## Search found 321 matches

- Mon May 21, 2018 5:25 pm UTC
- Forum: Forum Games
- Topic: My number is bigger and smaller!
- Replies:
**37** - Views:
**5732**

### Re: My number is bigger and smaller!

5 in a pentagon using Steinhaus-Moser notation

- Sat May 19, 2018 5:41 pm UTC
- Forum: Forum Games
- Topic: My number is bigger and smaller!
- Replies:
**37** - Views:
**5732**

### Re: My number is bigger and smaller!

2^2^2^2^2

- Sat May 19, 2018 4:59 pm UTC
- Forum: Forum Games
- Topic: My number is bigger and smaller!
- Replies:
**37** - Views:
**5732**

- Sat Aug 26, 2017 6:22 am UTC
- Forum: Forum Games
- Topic: My number is bigger!
- Replies:
**1590** - Views:
**412737**

### Re: My number is bigger!

Wardaft, a few questions: 1. Is the k in " decrease l until you reach the first (IE highest) k+1:v such that k:v is an empty cell" a different variable from the k in "order k worm"? 2. The "inspection vector" is just the vector for the current global head, right? 3. I d...

- Wed May 18, 2016 12:08 am UTC
- Forum: Forum Games
- Topic: My number is bigger!
- Replies:
**1590** - Views:
**412737**

### Re: My number is bigger!

Oddly enough, I believe the Four Color Theorem to be much more simple - both in depth and in proof complexity than Kruskal's Tree. Consider that those "short" proofs of Kruskal's refer to numerous results before them, such as König's Lemma (we're talking about Friedman's finite version, a...

- Tue May 17, 2016 8:56 am UTC
- Forum: Forum Games
- Topic: My number is bigger!
- Replies:
**1590** - Views:
**412737**

### Re: My number is bigger!

Well, certainly - but it doesn't have to be here. I'd be satisfied with a reference as well, as long that seems conclusive for the question here. Well, I don't know if I can find a reference - this sort of question doesn't seem to generate much interest in the general mathematical community. Howeve...

- Tue May 17, 2016 7:50 am UTC
- Forum: Forum Games
- Topic: My number is bigger!
- Replies:
**1590** - Views:
**412737**

### Re: My number is bigger!

Especially since you can write a Turing Machine that generates all possible proofs in ZFC. It would therefore seem to me, intuitively, that Busy Beavers should be stronger than proof-length base functions. Given how convinced the experts are on this, I'm pretty sure that my intuition is wrong. But ...

- Mon May 16, 2016 6:23 pm UTC
- Forum: Forum Games
- Topic: My number is bigger!
- Replies:
**1590** - Views:
**412737**

### Re: My number is bigger!

First, I want to recommend this page by Robert Munafo to anyone interested in the subject.It provides a very understandable, yet far reaching survey of the topic (even though it goes well into the uncomputable range). Second, I specifically find interesting a briefly mentioned reference regarding B...

- Mon May 16, 2016 4:24 pm UTC
- Forum: Forum Games
- Topic: My number is bigger!
- Replies:
**1590** - Views:
**412737**

### Re: My number is bigger!

I've actually been reading through the whole thread, and I submit a different concern regarding numbers submitted as some sort of proof enumeration in ZFC or stronger. So far, as I can see, I've only seen handwaving regarding proof length. Note that proof length is crucial since the limit on proof ...

- Sun Aug 16, 2015 11:49 pm UTC
- Forum: Mathematics
- Topic: Hard combinatorics
- Replies:
**24** - Views:
**3492**

### Re: Hard combinatorics

Here's a solution with 13 quintuplets:

1,2,3,4,5

1,2,6,7,8

1,2,9,10,11

1,2,12,13,14

1,3,6,9,12

1,3,7,10,13

1,3,8,11,14

1,4,6,10,14

1,4,7,11,12

1,4,8,9,13

1,5,6,11,13

1,5,7,9,14

1,5,8,10,12

1,2,3,4,5

1,2,6,7,8

1,2,9,10,11

1,2,12,13,14

1,3,6,9,12

1,3,7,10,13

1,3,8,11,14

1,4,6,10,14

1,4,7,11,12

1,4,8,9,13

1,5,6,11,13

1,5,7,9,14

1,5,8,10,12

- Sun Apr 12, 2015 8:59 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

Shouldn't ([()]) reduce to (<[]>)? Wouldn't that make it e_w, not z_0? Edit in general I think there is a problem with this rule: X[Y()]Z-- = X<[Y]>Z, as this is using the fundamental sequence W^(a+1)[n] = W^a n Also as a nitpick I think you need to specify Y is balanced. Thanks for the input, mike...

