Kill the Mathematical Hydra | Infinite Series
Why Mathematics Stack Exchange recommends it
Mathematics Stack Exchange askers cite it as the accessible introduction to the hydra game and to ordinals when posting follow-up questions about the proof.
Recommended by 0 people across 0 different months, 2017 to 2019.
What it is
A PBS Infinite Series episode in which mathematician Kelsey Houston-Edwards explains the Kirby-Paris hydra game, where a monster grows new heads each time you cut one off. She shows why the hydra can always be defeated, using a proof based on ordinal numbers.
Cited in 1 Wikipedia article, including Hydra game.
What people said
Cites mgold · 2017-01 · Mathematics Stack Exchange · on “Fastest way to defeat the Kirby-Paris hydra?”
The Kirby-Paris hydra, previously seen on math.stack here [LINK] and in this layperson-targeted video [this video] , is a one-player game that models the mythical hydra as a tree. Each node/vertex of the tree is considered to be a "head" of the hydra, except for the root which is its "body". A player "moves" by selecting a leaf node L and removing it. Then, the hydra "regrows" n copies what is left of L's parent's subtree, all rooted on L's grandparent. For our purposes, let n=2. The goal is to reduce the hydra to its body. The surprising result is that, not only is the game winnable, it is impossible to lose. Any hydra will be "defeated" by a sufficiently long but finite sequence of moves. Howev
Cites Blitzcraig · 2017-06 · Mathematics Stack Exchange · on “Why does slaying the hydra require infinite ordinals?”
I have recently come across the problem of the mathematical hydra. More information can be found here [LINK] or a video here [this video] . I have to say that I don't fully understand why infinite ordinals are required to prove the hydra can always be killed. For example, here is how I would go about it. Represent the hydra as a decorated tree so that leaf nodes are grouped together and represented by a single number on an edge leading to a representative node as shown below. [a bit awkward to represent in ASCII, but here goes] (r denotes the root, 1-multiplicity labels are omitted so o-[1]-o = o-o) r o-o-o \ o = r o-o-[2]-o since there are 2 leaf nodes in that branch. Then by repeatedly applying
Cites WillG · 2019-11 · Mathematics Stack Exchange · on “Why does the Well Ordering Theorem imply decreasing ordinals go to zero?”
The Well Ordering Theorem states that any set can be well ordered. But in this PBS Infinite Series video [this video] , Kelsey states the theorem as, "Any decreasing sequence of ordinals eventually goes to zero." This isn't an obvious equivalence to me. Does deeper work need to be done to prove it, or is it a one-liner? It may be hard for me to see the connection because I don't know the rigorous definition of ordinals, just the intuitive idea from the above-linked episode.