We explore extremely long Infinite Chess games, starting with Mate in Omega, and progressively climbing to higher and higher transfinite ordinals!
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\/ Credits for the positions shown, along with links to learn more about them! \/
Omega, Omega^2, Omega^3 by Joel Hamkins and C. D. A. Evans: http://math.colgate.edu/~integers/og2/og2.pdf
Omega*2 by Noam. D. Elkies: https://mathoverflow.net/a/63841/514442
Omega^4 by C. D. A. Evans, Joel Hamkins, and Norman Lewis Perlmutter: http://math.colgate.edu/~integers/rg4/rg4.pdf
Omega_1 Proof by Matthew Bolan: https://mathoverflow.net/questions/63423/checkmate-in-omega-moves/440147#440147
Subscribe to his channel! https://www.youtube.com/channel/UCB4XuRBJZBOpnoJSWekMohw
Some clips are from Joel Hamkins presentation, “The theory of infinite games, including infinite chess”, watch it here! https://www.youtube.com/watch?v=1_bVL1FG2gI
See more presentations on The Infinite Games Workshop: https://www.youtube.com/channel/UCZP2ve0DNfwAq8HrjBANc0Q
Note for the chapter “Climbing the Ordinal Tower”: I often mention “this to the power of this infinitely…”, or “infinite epsilons above this…”. When I word things like this, what I really mean is to take the limit, or supremum, as we approach doing that operation infinitely. I have chosen this wording to make things more intuitive to the viewer and those who may have zero ordinal arithmetic knowledge. This chapter is not meant to be a detailed explanation of ordinal arithmetic.
🎶 Music used (in order) 🎶
Background Check by David Robson: https://www.youtube.com/watch?v=r13MWpaUnVQ
Looking For Clues by David Robson: https://www.youtube.com/watch?v=YCi6znptiH8
Witness Testimony by David Robson: https://www.youtube.com/watch?v=C5US0Ycu7QU
Viking by Aakash Gandhi: https://www.youtube.com/watch?v=XkIhUzHehXY
Mario Bava Sleeps In a Little Later Than He Expected To by Chris Zabriskie: https://www.youtube.com/watch?v=YV2x_FUpasQ
This is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/by/4.0/
Source: http://chriszabriskie.com/vendaface/
Artist: http://chriszabriskie.com/
Anime Song 2 by Montage of Chips: https://youtu.be/iNSuzDPbpno?si=lzoqtrDKQ9LXOou3
Music: True Detective by Soundridemusic
Link to Video: https://www.youtube.com/watch?v=iCI-yY8QZIs&t=0s
White Hats by Wayne Jones: https://www.youtube.com/watch?v=VcZmSNhxC4o
Artemis by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au
Video: https://www.youtube.com/watch?v=vqb0XsZ04Og
Needle In A Haystack by David Robson: https://www.youtube.com/watch?v=xJiD6LvjFag
Signal to Noise by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au
Video: https://www.youtube.com/watch?v=x43OJXk8idI
Weirder Stuff by Geographer: https://www.youtube.com/watch?v=PeWgLUs40Pw
Many of these songs are provided by FesliyanStudios! Make a donation to also gain access to their music :) https://www.fesliyanstudios.com/policy
Chapters:
0:00 Introduction
1:18 The Game Tree
3:21 Omega
5:17 Why Ordinals?
8:04 Omega+1
8:58 Omega*2
10:41 Omega^2
12:27 Omega^3
14:48 Omega^4
20:20 Matthew’s Proof
24:09 Climbing the Ordinal Tower
26:53 Conclusion
Other stock footage is from Pixabay.com
If I missed, or incorrectly gave credit, please don’t hesitate to contact me using my email! Located in the "About" section of my channel, click "View email address".