- Wed Mar 25, 2015 12:59 pm UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

Sorry again for lateness. {X}-- = {X--}, X does not contain any terms a-b I'm not sure what "X does not contain any terms a-b" means. n,0,{0-0} = n+1,{0-0}, {{0}<,{0}>}<,0> n,{X,0},{0-0} = n+1,{X},{0-0},{{0}<,{0}>}<,0> n,0,0,{0-0} = n+1,<{>0<,{0-0}}><,0> n,{0-0},{0-0} = n+1<,0>{0-0}<,0> n,...

- Thu Mar 19, 2015 6:38 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

Now, I'll admit defeat. Looks like creating a bottom-top notation only can get one so far, because by the next hierarchy it didn't matter how one jumped at the base (everything before then becomes an offset) and it's all about the rules that you use by then, and all amount of weaker rules and exten...

- Wed Mar 18, 2015 9:23 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

n,{X,0},0,{0} = n+1,{X},0,<,{0}><,0> = phi(a@w) n,0,0,{0} = n+1,<{>0<,{0}}> = phi(a@w+1) This doesn't quite work, as n,0,0,{0} is diagonalized over over {X, {0}} which is a lower level operation. It will be silightly bigger than phi(1@w), but not much (something like phi(2,(phi(1@w)+1)) n,{0-0} = n...

- Mon Mar 16, 2015 6:17 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

Sorry for the late reply. case 2: all the variables are {0} n,{0},O -- = = <n+1,{><,O}><,0> This seems suboptimal - better would be n,{0},{0},O -- = n+1, <{>{{0},O}<,O}><,0> case 4: b = {0} n,O1,{a,0},{0},O2 -- = <n+1,{O1,{a},>{0}<,O2}><,0> I think what you have written doesn't work - you have n+1 i...

- Fri Mar 13, 2015 10:19 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

Okay, I had thought that once I managed to reach ψ(Ω^Ω^x) I could just plug any ordinal less than ψ(Ω^Ω^w) in x, and get ψ(Ω^Ω^x) where x is that ordinal. So, um, I should be able to hold rules strong enough that, say: {X[0]-0[0]0} Holds that ordinal, so: {{{0-0-0-0-0-0-0-0-0}}[0]-0[0]0} Reaches ψ(...

- Fri Mar 13, 2015 8:53 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

For the remaining cases, let b be the rightmost variable and a be the rightmost nonzero variable to the left of a. case 4: b = 0 {O1 - a,0 - 0 - O2}-- = {O1 - a -> 0 <-O2},<{O1 - a -> 0 <-O2}> Could this be a typo? I don't know what {O1 - a -> 0 <-O2} means. case 5: a and b both end in ,0 {O1 - a,0...

- Tue Mar 10, 2015 7:52 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

One correction to my notation: The line A(B(m)C n)D -- = A(<B(>m-1<)C n>)D, where n < m, C and D do not contain any )'s and C does not contain any x) with x < m. should read A(B(m)C n)D -- = A(<B(><m-1)C> n)D, where n < m, C and D do not contain any )'s and C does not contain any x) with x < m. I se...

- Tue Mar 10, 2015 5:16 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

Okay, we're waiting for WarDaft to respond, but in the meantime I figured that I would increase my number again. We will define entries to be strings of the form (x1 x2 ... xn m), where m is a nonnegative integer and the xi are entries (Call m the level of the entry). 0-entries are entries with leve...

- Tue Mar 10, 2015 2:25 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

Now, I wonder if this just gets me into a hole, because I'm nesting with weaker rules here. n,0,{0-0},{Y} follows the rules of above, but nesting {{0,{0-0},{Y}}} instead of {{{0-0},{Y}}} n,0,{0-0},{0-0},{Y} follows the rules of above, but nesting {{0,{0-0},{0-0},{Y}}} instead of {{0,{0-0},{Y}}} n,0...

