Projectively correct mice

In a previous post we proved that whenever a countable mouse M has n Woodins it understands \bf\Pi^1_{n+1} sets, implying that whenever A is such a set it holds that A\cap M\in M. As we mentioned back then, this is not as good as being correct about these sets, which would mean that A\cap M\neq\emptyset whenever A of course is non-empty as well. Another way to phrase this is to say that V\models\sigma iff M\models\sigma for every \bf\Pi^1_{n+1}-sentence. Now, what does it then take for a mouse to be projectively correct?

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Categorification of large cardinals?

When working with most of modern set theory we tend to transcend ZFC, always working with some strong background hypothesis, whether it being the existence of some elementary embedding, a colouring for some partition property, a generic for some uncountable poset or something completely different. When it comes to using these strong hypotheses in mainstream mathematics it seems that we hit a brick wall, as most of our strong hypotheses don’t easily translate to the language of everyday mathematics.

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Pcf scales and squares

The last few posts I’ve been covering a characterisation of pointclasses that admit scales. To make scale theory even more confusing there’s a completely different notion of scale, which really has nothing to do with our previous one — this one being of a more combinatorial nature. To avoid unnecessary confusion I’ll call these new objects pfc scales (but usually they’re simply called scales as well, however).

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