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In the paper ``Berkeley Cardinals and the Structure of $L(V_{\delta+1})$", by Raffaella Cutolo, a use is made of the notation $\alpha < < \beta$ for a pair of ordinals $\alpha, \beta$, without any explanation given for the meaning of the notation. Is this a standard notation?

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  • $\begingroup$ I think it might mean that the leading exponent of the Cantor normal form of $\beta$ is larger than the leading exponent of the Cantor normal form of $\alpha$. $\endgroup$ – James Hanson Mar 31 at 15:30
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    $\begingroup$ Rupert, could you mention where in the paper is the notation used, please? $\endgroup$ – Andrés E. Caicedo Mar 31 at 17:00
  • $\begingroup$ @AndrésE.Caicedo Proof of lemma 2.1, third paragraph, first line. $\endgroup$ – Wojowu Mar 31 at 18:37
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    $\begingroup$ You will have to check the details of the argument to see if something like this works here, but unless indicated otherwise, the notation is usually informal, meant to denote that $\alpha$ is "much" smaller than $\beta$, that is, $\alpha$ is not only less than $\beta$, but $\beta$ is indeed large enough that $V_\beta$ (or $L_\beta$, or whatever appropriate structure is under discussion) sees all relevant objects definable from $\alpha$ that one needs for the argument at hand to carry through. $\endgroup$ – Andrés E. Caicedo Mar 31 at 20:29
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    $\begingroup$ I would suggest to think of it as simply saying $\alpha<\beta$ until one finds a place in the argument where more is needed (say, $\alpha+\omega<\beta$) and iterate. Typically the process converges rather quickly. I agree it is not ideal. $\endgroup$ – Andrés E. Caicedo Mar 31 at 20:30

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