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My Number is Bigger (Crazy Edition)

Posted: Wed Dec 16, 2015 9:50 pm UTC
'Ello.

This version is not for the faint of heart.

There are three rules.

1. Unless it is blatantly obvious, you must show that your number is finite when you post it.
2. Unless it is blatantly obvious, you must show that your number is winning, otherwise it is not.
3. You must define a specific number that remains well defined no matter what anyone else posts later.

To get the ball rolling...

BB(5), the smallest number that we don't know how big it is. Well, probably.

Re: My Number is Bigger (Crazy Edition)

Posted: Wed Dec 16, 2015 9:55 pm UTC
Drat. Nonrecursive googology isn't my strongpoint.

The largest integer definable in at most 160 symbols in ZF.

Edit: I've not got a proof of largeness, so username's is winning.

Re: My Number is Bigger (Crazy Edition)

Posted: Wed Dec 16, 2015 9:55 pm UTC
EDIT: Ninja'd.

The largest number not definable using emlightened's number of characters in ZF.

Beat that!!!!!!

Re: My Number is Bigger (Crazy Edition)

Posted: Wed Dec 16, 2015 10:08 pm UTC
Well escalated quickly.

Or, you know, not.

As posted, username5243's number is undefined, because there is no largest number not definable in a finite number of characters in ZF.

As far as emlightened, I don't actually know how many characters it takes to express the operation of a Turing Machine in ZF. Without that, it would be a tremendous amount of work to prove that there existed a formula of size at most 160 symbols. You could try to actually encode the operation of a TM into a ZF formula, and then show that you have enough leftover symbols to specify a 5 state Turing machine for that formula to operate on, or you could just pretend you had two zeros in there, because then it's obviously bigger.

Re: My Number is Bigger (Crazy Edition)

Posted: Wed Dec 16, 2015 10:09 pm UTC
username, that's infinite. If the largest number you can define using k characters in ZF is x, then all values >x are not definable using k number of characters in ZF; hence there is no largest number.

I'll choose the "pretend you had two zeros in there, because then it's obviously bigger" option.

Define {System}0(k) = The largest integer definable in at most k symbols in System, and reduce {System}α(k) as a FGH (specify if not Wainer Hierarchy).

Send: {ZF}0(7777)

Re: My Number is Bigger (Crazy Edition)

Posted: Wed Dec 16, 2015 10:12 pm UTC
You got me, I guess I meant the largest definable number (or the smallest non-definable number in case that's infinite). I don't know what that means, so I give up and request that a game like this with no uncomputable functions be made soon.

Re: My Number is Bigger (Crazy Edition)

Posted: Wed Dec 16, 2015 10:14 pm UTC
We could just run two games simultaneously: a 'recursive largest number' and a 'uncomputable largest number'.

In the case of the latter, send: fε_0(12).

Re: My Number is Bigger (Crazy Edition)

Posted: Wed Dec 16, 2015 10:15 pm UTC
Oh yeah?

f_Gamma_0(Googolplex)

Re: My Number is Bigger (Crazy Edition)

Posted: Wed Dec 16, 2015 10:18 pm UTC
To stop this from turning into a 'name the largest ordinal' game, does it sound fair to say that all ordinals >Γ0 need explicitly defined fundamental sequences?

Also: fΓ_0(fΓ_0(2))

Re: My Number is Bigger (Crazy Edition)

Posted: Wed Dec 16, 2015 10:20 pm UTC
Fine...

f_G_0+(Graham's Number)(googol)

Re: My Number is Bigger (Crazy Edition)

Posted: Wed Dec 16, 2015 10:26 pm UTC
On the other hand, it wouldn't be hard to just copy definitions from deedlit on the wiki, and the Wainer hierarchy covers all cases up to ε_(Γ0+1). Or you could just send (say) {ZF}Ψ(Ω^Ω^ω)(10100) in the other category.

f_(Γ0ω)(14)

Edit: I feel like I've completely derailed WarDaft's thread. Sorry.

{ZF}2(7777)

Re: My Number is Bigger (Crazy Edition)

Posted: Wed Dec 16, 2015 10:50 pm UTC
{ZF}3(7777)
(which is probably more laughable then posting "Graham's Number plus 1" in reply to Graham's Number. But hey, everybody seems to be doing it and it's perfectly legal, so why not?)

Re: My Number is Bigger (Crazy Edition)

Posted: Wed Dec 16, 2015 11:00 pm UTC
{ZF}3(7777)

This is Graham's Number +1. I think you were at Graham's Number3.

Re: My Number is Bigger (Crazy Edition)

Posted: Wed Dec 16, 2015 11:09 pm UTC
{ZF}ω^77(77↑7777)

(I personally feel that applying these simple recursions to the ZF thing is actually far more naïve than saying "Graham's plus one" - relatively speaking. I wonder if there's an objective way to compare such things on such vastly different scales)

Re: My Number is Bigger (Crazy Edition)

Posted: Wed Dec 16, 2015 11:19 pm UTC
The only thing I can think of is comparing proof-theoretic ordinals (or their nonrecursive equivalent), but notations that go that large don't yet exist.