- Mon Mar 09, 2015 8:05 pm UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

Well, there's just one issue, and that is you want to insert notations into other notations without creating ambiguity. For example, I want to be able to have {0}, {{0}}, A, 0 with A = {0-0}, {{0-0},{{0,0}}}, but of course we want {0}, {{0}}, {0-0}, {{0-0},{{0,0}}}, 0 to be something else entirely. ...

- Mon Mar 09, 2015 6:49 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

Okay, no shortcuts here because from what I read shortcuts mean getting rid of everything I have so far and "going for something stronger" or something. So I'll make: n,{0[0]0} = n,{0<-0>} So that it reaches the SVO. Now {0[0]0} can be used in any place of n,{0<-0>} for the same rules. n,...

- Sun Mar 08, 2015 11:21 pm UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

Very well then, n|0 = n n|X = n+1|X-- (b)-- = b (X,0,0)-- where X contains no non-zero terms = <(X,>0<)> (X,0,b,Y)-- where X contains no non-zero terms = <(X,>0<,b--,Y)> (a,X,0,b,Y)-- where X contains no non-zero terms, and b does not contain square brackets = <(a--,X,>((a--,X,0,b,Y))<,b--,Y)> (a,b...

- Sun Mar 08, 2015 4:57 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

Vytron wrote:Question:

Does it continue:

([[[[]]]])=ψ(Ω^Ω^Ω^Ω)

([[[[[]]]]])=ψ(Ω^Ω^Ω^Ω^Ω)

([[[[[[]]]]]])=ψ(Ω^Ω^Ω^Ω^Ω^Ω)

Making the limit ψ(ε_{Ω+1})?

Yup!

- Sun Mar 08, 2015 4:39 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

b+1: ((x^ε0^ω)^ε0^ω) = x^ε0^ω^2 b+2: ((x^ε0^ω^2)^ε0^ω) = x^ε0^ω^3 limit 0¡0!0¡0!0 = x^ε0^ω^ω c¡0!b¡0!a a: f#^ε0 limits at b+1 #^ε0^ω b: f#^ε0^ω c: f#^ε0^ω^ω Actually, ((x^ε0^ω)^ε0^ω) = (x^(ε0^ω)*(ε0^ω)) = (x^(ε0^(ω+ω))) = x^ε0^(ω2), not x^ε0^ω^2. so b+2: ((x^ε0^(ω2))^ε0^ω) = x^ε0^(ω3) limit 0¡0!0¡0...

- Sun Mar 08, 2015 4:35 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

I can't say about Daggoth's notation, because I don't see a definition, it must be pretty far back in the thread. Okay, now seems like a good time to challenge WarDaft's leading notation, which I believe is currently at the level psi(Ω^Ω^Ω^2). Okay, so this time I'll avoid the use of numbers in the ...

- Sun Mar 08, 2015 3:13 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

Oh, and Daggoth, sorry that I haven't commented on your notations, but I haven't had the energy to try and decipher them in addition to Vytron's notation.

- Sun Mar 08, 2015 2:52 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

Woah. Vytron, I intentionally didn't give you precise rules for how to make your notation reach the SVO, not because I didn't want you to reach it, but because I figured you wanted to make your own notation, and giving you the exact rules might get in the way of that. I would certainly have given yo...

- Sun Mar 08, 2015 12:21 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

Hmm, it looks like in that post mike-l never describes how he handles [X | Y,0] when X is anything other than 0, and how he handles that is key to whether his notation reaches Gamma_0 or just zeta_0. If he uses the same rule for X as for 0, i.e. [X | Y,0] (n) = [X | Y]^n (n), then he'll only get to ...

- Sat Mar 07, 2015 11:44 pm UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

No, I'm afraid not. Under the new rules if {0-X} has ordinal a, then {0-X,0} has ordinal a+1. This is actually weaker than before. So we have {0-0} = z_0 {0-0,0} = z_0 + 1 {0-{0}} = z_0 + w {0-{{0}}} = z_0 + e_0 {0-{0-0}} = z_0 2 {0-{0-{0-0}}} = z_0 3 {0,0-0} = z_0 w {0,0-0,0} = z_0 w + 1 {0,0-{0,0-...

- Sat Mar 07, 2015 1:20 pm UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

Not quite general enough, but it's a good start. Of course, the notion of "bigger" will have to be precisely defined, but usually I have a pretty good idea of what is bigger. [quote} And, generally, n,{0,0-X} = n,<{0->{0,0-X}<}> where X is a reduced string, n,{Y-X} = n,<{y->{Y-x}<}> where ...

