What? W takes an array and returns an integer.

W([1, 1]) = W([W([1, 0]), 0]) + 1.

- Yesterday, 8:18 pm
- Forum: The Sandbox
- Topic: Largest total computable function competition
- Replies:
**151** - Views:
**14918**

- Yesterday, 9:24 am
- Forum: The Sandbox
- Topic: Largest total computable function competition
- Replies:
**151** - Views:
**14918**

- Yesterday, 9:21 am
- Forum: The Sandbox
- Topic: Largest total uncomputable function competition
- Replies:
**24** - Views:
**1446**

Let S_i be a set theory. S_0 = ZFC, and S_i is the axiom set that can prove S_{i-1} well-founded. Assume that S_i will always be provably well founded, since otherwise ZFC isn't a well founded set theory. My function is the Rayo-analogue function. but instead of FOST we have S_{10^100}. Has got to ...

- Yesterday, 2:05 am
- Forum: The Sandbox
- Topic: Largest total uncomputable function competition
- Replies:
**24** - Views:
**1446**

Let S_i be a set theory. S_0 = ZFC, and S_i is the axiom set that can prove S_{i-1} well-founded. Assume that S_i will always be provably well founded, since otherwise ZFC isn't a well founded set theory. My function is the Rayo-analogue function. but instead of FOST we have S_{10^100}. Has got to b...

- November 12th, 2019, 9:16 am
- Forum: Website Discussion
- Topic: Like button in forum request
- Replies:
**33** - Views:
**10403**

Aha. Doesn't that automatically prove a dislike button should exist? Otherwise why would they use it?

- November 11th, 2019, 10:13 pm
- Forum: The Sandbox
- Topic: Ordinals in googology
- Replies:
**121** - Views:
**2171**

Since my last attempt at a FPT was trash, I made another:

https://googology.wikia.org/wiki/User_b ... udop/FPT_2

https://googology.wikia.org/wiki/User_b ... udop/FPT_2

- November 11th, 2019, 10:11 pm
- Forum: Website Discussion
- Topic: Like button in forum request
- Replies:
**33** - Views:
**10403**

I disagree with Goldberg, who disagrees with Calyman, who is agreed by me; hence, Calcyman disagrees with Goldberg.

- November 7th, 2019, 2:19 am
- Forum: Scripts
- Topic: Geneascopy
- Replies:
**6** - Views:
**2363**

Oh wow you're back? So far, the only rule to show similar behaviour to CGoL is B37e/S23, which, beyond the collective [linear growths]*P(linear growths) after gen 5470, shows a similar dip followed by a rapid increase with loads of fine structure, although obviously reaching higher populations. A le...

- November 6th, 2019, 10:07 am
- Forum: The Sandbox
- Topic: Largest total uncomputable function competition
- Replies:
**24** - Views:
**1446**

Ascending TMs Consider a "Universe ascend" which can reach another Universe of dimensions. You can't. There isn't a next universe, we have already took up all the space... ...unless you index dimensions by ordinals, and invent some method of fixed-point leap. i.e. we originally had 1d, 2d, ... nd, ...

- November 5th, 2019, 8:54 am
- Forum: The Sandbox
- Topic: Largest total uncomputable function competition
- Replies:
**24** - Views:
**1446**

OMG I KNOW I believe the fastest growing known computable functions are of the following form: n ↦ The combined running time of all Turing machines (with no input) such that there exists a proof that they halt in second-order PA using fewer than n symbols FIrst order set theory is much stronger, and...

- November 4th, 2019, 11:30 pm
- Forum: The Sandbox
- Topic: Ordinals in googology
- Replies:
**121** - Views:
**2171**

Is the well ordering of w_1 provable? Isn't well-orderedness part of the definition of an ordinal? If I'm missing something, then see this proof (set X to be the set of natural numbers to get the conclusion that ω_1 exists and is well-ordered) Several things: The well ordering of w_1 implies the we...

- November 3rd, 2019, 12:29 am
- Forum: The Sandbox
- Topic: Random posts
- Replies:
**3285** - Views:
**396481**

- November 2nd, 2019, 11:48 pm
- Forum: The Sandbox
- Topic: Largest total uncomputable function competition
- Replies:
**24** - Views:
**1446**

No it doesn't solve its own halting problem. It executes on a higher level as soon as it ascends; it can't descend. Hence, it can never solve its own halting problem, since to solve the halting problem on its current level it needs to ascend to the next level.

- November 2nd, 2019, 11:47 pm
- Forum: The Sandbox
- Topic: Ordinals in googology
- Replies:
**121** - Views:
**2171**

Is the well ordering of w_1 provable?