{ZF}ω↑↑77(77)

Re: My Number is Bigger (Crazy Edition)

Posted: Thu Dec 17, 2015 12:21 am UTC
PsiSquared wrote:{ZF}ω^77(77↑7777)

(I personally feel that applying these simple recursions to the ZF thing is actually far more naïve than saying "Graham's plus one" - relatively speaking. I wonder if there's an objective way to compare such things on such vastly different scales)

In a sense, I agree. When you do X+1, you know you're not really contributing to the size, but trying to square or cube something like Grahams number implies you think you're contributing meaningfully to the enhugemousness.

To that end...

{ZF+Con(ZF)}10(10) will handily beat {ZF}a(n) for just about any practically definable a or n that could be proven well founded by ZF.

Re: My Number is Bigger (Crazy Edition)

Posted: Thu Dec 17, 2015 12:52 am UTC
Well, if a person thinks he is actually making a difference when cubing Graham's number, that's only because he doesn't understand Graham's number. The question is whether there's an objective way to compare various levels of "naiveness" across different scales.

As for the game:

While I could post something like {ZF+Con(ZF)}11(10), I think this kind of stunt had already outlived its welcome on this thread. So I'll just shut up and try to learn from what the others are doing, before submitting my own two cents.

Re: My Number is Bigger (Crazy Edition)

Posted: Thu Dec 17, 2015 4:26 am UTC
Hmmm, let me try this:

Define a super machine as a machine that can solve the halting problem* for BB(N).

Define a BB2(N) to be the maximum number of steps a super machine with N rules can make before halting.

Define a super super machine as a machine that can solve the halting problem for BB2(N).

Define a BB3(N) to be the maximum number of steps a super super machine with N rules can make before halting.

Generally:

Define a super...for k "super"s...super machine as a machine that can solve the halting problem for BBk(N).

Define a BBk+1(N) to be the maximum number of steps a super...for k "super"...super machine with N rules can make before halting.

*-A super machine would know if a program of length N halts by simulating it for BB(N) steps, throwing away the ones that halt and returning the maximum score of one that didn't.

Define:

BBω(n) = BBk(n)

And define:

BBFGH(n)

As a hierarchy of functions that collapse in the same way that the FGH does by repeating this process of super machines (where you'd replace FGH with the symbol expanded).

Send:

BBΓ_0({ZF+Con(ZF)}10(10))

Re: My Number is Bigger (Crazy Edition)

Posted: Thu Dec 17, 2015 12:22 pm UTC
Aha, the oracle machines show up. Oracle is the word your looking for, when you give a Turing Machine the ability to solve the halting problem.

I'm very sorry Vytron, but that's actually smaller than {ZF+Con(ZF)}11(10). {ZF}10(10) already includes statements which describe oracle Turing machines. In fact {ZF}0(7777) will already be able to define oracle machines that complicated.

Actually, now that I think about it, I was wrong, {ZF+Con(ZF)}10(10) isn't any bigger than {ZF}10(10). Since the +Con(ZF) only adds to our ability to prove things true rather than our ability to define things, and the wording of the {A}x(B) system does not constrain to statements which can be proven true.

The more that I think about it, the less sure I am that this is well defined at all! If we have ZF+A and ZF+B, where A and B are each independent of ZF, it is possible for different statements to be true in each. With a different set of true statements, the numbers that you can define in a given set of symbols will be different. Thus, without requiring that they are provable in ZF, I think we have allowed ourselves to include statements whose truth is not decided.

But I'm not actually a mathematician so...

Let [A|k] be the largest recursive ordinal provably well founded in system A that can be defined in at most k symbols.

{ZF}[ZF|1010](10)

Re: My Number is Bigger (Crazy Edition)

Posted: Thu Dec 17, 2015 12:40 pm UTC
I see...

Well, oracles have to do something somewhere, don't they?

{ZF}[ZF|BBΓ_0(10)](10)

Just because, it'd be mind-boggling if this wasn't much bigger than your number.

Re: My Number is Bigger (Crazy Edition)

Posted: Thu Dec 17, 2015 12:44 pm UTC
Oh, yes, that's defnitely bigger. The thing was you used one on an older number that had a 10 in it rather than an 11, and you applied it to the final value, which wasn't enough to overcome the bump to 11.

Re: My Number is Bigger (Crazy Edition)

Posted: Sun Dec 20, 2015 7:33 pm UTC
Well, in the interest of keeping things rolling...

Since {ZF}0(n) grows faster than BBa(n) for any recursively defined, naturally the next stop on the crazy train is {ZF}[ZF|{ZF}0(1010)](10)

Re: My Number is Bigger (Crazy Edition)

Posted: Sun Dec 20, 2015 7:54 pm UTC
Send: {ZF}[ZFC|7712](1277)

Much smaller than {ZFC}0 (asymtotically).

Re: My Number is Bigger (Crazy Edition)

Posted: Thu Jan 14, 2016 7:58 pm UTC
{PRA}[ZFC|{ZFC}g_65(100)](100000000000000000000000000000)!

Where PRA stands for primitive recursive arithmetic. Because why not

Re: My Number is Bigger (Crazy Edition)

Posted: Thu Jan 14, 2016 8:46 pm UTC
Coming back here, I feel disappointed that I missed out Z2 and KP.

I initially missed that your number was lower-bounded by {ZFC}g_65(100), and started explaining why it was so small. My bad.

Also: welcome to the forums!

{ZFC}[PRA|200](200)

Re: My Number is Bigger (Crazy Edition)

Posted: Sun Jan 17, 2016 6:07 pm UTC
{ZFC}[PRA|200](200)!

Yes, that's factorial.

@emlightened Thanks!