribable if it is ${\displaystyle \Pi _{m}^{n}}$-indescribable for all positive integers m and n.

If ${\displaystyle \alpha }$ is an ordinal, the cardinal number ${\displaystyle \kappa }$ is called ${\displaystyle \alpha }$-indescribable if for every formula ${\displaystyle \phi }$ and every subset ${\displaystyle U}$ of ${\displaystyle V_{\kappa }}$ such that ${\displaystyle \phi (U)}$ holds in ${\displaystyle V_{\kappa +\alpha }}$ there is a some ${\displaystyle \lambda <\kappa }$ such that ${\displaystyle \phi (U\cap V_{\lambda })}$ holds in ${\displaystyle V_{\lambda +\alpha }}$. If ${\displaystyle \alpha }$ is infinite then ${\displaystyle \alpha }$-indescribable ordinals are totally indescribable, and if ${\displaystyle \alpha }$ is finite they are the same as ${\displaystyle \Pi _{\omega }^{\alpha }}$-indescribable ordinals. There is no ${\displaystyle \kappa }$ that is ${\displaystyle \kappa }$-indescribable, nor does ${\displaystyle \alpha }$-indescribability necessarily imply ${\displaystyle \beta }$-indescribability for any ${\displaystyle \beta <\alpha }$, but there is an alternative notion of shrewd cardinals that makes sense when ${\displaystyle \alpha \geq \kappa }$: if ${\displaystyle \phi (U,\kappa )}$ holds in ${\displaystyle V_{\kappa +\alpha }}$, then there are ${\displaystyle \lambda <\kappa }$ and ${\displaystyle \beta }$ such that ${\displaystyle \phi (U\cap V_{\lambda },\lambda )}$ holds in ${\displaystyle V_{\lambda +\beta }}$.^[3] However, it is possible that a cardinal ${\displaystyle \pi }$ is ${\displaystyle \kappa }$-indescribable for ${\displaystyle \kappa }$ much greater than ${\displaystyle \pi }$.^{[1]Ch. 9, theorem 4.3}