- November 2nd, 2019, 4:14 am
- Forum: The Sandbox
- Topic: Largest total uncomputable function competition
- Replies:
**24** - Views:
**1446**

Consider a Turing machine with another added state, "ascend", which makes a new tape called the "ascended tape" and all previous tapes are called "lower tapes". The head is allowed to read/write to the ascended tape and the lower tapes, and the ascended tape has the ability to solve the Halting pro...

- November 2nd, 2019, 4:08 am
- Forum: The Sandbox
- Topic: Largest total uncomputable function competition
- Replies:
**24** - Views:
**1446**

Consider a Turing machine with another added state, "ascend", which makes a new tape called the "ascended tape" and all previous tapes are called "lower tapes". The head is allowed to read/write to the ascended tape and the lower tapes, and the ascended tape has the ability to solve the Halting prob...

- November 2nd, 2019, 3:11 am
- Forum: The Sandbox
- Topic: Ordinals in googology
- Replies:
**121** - Views:
**2171**

sup{a<w_1^CK:a}Moosey wrote: ↑November 1st, 2019, 5:35 pmThe binary Veblen phi function.There is no such thing. If you say that a is the largest countable ordinal, then I can point out that a+1 is larger.

- October 31st, 2019, 11:14 pm
- Forum: The Sandbox
- Topic: Ordinals in googology
- Replies:
**121** - Views:
**2171**

Isnt T0 the result of the diagonalization of e0, z0, n0, etc? i.e. e0 = w^^2 [sic], z0 = w^^^2, n0 = w^^^^2 I remember you said somewhere T0 = w^^w, obviously that's a large underestimation T0? Do you mean gamma_0? gamma_0 is certainly not the supremum (which is what you mean-- I won't argue with P...

- October 31st, 2019, 8:44 pm
- Forum: The Sandbox
- Topic: Ordinals in googology
- Replies:
**121** - Views:
**2171**

e0 is the first solution to a = w^a, or a transfinite power stack of w's. Adding another w will not change the number.

- October 31st, 2019, 9:13 am
- Forum: The Sandbox
- Topic: Ordinals in googology
- Replies:
**121** - Views:
**2171**

I have defined an OCF, a fat version of it, and an accompanying fundamental sequence system. ... See my blog post here: https://testitem.github.io/colg/fpt.html That's not a traditional OCF-- OCFs are defined differently, like this: C_0(x) = set C_n+1(x) = C_n(x) U {operations with a and b in C_n(x...

- October 31st, 2019, 2:27 am
- Forum: The Sandbox
- Topic: Ordinals in googology
- Replies:
**121** - Views:
**2171**

I have defined an OCF, a fat version of it, and an accompanying fundamental sequence system. I have took the limit of the fat OCF function and defined the fundamental sequence for it as well. (Note that the fat OCF system isn't exactly "more powerful", but instead "simpler".) You can see where this ...

- October 25th, 2019, 11:16 pm
- Forum: The Sandbox
- Topic: Ordinals in googology
- Replies:
**121** - Views:
**2171**

What is the cardinality of the set of: 1) all ordinals; 2) all countable ordinals; 3) all computable ordinals. 3) ℵ_0 That does not make sense; you can infinitely exponentiate w and add them together (this is further enlargened by the branching possibilities of each exponent and coefficient), so un...

- October 25th, 2019, 7:34 am
- Forum: The Sandbox
- Topic: Ordinals in googology
- Replies:
**121** - Views:
**2171**

My argument for Latex is with such a complicated definition, with no formatting it's difficult to understand.

What is the cardinality of the set of:

1) all ordinals;

2) all countable ordinals;

3) all computable ordinals.

1 is at least beth_2.

What is the cardinality of the set of:

1) all ordinals;

2) all countable ordinals;

3) all computable ordinals.

1 is at least beth_2.

- October 24th, 2019, 8:45 pm
- Forum: The Sandbox
- Topic: Ordinals in googology
- Replies:
**121** - Views:
**2171**

PLEASE USE LATEX

And to say something constructive I'm going to use p-adic's fundamental sequence argument

And to say something constructive I'm going to use p-adic's fundamental sequence argument

That does not work with your recursive definition at all.8a. ah_n((0@a)#{b,$_6}) = ah_n((0@(a[n])#{b,$_6}), a a lim ord

- October 22nd, 2019, 7:13 am
- Forum: The Sandbox
- Topic: Random posts
- Replies:
**3285** - Views:
**396481**

in googology wiki I basically put a lot of pages in the "-thoth" and "-logue" category and now i have around 400 points