Infra-Exercises, Part 1

2.3: I claim that the definition "m is the infimum of S := ∀x:x≤m⇔x≤S" is better: It should make for prettier proofs, and expresses "m eliminates ∀s∈S for purposes of testing ≤".

Have this marvelously pretty proof of 3b:
Theorem: $\forall p\in${monotone, 1-Lipschitz, 0-increasing, concave}$:p\Phi ⟹pinf\Phi$
Proof:

5c:

pg 6 "there exist" -> "there exists"

pg 13 maybe specify that you mean a linear functional that cannot be written as an integral (I quickly jumped ahead after thinking of one where you don't need to take any integrals to evaluate it)

I've posted complete, Moore-method-style solutions over on my blog. Spoilers, natch. Please nitpick them to death.

In section 1: When the 'affine' word is commonly used, the mixture coeffecient (denoted there by $p$) can be any real number. When $0\le p\le 1$, then it's still an affine combination, but a more precise tern could be 'convex combination'.

However, as you work with function spaces, convexity in a function space is different then being a convex function, so maybe some new notation should be introduced.

Video link in the pdf doesn't work

Nitpicks: Def 2.4 builds on the obvious generalisation of Def 2.3, not Def 2.3 itself. Also, Def. 2.3 is a little poorly worded, and should make clear that "not all sets ^[1]all sets of reals which are bounded below have an infimum". I think it would be helpful for those with no background in proofs if you had an exercise to justify the use of "the" when talking about the supremum of a set. Or even link it to the idea of a lower bound at all. 

More to come as I read through this.

EDIT: I think you probably want to point people to guides on how to prove things so that people without maths backgrounds know what task they're meant to be accomplishing. Otherwise they might go through this text and be hopelessly stuck or think they've proven something when they really haven't. Plus, having the ability to notice the difference between a proof and the intuition behind a proof is quite important for doing mathsy stuff, and Vanessa's agenda is amongst the more mathy alignment agendas. 

1. ^{^}

   Ignore the struck through text.

I have two questions that may be slightly off-topic and a minor remark:

• Is a list of open and tractable problems related to Infra-Bayesianism somewhere available?
• Do you plan to publish the results of the Infra-Bayesianism series in a peer-reviewed journal? I understand that there are certain downsides; mostly that it requires a lot of work, that the whole series may be too long for a journal article and that the peer review process takes much time. However, if your work is citeable, it could attract more researchers, who are able to contribute.
• On page 22, you should include the condition a(bv) = (ab)v into the definition of a vector space.