Step into the world of the study of numbers that get increasingly incomprehensible at nearly every single step of the way, also known as googology, with the ultimate number comparison from zero to infinity and beyond (and I mean that in the literal sense)!
After the better part of a year working on and off on the project, my version of the ultimate number comparison IS FINALLY HERE! And it's one of the better versions too, featuring a gradual change in tone as the video progresses and use of special software to generate visuals programmatically, and with the entirety of the video being in HD at 60fps! The aforementioned software used is manim (by @3blue1brown, used for his animated math visualizations) to create the entries and Shadertoy (by @InigoQuilez, the platform for shader art) to make special backgrounds. All in all this means this video provides a unique touch to this number comparison format that viewers familiar with the study of large numbers will certainly enjoy!
@MathisRV was the first to popularize the modern Numbers 0 to Absolute Infinity format so I'd like to give a MASSIVE shout-out to them as this video was heavily influenced by theirs. The original https://youtu.be/j98suMfW1tc and remake https://youtu.be/d9QJHaFVh0Y were published when I started working on my version, and version 3 https://youtu.be/aqtrYVrRLYw which did not influence this video as much but really blew it out of the water with the visuals was published in the middle of the creation of this version.
Due to the sheer quantity of significant numbers in this video I had to resort to using others' videos, the videos in question being https://youtu.be/kr7Wx0f3Xpg by @notnt2048, itself being a compilation of the Bird's array notation Large numbers gradually increasing series from @douglasshamlinjr.392 in addition to https://youtu.be/NNr20-171PA, the forward-slash version of https://youtu.be/_rNm88z6lIw, from @douglasshamlinjr.392.
Sources used for this video regarding numbers include: https://www.mrob.com/users/chrisb/Fast-Growing_Hierarchy.pdf by Chris Bird (See Chris Bird's other documents at: https://www.mrob.com/users/chrisb/index.html ), https://googology.fandom.com, https://sites.google.com/view/xrq-golapulus-numbers/home by XRQ CORPORATION, https://youtu.be/5hfkzo_ojGE by @douglasshamlinjr.392, and https://en.wikipedia.org/wiki/Fixed-point_lemma_for_normal_functions#Example_application. This video's chapters are based on @notnt2048's Ordinal Levels, the information for which was retrieved from https://youtu.be/9e17ykV_Nzk and https://youtu.be/pXriF7FzWoU. Note that the limit of OL(16) was changed from ε₀ to encapsulate all countable ordinals.
Music credit: Joakim Karud (@JoakimKarud) - Clouds: https://youtu.be/9O13PpAoVdI, Robin Hustin (@RobinHustin) - On Fire [NCS Release]: https://youtu.be/S0STHSOPPSM, French Fuse (@frenchfuse) - Straight Fuse: https://youtu.be/6ozdaINBLhQ, Nico Staf - Fast and Run: https://youtu.be/cXtit9Ux8N0, half.cool (@halfcoolmusic) - Virtual Roaming Charges: https://youtu.be/pY1ZmluAYGM, Godmode - Whole Tone Limbo: https://youtu.be/K_FaVVRAsnE, Unicorn Heads (@UnicornHeads) - Psychic Magic: https://youtu.be/rfOA-tmBuPM, Jingle Punks - Ambient Ambulance: https://youtu.be/GFH1jjzqeWI, @Serg66260 - Сравнение размеров звёзд и планет(Захватывает).flv: https://youtu.be/SQrnzZGuwZM (Careful, as the description with this one contains a scam)
This video's creation concluded on May 27, 2024
0:00 Growth rate 4 / NO! Ordinal Level 1
2:15 Growth rate ω / NO! Ordinal Level 2
3:01 Growth rate ω+1 / NO! Ordinal Level 3
3:32 Growth rate ω² / NO! Ordinal Level 4
5:34 Growth rate ω^ω / NO! Ordinal Level 5
7:15 Growth rate ω^ω^2 / NO! Ordinal Level 6
8:34 Growth rate ω^ω^ω / NO! Ordinal Level 7
9:58 Growth rate ω^ω^ω^ω / NO! Ordinal Level 8
11:56 Growth rate ε₀ / NO! Ordinal Level 9
13:49 Growth rate ζ₀ / NO! Ordinal Level 10
15:14 Growth rate φ(ω,0) / NO! Ordinal Level 11
15:54 Growth rate ψ₀(Ω₂^ω) / NO! Ordinal Level 12
16:58 Growth rate ψ₀(Ω_ω) / NO! Ordinal Level 13
17:41 Growth rate ψ₀(ψ_I(0)) / NO! Ordinal Level 14
21:57 To infinity / NO! Ordinal Level 15
24:38 א₀ / NO! Ordinal Level 16
29:03 Uncountable infinity and beyond / NO! Ordinal Level 17
32:57 Conclusion
33:22 Outro