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I would like to know how Mahlo cardinals are used - as such examples may help me understand why they were created and so on.

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How should they be used? With caution. Why they were created is quite simple, as soon as you know what is hyper-inaccessible you can ask "Okay, now I want to have a stationary limit of inaccessible cardinals", and it turns out this is a properly stronger notion. –  Asaf Karagila Feb 22 '13 at 12:43
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Sometimes a Mahlo cardinal $\kappa$ is used to get an elementary substructure of $V_\kappa$ of the form $V_\alpha$ where $\alpha<\kappa$ is regular. –  Trevor Wilson Feb 22 '13 at 19:25

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Here is a neat use of Mahlo cardinals that I have recently learned about:

Denote, for an uncountable cardinal $\kappa$, by $\square_\kappa$ the statement asserting the existence of a sequence $\langle C_\alpha\mid\alpha<\kappa^+\rangle$ with the following properties:

  1. $C_\alpha\subseteq\alpha$ is a club in $\alpha$.
  2. The order type of $C_\alpha$ is less or equal to $\kappa$. (Sometimes you find the requirement that if the equality may occur only when necessary, i.e. $\operatorname{cf}(\alpha)=\kappa$.)
  3. If $\beta$ is a limit point of $C_\alpha$, then $C_\alpha\cap\beta=C_\beta$.

Jensen proved that $\square_\kappa$ holds in $L$ for every uncountable $\kappa$. He also proved that if $V\models\lnot\square_\kappa$, then $(\kappa^+)^V$ is Mahlo in $L$. Moreover, Solovay proved that if $\kappa$ is Mahlo, and $\lambda<\kappa$ is regular then there is a forcing extension in which $\kappa=\lambda^+$ and $\lnot\square_\lambda$ fails.

Therefore Mahlo cardinals are used to negate squares on regular cardinals.


One point to make is that the definition of a Mahlo cardinal predates these proofs by three or four decades. This has little to do with the history and the original definitions of a Mahlo cardinal, but it does point out an interesting use for them.

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You may want to say a bit more, indicating how it is that the fact that $\kappa$ is Mahlo enters into the argument. –  Andres Caicedo Aug 29 '13 at 19:46
    
Andres, do you know if Solovay's proof was ever published? I couldn't find a reference anywhere. –  Asaf Karagila Apr 20 at 2:17
    
I'll see if I can find something. Kanamori mentions it in his draft for a chapter of his probably apocryphal volume 2, but I do not recall at the moment whether he cites a reference. –  Andres Caicedo Apr 20 at 2:19
    
For what it's worth, in Jensen's paper from 1972 he recalls Solovay's result, but gives no citation. –  Asaf Karagila Apr 20 at 2:21
    
It is most likely unpublished, then. The only paper by Solovay that I am aware of dealing with square sequences is Abraham, Shelah, Solovay, Squares with diamonds and Souslin trees with special squares, but this result is not discussed there. –  Andres Caicedo Apr 20 at 3:10

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