Conway and Moser Revisited

Monday, February 28, 2011

Direct continuation of: Comparing Moser Notation With Conway Notation

When we last forayed into the ultra-esoteric land of comparing large numbers, I was taking two systems of notation — Moser Notation and Conway Chained-Arrow Notation and determining which one made larger numbers. A reader Louis Epstein had asked:

What about Moser polygons,considered the step beyond Conway arrows at Susan Stepney’s site?…I’d like to see a good comparison between 3→3→3→3 and
3[3[3[3]]]] (i.e. 3[3[27]]])…both are way bigger than Moser’s Number but are they both bigger than Graham’s?

 

The answer turned out that 3→3→3→3 was definitively bigger than 3[3[3[3]]]], and both were definitively bigger than Graham’s Number. But now we return with bigger and better questions, also from Louis Epstein. He asked a few others, but this is the question I intend to answer for this post:

The broader issue of the relative power of Conway and Moser notations is not yet clear,though.Does the relationship hold for all values of x[x[x[x]]] vs x->x->x->x or for all numbers of arrows vs. polygon-nestings of repetitions of a given x?

 

Setting Up the Problem

So using the notations and methods I established in the previous post, I will now compare the following two functions:

M(x) = \underbrace{x[x[x[x[\cdots[x]]]]\cdots]}_{\text{width of x-1}}

C(x) = \underbrace{x \to x \to x \to \cdots \to x}_{\text{width of x-1}} = x \to_2 x

 

If you’re familiar with my large number series, you should recognize the second one as Extended Conway Chained-Arrow Notation.

 

Values and Examples

Here are some values to get an idea of how these functions work.

M(3) = 3[3] = 3^3 = 27

M(4) = 4[4[4]] = 4[4[3]4] = 4[(4[3]3)[3]] = 4[((4[3]2)[3])[3]] = 4[(((4[3])[3])[3])[3]] = 4[(((256)[3])[3])[3]] = … = ~4[256^256^256^256]

M(5) = 5[5[5[5]]]

M(10) = 10[10[10[10[10[10[10[10[10]]]]]]]]

 

C(3) = 3→3 = 27

C(4) = 4→4→4 = 4→(4→3→4)→3 = 4→(4→(4→2→4)→3)→3 = 4→(4→(4→4→3)→3)→3 = 4→(4→(4→(4→3→3)→2)→3)→3 = 4→(4→(4→(4→(4→2→3)→2)→2)→3)→3 = 4→(4→(4→(4→(4→4→2)→2)→2)→3)→3 = 4→(4→(4→(4→(4^4^256)→2)→2)→3)→3

C(5) = 5→5→5→5

C(10) = 10→10→10→10→10→10→10→10→10

 

The ultimate question to solve will be: which function, C(x) or M(x), has a larger growth rate?. Of course, the two functions generate numbers so large that they are impossible to compare directly, which is why I used a mutual and easy comparison in the previous function — turn both into functions into limit functions using ordinal notation, and then compare the limit functions.

 

Setting Up M(x)

At first glance, it looks like it would be intractable to get a value for such a function, but the previous post actually does all the work for us.

We found out in the previous post that 3[3] = 3^3 = m(3), where m(x) = x^x.

And we came up with the general principle that x[b] = mb-2(x).

Now, we’re looking for 3[3[3]]. What’s that? Well, we know what 3[3] is, so we can plug it in. 3[3[3]] = 3[m(3)] = mm(3)-2(3)

Now what about 3[3[3[3]]]? Well, we do it again.

3[3[3[3]]] = 3[3[m(3)]] = 3[m_{m(3)-2}(3)] = m_{m_{m(3)-2}(3)-2}(3)

 

And we can then do it for 3[3[3[3[3]]]]:

3[3[3[3[3]]]] = m_{m_{m_{m(3)-2}(3)-2}(3)-2}(3)

 

And we can then do it for any length:

\underbrace{3[3[3[\cdots[3]]]}_{\text{width of x}} = \underbrace{m_{m_{m_{\cdots_{m(3)-2}-2}\cdots}-2}(3)}_{\text{depth of x}}

 

Getting A Limit Function For M(x)

Now we want to turn something with a variable depth into a function that is more easily understood. Luckily we have just the thing. If we look at Mω(z), we can do that.

Specifically, mωx(3) will be larger than M(x) for all finite x. (Remember that M(x) = x[x[x[x[...[x]]]…]]]] and that m(x) = x^x).

For example, see what happens when we compare mω3(3) and 3[3[3]].

M(3) = 3[3[3]] = m_{m(3)-2}(3)

m_{\omega}^3(3) = m_{\omega}(m_{\omega}(m_{\omega}(3))) = m_{\omega}(m_{\omega}(m_{3}(3))) = m_{\omega}(m_{m_{3}(3)}(m_{3}(3))) = m_{m_{m_{3}(3)}(m_{3}(3))}(m_{m_{3}(3)}(m_{3}(3)))

 

That gives us the conclusion that mωx(x) > M(x).