- Sat Mar 07, 2015 9:03 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

Thanks for your effort, but it would really help if you could create a set of general rules for your notation. It looks like {X} means something vaguely like "reduce the least important thing and iterate at the next lower level", but that needs to be made precise.

- Sat Mar 07, 2015 5:31 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

I'm afraid I'm not seeing the general pattern here. For example, n,{{0,0-0}} = n,<{0->{0}<}>,<{0->{0}<}>,{0,0-0}> doesn't seem to follow the pattern of any previous example. I don't know what "Every lower line dominates a previous line" means. It may well be that what you have in your mind...

- Sat Mar 07, 2015 2:53 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

All right, you're going to have to explain your rules again, because before the entire definition for the - notation was n,[0-0] = n,<{>0<}> n,[0-x] = Follows the rules of above n,[0,0-0] = n,<[0->{0}<]> n,[0,0,0-0] = n,<[0,0->{0}<]> n,[x-0] = Follows the rules of above n,[0-0-0]= n,<[>{0}<-0]> n,[0...

- Thu Mar 05, 2015 11:24 pm UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

I don't think that can be right, if: {0,0-0} = phi(3,0) Then: {0,0-0},0 = phi(3,0)+1 {0,0-0},0,0 = phi(3,0)+2 {0,0-0},{0} = phi(3,0)+ω {0,0-0},{0,0-0} = phi(3,0)*2 0,{0,0-0} = phi(3,0)*ω {0-0},{0,0-0} = phi(3,0)*phi(2,0) {{0,0-0},0} = phi(3,0)^2 {{{0,0-0}}} = ε_phi(3,0) {{{{0,0-0}}}} = ε_ε_phi(3,0)...

- Thu Mar 05, 2015 4:09 pm UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

Sorry, I gave you the value of {0-{0,0-0}-0}. {0-{0,0-0}} is a weird expression, it's bigger than {0,0-0} but not much bigger; {0,0-0} resolves to <{0->{0}<}>, and {0-{0,0-0}} just has an extra layer of {0-x} to it, so it has the same limit. So {0-{0,0-0}} is also at phi(3,0) So {0-{0,0-0},0} is at ...

- Thu Mar 05, 2015 3:08 pm UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

with three arguments, each increment of the third argument increments to the next z-ordinal, each increment of the second argument increments to the next phi(3,x) ordinal, and each increment of the first argument increments to the next phi(4,x) ordinal. Since {0,0-0} = phi(3,0), {0-{0,0-0},0} = phi(...

- Thu Mar 05, 2015 2:50 pm UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

Hmm, based on your examples I'm afraid I don't think the notation will reach the SVO. More analysis: {0-0} = z_0 {0-0,0} = z_1 {0-0,0,0} = z_2 {0-{0}} = z_ω {0-{0-0}} = z_z_0 {0,0-0} = phi(3,0) {0,0-0,0} = z_(phi(3,0)+1) {0,0-0,0,0} = z_(phi(3,0)+2) (0,0-{0}} = z_(phi(3,0)+ω} {0,0-{0-0}} = z_(phi(3,...

- Thu Mar 05, 2015 11:49 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

I was wondering whether there could be an issue with the distinction between, say, 0-0 , 0-0 and 0 - 0.0 - 0 but I guess one can figure it out by looking at the structure of the {}'s. I'm not totally sure though. Okay, so continuing the analysis of Vytron's notation: {{0}, 0} = e_0^2 {{0}, 0} {{0}, ...

- Thu Mar 05, 2015 10:53 am UTC
- Forum: Forum Games
- Topic: Your number is, in fact, not bigger!
- Replies:
**1240** - Views:
**150453**

### Re: Your number is, in fact, not bigger!

Okay, here are ordinal equivalences for my SVO notation: n = n (0) = ω n(0) = ω+n (0)(0) = ω2 (0)(0)(0) = ω3 (1) = ω^2 (2) = ω^3 (3) = ω^4 ((0)) = ω^ω (0(0)) = ω^(ω+1) ((0)(0)) = ω^(ω2) ((0)(0)(0)) = ω^(ω3) ((1)) = ω^(ω^2) ((2)) = ω^(ω^3) ((3)) = ω^(ω^4) (((0))) = ω^(ω^ω) ((((0)))) = ω^(ω^(ω^ω)) (,0...