Beren

[Beren @berenthefirst]

(I think time is only guaranteed to be a good metaphor for normal functions: for a regressive function interpreting it in terms of time, would be very jumbled, as you might be moving very discontinuously and randomly up and down the mountain with a time interpretation: maps from the mountain to the mountain make only so much sense, but introducing gods on a subset of the mountain that have agency, seems to be the right metaphor here)

[Beren @berenthefirst]

Here's my metaphor for Fodor's lemma: imagine you are trying to climb to a regular uncountable cardinal k, and there are a bunch of gods on the mountain stationed to stop you. There's no god blocking k itself, as that'd be very unfair, but your legs are not strong enough to make it in one jump (but you may jump any lesser amount than k). So again, since k is regular, it must take time k to get to the top of the mountain.

But the gods are clever: they're guarding on a stationary set! Which is to say, you'll inevitably run into them trying to climb k. The gods have been stationed to defend the mountain by Fodor, and if they leave their posts, the penalty is exile from the mountain.

But! You somehow tricked them into thinking they are released from their duties, and every god decides to jump halfhazardly down the mountain to a lower spot, as they walk down (note: this just is the "regressive function" in the lemma)

Then it's revealed that they in fact are not released: surely you'll be able to make it now! Except: Fodor regrets that they will be exiled (which would make it easy for you to walk up to k), and is able to jump to one location on the mountain and pardon all the gods there, before the curse takes effect.

It turns out, no matter how halfhazardly the gods jumped down, they'll be some location Fodor can go, where if he pardons the gods there and they return to their posts... they'll still be able to block you, without the help of the other exiled gods!

So, in this case the "f" in the lemma below is not representing time, but rather the way in which the gods descend the mountain... but we're still understanding stationary sets w/ a time metaphor.

[Beren @berenthefirst]

@uncountableart Right now I'm trying to scale a weakly compact cardinal, I think it's going to take a bit! Maybe in a few days I'll figure out how

[Beren @berenthefirst]

Alright, here are my current thoughts. Please forgive any errors, I am very much an amateur! It will take a bit to build up to.

The setup is: k is a regular cardinal, and we have a normal function f : k -> k .

Think about the domain as "time", and the the range as "height", like you are climbing up a mountain with a step for each ordinal. While climbing, you only really have a choice of where to go when moving to a successor ordinal: naturally, where you'll be at time w (that is, omega) should be the limit of all finite times, and so on for limits in general. Since k is regular, as long as you climb it "fairly" and never jump straight to the top of the mountain, it will in fact take time k to climb it (unlike e.g. aleph_omega, which can be climbed "fairly" in just omega steps despite being a lot bigger).

Now, on this mountain, imagine a inacessible cardinal is always a sheer cliff: you could live your entire life below this cliff, and always think the top was the absolute infinite, not knowing the mountain actually continued higher!

A Mahlo cardinal k is inaccessible, so it will also be a sheer cliff. But it has the property, that if you try to climb up to a Mahlo cardinal "fairly" (which tells us, our path is determined by a normal function f : k -> k), you in fact *necessarily* also climbed some other inaccesible (call it v) along the way! That is, f(v) = v ! This is what it means for the inacessibles below k to be a stationary set.

If k is the smallest Mahlo cardinal, then standing on the edge of the cliff and looking down, you can see that any route up necessarily crosses k inaccessibles, k-hyper inacessibles, k hyper-hyper inaccesibles, even k utter-inaccessibles, etc... despite the fact that for any cardinal smaller than k, you can avoid scaling any cliffs at all! (besides possibly one, if k is inacessible).

The smallest Mahlo cardinal is pretty big!

That's my current metaphor I'm working with! I kinda left the waves metaphor behind. domain as time, range as height, is pretty satisfying.

[Beren @berenthefirst]

@littmath I don't say this a lot, but, based

[Beren @berenthefirst]

@JDHamkins Is there any theorem where this trick is very helpful?

[Beren @berenthefirst]

Quaternions are beautiful! If you're interested in a deep mathematical understanding, IMO you first want to understand on a gut level how multiplying complex numbers relates to rotations, and what went "right" for that to be possible. Part of the answer is, the space of rotations around a point in 2D is topologically a circle.

