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...
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...
Omega_1 Proof by Matthew Bolan: https://mathoverflow.net/questions/63...
Subscribe to his channel!    / @matthewbolan8154  

Some clips are from Joel Hamkins presentation, “The theory of infinite games, including infinite chess”, watch it here!    • TMWYF: The theory of infinite games, ...  
See more presentations on The Infinite Games Workshop:    / @infinitegamesworkshop-kd5et  

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:    • Detective Music For Investigation And...  

Looking For Clues by David Robson:    • Suspenseful Detective Background Musi...  

Witness Testimony by David Robson:    • Mysterious Detective Background Music...  

Viking by Aakash Gandhi:    • Viking - Aakash Gandhi  

Mario Bava Sleeps In a Little Later Than He Expected To by Chris Zabriskie:    • Mario Bava Sleeps In a Little Later T...  
This is licensed under a Creative Commons Attribution 4.0 license. https://creativecommons.org/licenses/...
Source: http://chriszabriskie.com/vendaface/
Artist: http://chriszabriskie.com/

Anime Song 2 by Montage of Chips:    • Anime Song 2  

Music: True Detective by Soundridemusic
Link to Video:    • Suspense Detective, Crime, Spy, Tensi...  

White Hats by Wayne Jones:    • White Hats - Wayne Jones  

Artemis by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au
Video:    • 'Artemis' [Inspirational Epic Orchest...  

Needle In A Haystack by David Robson:    • Investigation Crime Scene Music For D...  

Signal to Noise by Scott Buckley - released under CC-BY 4.0. www.scottbuckley.com.au
Video:    • 'Signal to Noise' [Cinematic Classica...  

Weirder Stuff by Geographer:    • Weirder Stuff - Geographer  

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".