Why "Ann believes that Bob assumes that Ann believes that Bob’s assumption is wrong" is paradoxical?

Question

In a paper(see here) by Adam Brandenburger and H. Jerome Keisler, they give a game-theoretic impossibility theorem akin to Russell’s Paradox:

Ann believes that Bob assumes that Ann believes that Bob’s assumption is wrong.

Their analysis is

By an assumption (or strongest belief) we mean a belief that implies all other beliefs.

To get the impossibility, ask: Does Ann believe that Bob’s assumption is wrong? If so, then in Ann’s view, Bob’s assumption, namely “Ann believes that Bob’s assumption is wrong”, is right. But then Ann does not believe that Bob’s assumption is wrong, which contradicts our starting supposition. This leaves the other possibility, that Ann does not believe that Bob’s assumption is wrong. If this is so, then in Ann’s view, Bob’s assumption, namely “Ann believes that Bob’s assumption is wrong”, is wrong. But then Ann does believe that Bob’s assumption is wrong, so we again get a contradiction.

Acutually, I don't understand the first part:

If "Ann believe that Bob’s assumptionis wrong", then in Ann’s view, Bob’s assumption, namely “Ann believes that Bob’s assumption is wrong”, is wrong.

Why?

What does it mean by "a belief that implies all other beliefs"?

Answer 1

The "paradoxical" configuration is :

Ann believes that Bob assumes that

Ann believes that Bob’s assumption is wrong.

We have here Bob's assumption : "Ann believes that Bob’s assumption is wrong".

Now consider Ann's belief attitude towards Bob's assumption; we can say that if $p$ is any statement, a "reasonable" principle will be :

$Believes(Ann,p) \lor \lnot Believes(Ann,p)$.

Thus, we have two possibilities :

(i) Ann believes that Bob’s assumption is wrong.

But this is exactly Bob's assumption : "Ann believes that Bob’s assumption is wrong", and thus we have to conclude that, in Ann’s view, Bob’s assumption is right, and this conclusion contradicts the initial supposition about Ann's belief.

Thus, we have to reject it, i.e. the supposition about Ann's belief, and we are left with the second possibility :

(ii) Ann does not believe that Bob’s assumption is wrong.

But this is the negation of Bob's assumption : "Ann believes that Bob’s assumption is wrong", and we have to conclude that, in Ann’s view, Bob’s assumption is wrong. But this means that Ann does believe that Bob’s assumption is wrong, and again we get a contradiction.

We have "exhausted" all possibilities and thus we have to cocnlude that :

the configuration of beliefs in bold is impossible.

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As stated by the author's, the argument is a paradox of self-reference about beliefs akin to Russell's Paradox "maufactured" on the principle that any meaningful "condition" $\varphi(x)$ can be used to specify a set, i.e. the set of all and only those "objects" satisfying the condition.

If we let $\varphi(x)$ stand for $x \in x$ and let $R = \{ x: \lnot \varphi(x) \}$, then $R$ is the set whose members are exactly those objects that are not members of themselves.

Thus we apply the reasonable principle that for any set $S$ and any "object" $x$ :

$x \in S \lor x \notin S$

with $R$ as both $x$ and $S$. We consider the two possibilities : (i) $R \in R$, and (ii) $R \notin R$, and the contradiciton follows.