# What is f\_ε0(3)?

**URL:** <https://boards.straightdope.com/t/what-is-f_-0-3/844543>\
**Category:** Factual Questions\
**Created:** [December 8, 2019, 2:58am UTC](https://boards.straightdope.com/t/what-is-f_-0-3/844543 "2019-12-08T02:58:17Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![ricksummon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ricksummon/32/3333_2.png) [@ricksummon](https://boards.straightdope.com/u/ricksummon)\
**Post date:** [December 8, 2019, 2:58am UTC](https://boards.straightdope.com/t/what-is-f_-0-3/844543/1 "2019-12-08T02:58:17Z")

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I’ve been studying the [Fast-growing hierarchy](https://en.wikipedia.org/wiki/Fast-growing_hierarchy) and I’m trying to figure out the value of **f\_ε0(3)**. This number is undoubtedly too large to directly calculate, but I should be able to reduce it to **f\_n(3)**, where n is a natural number.

Since ε0 is a limit ordinal, you take the 3rd term in its fundamental sequence, which is ω^ω. The 3rd term in _its_ fundamental sequence is ω^3, then ω^2\*3. However, I can’t figure out how to go further from here.

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**Author:** ![ricksummon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ricksummon/32/3333_2.png) [@ricksummon](https://boards.straightdope.com/u/ricksummon)\
**Post date:** [December 8, 2019, 11:56pm UTC](https://boards.straightdope.com/t/what-is-f_-0-3/844543/2 "2019-12-08T23:56:22Z")

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I’m guessing that the 3rd term in the fundamental sequence for ω^2_3 is the same as the 3rd term in ω^2 (ω_3) multiplied by 3, so it would be ω\*9. If so, the 3rd term in the fundamental sequence for ω is just 3, making **f\_ε0(3)** equal to **f\_27(3)**. That’s about 2 ↑(26) 3, so way too large to directly calculate.

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**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\
**Post date:** [December 9, 2019, 1:11am UTC](https://boards.straightdope.com/t/what-is-f_-0-3/844543/3 "2019-12-09T01:11:58Z")

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Did you just watch the Numberphile video on TREE(Graham), too?

Of course, the fun part about this is that all of these numbers, even the ones that need transfinite ordinals to express their indices, and all perfectly finite.

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**Author:** ![ricksummon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ricksummon/32/3333_2.png) [@ricksummon](https://boards.straightdope.com/u/ricksummon)\
**Post date:** [December 9, 2019, 2:55am UTC](https://boards.straightdope.com/t/what-is-f_-0-3/844543/4 "2019-12-09T02:55:58Z")

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I _did_ watch that, though I was a bit disappointed that they didn’t even _mention_ the actual position of TREE(n) on the fast-growing hierarchy (which, according to [this link,](https://math.stackexchange.com/questions/1950116/where-does-treen-sit-on-the-fast-growing-hierarchy) is **F\_θ(Ω^ω\*ω)(n)**.)

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**Author:** ![Derleth](https://avatars.discourse-cdn.com/v4/letter/d/b9e5f3/32.png) [@Derleth](https://boards.straightdope.com/u/Derleth)\
**Post date:** [December 9, 2019, 3:45am UTC](https://boards.straightdope.com/t/what-is-f_-0-3/844543/5 "2019-12-09T03:45:02Z")

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You might get a better answer at [MathOverflow](https://mathoverflow.net/), or at [/r/math on Reddit](https://old.reddit.com/r/math).