No it’s not the Redline talking. What *is* hyper in terms of mathematical context? Suppose Alice lives in Universe A and Bob in parallel universe B. If Alice draws a box around a stick figure of Bob and Bob the same of Alice, does it make sense to ask if set A contains set B or vice versa?

Is there a generalization of ‘hyper’ such as if hyper in hyper-Mahlo cardinal same as hyper of hypercohomology? Or hypergame paradox for that matter.

Going back to the original discussion we will shortly introduce Zwicker’s hypergame paradox. But first let me give you this question of MathOverflow: Can we have A={A}?

My answer -the one with the lowest vote- points to Ben Goertzel’s link shows that the concept of non-well founded set theory has also been given serious thought in Situation Theory of Barwise(& Etchemendy). Wikipedia proclaims:

Within this framework, it can be shown that the so-called Quine atom, formally defined by Q={Q}, exists and is unique.

Borrowing from here we get a feel for how hypergame works:

A

finite gameis a two-player game which is guaranteed to end after a finite number of moves. It is ok for a finite game to have an infinite number of possibilities; for example, the game where player one names any number and immediately wins is a finite game. It is also ok for a game to have an unbounded number of moves, as long as it always ends after finitely many. For example, the game where player one names any numberN, thenNmoves pass with nothing happening before player one wins. This game is still a finite game, since player one will eventually win in a finite number of moves no matter whichNwas chosen.To play hypergame, player one names any finite game. Player two then makes the first move in that game, and play continues as usual. Whoever wins the game also wins the hypergame.

To draw an example suppose Alice starts her first legal move

Let’s play a hypergame.

to which Bob replies

Let’s play a hypergame.

giving the following structure

Let’s play a hypergame.

- Let’s play a hypergame.

begging the obvious question if hypergame is finite or not. More interestingly if one takes a pot and throw in Ehrenfeucht–Fraïssé, Axiom of Determinacy and Newcomb’s Paradox, what stew will she be able to concoct?

Hyped up yet? Well the game has just began! Here are some links to stay busy for a while:

- Playing games with games: the hypergame paradox
- Translating the hypergame paradox: Remarks on the set of founded elements of a relation
- Hypergame and Infinity
- Hypergame Paradox and Cantor’s Theorem