If you grok that and reflect on rotations in 3D, you'll realize the space of such rotations *is* 3D, topologically (2 dimensions go into specifying rotation axis, 1 into angle to rotate).

This makes one suggest to describe 3D rotations, you want a 4d number system, as the "unit circle" (really, sphere) in n dimensions is n-1 dimensional. So as there's a 3d space of 3d rotations, you want 4D.

Now, ideally the space of rotations in 3D would be topologically the surface of a sphere in 4d, so it'd all go as smooth as the 2d complex numbers did... But if you think carefully, you see it's not. But it's *close* to right: it's topologically a sphere in 4d, with every pair opposite points *glued together*. So you want to somehow*undo* that gluing process for it to work right. You can do that by taking the "double cover".

All the details and how you get from here to normal multiplication rules for quaternions and how they concretely let you do 3D rotations, can be done elegantly from this perspective!

(Of course, it's not necessary to understand this route to use quaternions in practice... But it's deeply beautiful this way imo.)

[Beren @berenthefirst]

@DoozerDiffuser Another reason: Q is homeomorphic to Q^2, but R isn't homeomorphic to R^2. There's no topological notion of dimmension for Q^n, they're all the same!

[Beren @berenthefirst]

@DoozerDiffuser Take 1/(x^2-2) on the interval [1,2]: continuous on Q, but unbounded. Pretty bad, isn't it?

[Beren @berenthefirst]

@uncountableart Oh also i like this description of the ordinals as a "skyscraper fractal": https://t.co/lcWJkUMK9k

(And I really want to learn how large cardinals relate to Laver tables, as there are really neat fractal phenomena there: the sequence 0,1,0,2,0,1,0,3,0,1,0,2,0,1,0,4... appears!)

[Beren @berenthefirst]

@uncountableart Right now, I'll point you to https://t.co/zmWZ9o6CIb (free!): if you search for "Mahlo", that section was extremely helpful for me.

I'll put it in my own words once I do a bit more thinking about it. But as a hint for the metaphors I'm working with: https://t.co/e9LKKfIPvK

[Beren @berenthefirst]

@suzania Even I am amused at this point (!) . 166 to 167 with a comeback! I hear people cheering from a spot in FiDi I have a view of

[Beren @berenthefirst]

@QiaochuYuan Set theorists care about philosophy e.g. justifying the truth or plausibility of large cardinal axioms (Have you read "Believing the Axioms"?) And this connects to the foundation questions, as (ideally) large cardinal hypotheses are the right "ladder of consistency strength"

[Beren @berenthefirst]

re: the Julius Caesar argument, lots of rationalists do accept the idea that "if LLMs are conscious, they are in an extremely weird way that is not remotely like them "being like" the supposed assistant character, but rather the conciousness is extremely alien and is play acting both Julius Caesar and the assistant persona.

(but most people falling for "ai conciousness" aren't thinking that way, indeed!)

[Beren @berenthefirst]

@theistinthought There is a grocery store ~.1 miles from me. It's extremely easy to hit on the way home from my 15m walk from work.

[Beren @berenthefirst]

Are you suggesting people leave their current local church that they are active and happy members of, if that would switch their denomination PCA->PCUSA? Or just that if someone is moving in the future/deciding on a church, they should prefer PCUSA? The former seems crazy. I am far more attached to the particular local church I'm at than to the exact denomination it's in, and I think this is healthy.

[Beren @berenthefirst]

@suzania i want to access more lattes. it's me.

[Beren @berenthefirst]

I hate to say this, but I believe the screenshoted section of the encyclical above was AI assisted. I think that both based on vibes (as someone who works with/keeps up on AI and is familiar with the tells), and ran Pangram (an AI detector that, for the moment, has a low false positive rate) on it, which flags it as AI assisted.

Beyond this, I also gave it the first three sections of the enyclical as a control: that part did not feel AI generated to me, and indeed, Pangram says 100% human generated for that part.

When I first read the substack post (by linch) claiming parts of the encyclical were AI assisted, I was not entirely convinced, but thinking about it more, I agree with the analysis there. this is sad :(