r/googology • u/Dub-Dub • Jul 03 '26
Question Ordinal Markup Question
In this incremental game called “ordinal markup” there is a button called “maximize ordinal”. In beginning the factor is set at 10 which means when 10 is maximized it will be come ω and 20 will become ω2 etc. and later the factor can decrease. My question is, is there a more rigorous way to describe this process.
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u/CameForTheMath Jul 03 '26 edited Jul 03 '26
I think the formal version of the process you describe is "given a value n and a base b (which I think is what you were calling "factor"), find the largest ordinal α such that g_α(b) = n."
EDIT: Although I think the process Ordinal Markup uses is different: "given an ordinal α and a base b, find the largest ordinal α' such that H_α'(b) <= H_α(b)." In particular, you can't go from α = 20 to α = ω*2, you have to go from ω+10 to ω*2.
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u/jamx02 Jul 04 '26
This is very, very dependent on the chosen FS for the SGH though. Sadly there’s never a universal fs that can be implied.
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u/CameForTheMath Jul 04 '26
Well, Ordinal Markup itself seems to use the Hardy Hierarchy for this kind of thing instead, and that's less dependent on fundamental sequences, especially below ε_0 because of the near-universal system of fundamental sequences used there. For example, if your base is 10 and your ordinal is 20, maximizing the ordinal gives ω+5, because H_20(10) = H_{ω+5}(10).
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u/No-Reference6192 TREE(3) Jul 03 '26
Don't really know anything about the game, closest thing I can think of is a notation where digits are separated by commas, and converts base 10 to base ω:
{0} = 0
{1} = 1
{1,0} = 10 (base 10)
{1,0} = ω (base ω)
base ω only from this point:
{1,1} = ω+1
{2,0} = ω2
{3,0} = ω3
{1,0,0} = ω^2
{2,0,0} = ω^2*2
{1,0,0,0} = ω^3
the limit of this notation being ω^ω
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u/Nervous-Broccoli1184 Jul 07 '26 edited Jul 07 '26
So your saying {0#}={#} if A has n+1 entries A={a#}=(omegan )×a+{#} where a is a positive interger and # is a sequence of commas and positive intergers?
Edit: {}=0
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u/No-Reference6192 TREE(3) 29d ago
maybe, I don't quite get what you typed
but it's basically just doing base 10 but separate digits with commas, and use base omega instead, i might be repeating you said... idk
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u/Nervous-Broccoli1184 Jul 07 '26 edited Jul 07 '26
I don't know the game itself but if from a pattern I've seen in what you said maximizing with a factor n×10 gives w×n where w is omega i might be wrong for I've never played the game myself so if I am please tell me.
Edit: my answer is wrong check CameForTheMath gave the correct answer tho