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Does the term mouse stand for "Model Of Universe with Sequences of Extenders" or perhaps mice stands for "Model Including Cardinal Extensions"?

A quick review:

Gödel's L-universe is a core model for ZF, defined via transfinite induction as follows:

  • $L_0 := \emptyset$
  • $L_{\alpha + 1} := Def(L_\alpha)$
  • $L_\beta := \bigcup \{ L_\alpha | \alpha < \beta \}$ ($\alpha$ precedes $\beta$)

If we wish to examine a natural inner model for a strong cardinal, we add extender sequences (i.e., sequences of compatible ultrafilters) to our universe so that we can deal with large cardinals. The universe $L$ with this additional axiom of extender sequences is denoted $L[E]$.

It's my understanding that we call the structure $L_\alpha [E]$ a mouse if our $E$ is in some way canonical (a "coherent/good extender sequence"). ($L_\alpha[E]$ without this added condition of "canonicality" is called a "premouse").

Apologies if the above is off-kilter, I've likely got some of the model-theoretic vocabulary mixed-up.

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    $\begingroup$ As far as I know it is something of a running joke that Jensen keeps changing his story about how he chose the name mouse. It definitely isn't some sort of acronym. I think the most widely accepted version is that he deemed the concept so important that he wanted a completely new term for it, without any existing mathematical connotations. $\endgroup$
    – Miha Habič
    Aug 18, 2015 at 12:16
  • $\begingroup$ At the time, we were told that the name was picked, more or less, at random. $\endgroup$
    – Will Jagy
    Aug 18, 2015 at 18:10
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    $\begingroup$ I have been told by several people that it was as a result of a typo of `nice'. Incidentally, I always thought that the key something being a mouse was that you could iterate the ultrapower using the extender sequence through the ordinals whilst keeping the ultrapower well-founded. I'm a relative newbie to this too though! $\endgroup$
    – Neil Barton
    Oct 11, 2015 at 22:59
  • $\begingroup$ The OP is probably aware of this paper, but for readers who may not be, I recommend Schimmerling's paper The ABCs of mice as a relatively readable introduction to the topic. $\endgroup$
    – Noah Schweber
    Mar 6, 2021 at 8:01
  • $\begingroup$ Perhaps $\text{mouse}_\text{set}=\text{model of ZF}$ in the same way that $\text{mouse}_\text{biology}=\text{model for experiments}$... $\endgroup$
    – TheSimpliFire ♦
    May 29, 2021 at 15:04

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