A toolkit for the Dodd-Jensen core model

I just finished a note with some black boxes for the Dodd-Jensen core model $K^\text{DJ}$, the core model that cannot have a measurable cardinal. I am not an expert on core model theory, so this is just a  presentation of $K^\text{DJ}$, like painter A. Dürer’s presentation of a Rhinoceros (in the featured image), just by the descriptions of others, never having seen one himself. And perhaps my note is not as detailed as Dürer’s painting is, but I hope it’s a sufficient presentation for someone who just wants to get consistency strength from $K^{\text{DJ}}$ without having to go through all its glorious detail. One could say this is a note on applied set theory.

This note includes the trick to construct $K^{\text{DJ}}$ inside a model of ZF, by using the fact that  for any set of ordinals $x$, $(K^\text{DJ})^{\text{HOD}}=(K^\text{DJ})^{\text{HOD}[ x] }$ and this equality holds for every level of the $K^\text{DJ}$-construction.

There’s also a running example,  the proof that if $V$ is a model of ZF in which the Chang conjecture $(\omega_5,\omega_4)\twoheadrightarrow (\omega_3,\omega_1)$ holds,  $\omega_4$ is a regular cardinal, and cf$\omega_3>\omega$ then in $(K^\text{DJ})^{\text{HOD}}$, an Erdős cardinal exists, and in particular, $(K^\text{DJ})^{\text{HOD}}\models\omega_5^{V}\to(\omega_3^V)^{<\omega}_2$. This is an example of how to use these black boxes, and I think it helps clarify how this Rhinoceros model looks like. As always, there are pictures included – now in pgfpicture.

I’d be happy to get suggestions, corrections, and more $K^\text{DJ}$-black boxes and examples.

2 thoughts on “A toolkit for the Dodd-Jensen core model”

  1. I look forward to reading the note! Since I have given up on making the time to understand all the “glorious details” of core models, this might be the perfect introduction :). Great to know that you will be at the Colloquium Logicum! I thought I wouldn’t know anyone there.

    1. I’m really happy to see that this might be useful to you! Please do send me comments, suggestions, and tell me if something is unclear. And if a measurable cardinal is too weak for your assumptions, I’m planning of making something similar with the core model for a Woodin cardinal, but I have to finish a proof where I’m trying this model out first… (what will be the running example).

      I’m looking forward to meeting you in Munich!

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