 

Using Limit Functions For C(x)

Building on the previous part again, we can also construct a representation of C(x) in terms of m(x).

Earlier we established that 3→x = m(x), that 3→x→3 = mx(3) and that 3→3→x→3 = mω+2(x).

From this, we can actually see a pattern that can be carried out: 3→3→3→x→3 = mω2+2(x), 3→3→3→3→x→3 = mω3+2(x), 3→3→3→3→3→x→3 = mω4+2(x), etc.

Therefore, C(x) ≈ mωx+2(x).

 

And since mωx+2(x) is much greater than mωx(x), we have our conclusion:

C(x) ≈ mωx+2(x) > mωx(x) > M(x), therefore, C(x) > M(x).

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13 Comments (RSS)

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  1. Louis Epstein says:
    Thursday, March 3, 2011 at 4:28 pm

    So,Susan Stepney’s characterization of Moser as more extensible than the more recently devised Conway notation was apparently off base,but I still don’t see changing the popble sequence.I’m looking forward to my Titled and Alphabet Number pages going live soon,I just have to do some more defining there before I unleash some really complex functions.

  2. Deedlit says:
    Saturday, April 16, 2011 at 3:31 am

    Another minor correction:

    3→3→x→3 is approximately m_(w+2) (x)

    more generally, 3→3→x→(y+1) is approximately m_(w+y) (x)
    3→3→3→x→3 is approximately m_(w*2+2) (x)
    3→3→3→x→(y+1) is approximately m_(w*2+y) (x)

    and so on.

    So, 3→3→3→…→3 is with x+3 3′s is approximately m_(w*x+2) (3). So your final claim is basically correct.

  3. Giga Gerard says:
    Thursday, June 16, 2011 at 8:28 pm

    Hi Peter, cleanup?
    Did you discard of: “Intro to Ordinals and Epsilon Naught” (part 9) altogether now?
    I can’t seem to find part 6 to 9 of the large number series?
    Hé! Why should that be?
    Regards Giga Gerard

  4. Peter says:
    Friday, June 17, 2011 at 1:04 am

    Hello.

    Yes, I did discard my old “Intro to Ordinals” essay and the entire old series because I felt the old series was long-winded, improperly organized, and had too little information per part. I resolved to fix this by starting the new large number series, which begins at http://www.greatplay.net/essays/large-numbers-part-i-magnitude-and-simple-functions. The series that exists now is a new one, entirely rewritten and reorganized — it communicates the same amount of information in much less text and with much better clarity.

    Right now the series has gotten to ω^3, but the next part (part 4) will bring this all the way to epsilon naught, matching with the entire previous series. You can expect this essay to appear on June 22.

  5. Giga Gerard says:
    Friday, June 17, 2011 at 6:08 am

    Yes Peter, your explanation of the gresat matter looks very striaghtforward. I’ll adjust my link to the part 1 page you mention.
    That continues to part 2 and then part 3 below the articles, but then the Conway and Moser story has no such connection. Ok, I’ll add a separate link for that.
    Thank you.

  6. Peter says:
    Friday, June 17, 2011 at 2:36 pm

    No problem, thanks for the link!

  7. Louis Epstein says:
    Friday, December 30, 2011 at 11:16 pm

    I’ve put off working on the Titled Number and Alphabet Number files too long,but rest assured that they and the Epstein Numbers beyond them are still planned.

  8. Peter Hurford says:
    Saturday, December 31, 2011 at 3:15 am

    Let me know when you finish it. I’m going to hopefully finish up the Large Number series in 2-4 more installments someday.

  9. Giga Gerard says:
    Saturday, December 31, 2011 at 10:11 am

    Wondeful! I keep watching you guys closely ;-)

    http://iteror.blogspot.com/ (for data persistence = learn from your DNA)

  10. Louis Epstein says:
    Saturday, December 31, 2011 at 11:06 pm

    Looks like I’ll finish 2011 having defined the Great Big Number,with a dozen “Titled Numbers” to go before setting up the Alphabet Numbers,their bracket-colon notation,and the Epstein Numbers and their multifarious series…which have been in my head for a while.

  11. Louis Epstein says:
    Saturday, December 31, 2011 at 11:53 pm

    Got the Majestic Dominion Number done by midnight…11 to go!

  12. Giga Gerard says:
    Sunday, January 1, 2012 at 3:53 am

    Sounds good mister Epstein, where is it, the page?

    I read here that you use natural number variables (sign 1…) then the usual commas for the function , and then you make them countable within brackets ` (nesting needs 1 bracket usually) like Bowers did, and then you use a colon to express dimensionality function-like (like Bowers did).

    So 4 characters in all, that’s close to counting characters themselves (the next step in the algorithm).

  13. Louis E. says:
    Sunday, January 1, 2012 at 9:22 pm

    The page with the Titled Numbers isn’t live yet,it will link from the one my name links to when it is.

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