
Really big numbers - Petiver
http://blog.oup.com/2017/01/really-big-numbers/
======
tromp
The write-up manages to misquote Rayo's huge entry

"The smallest number bigger than any finite number named by an expression in
the language of set theory with a googol symbols or less"

into the relatively minute

"The least number that cannot be uniquely described by an expression of first-
order set theory that contains no more than a googol (10^100) symbols."

by leaving out the crucial "bigger than". The latter is no more than #symbols
^ googol since any N descriptions cannot describe all of the first N+1
numbers...

Similarly, the proposed function F(n) is merely exponential in n.

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ggchappell
The post takes comments. Write one?

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tromp
Thanks for pointing that out.

Awaiting moderation... (the lack thereof makes HN's commenting much more
satisfying:-)

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lisper
The ultimate discussion of big numbers:

[http://mrob.com/pub/math/largenum.html](http://mrob.com/pub/math/largenum.html)

This too:

[https://johncarlosbaez.wordpress.com/2016/06/29/large-
counta...](https://johncarlosbaez.wordpress.com/2016/06/29/large-countable-
ordinals-part-1/)

Literally to infinity and beyond! :-)

~~~
tromp
Notably absent from that first link is mention of the elegantly fast-growing
Goodstein sequences [1]

[1]
[http://tromp.github.io/pearls.html#goodstein](http://tromp.github.io/pearls.html#goodstein)

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Patient0
Also worth a read (and also in the original articles comments):
[http://www.scottaaronson.com/writings/bignumbers.html](http://www.scottaaronson.com/writings/bignumbers.html)

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gamina
Also worth a read (it makes you try to grasp how enormous is Graham's number):
[http://waitbutwhy.com/2014/11/1000000-grahams-
number.html](http://waitbutwhy.com/2014/11/1000000-grahams-number.html)

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brownbat
The use of constants without any actual representation is a bit sneaky.

You might apply a ceiling assuming constants are limited to, say, 16x16
pixels, with each pixel marked or unmarked.

That gives you 2^256 constants to play with (minus those that are already
taken), probably many more than the human mind can effectively remember or
distinguish without confusion. I could be bargained into more pixels per
symbol, but let's consider this case to start...

H(2^256, 10^100) would still be quite large. Considerably less so than
G(10^100), but I think it might be more fair (for some arbitrary definition of
fair).

Of course, pixels per character can vary wildly, based on display size, and
even on simple displays often contain more than 256 pixels. A more reasonable
assumption for very simple displays might be 32x48. At that large, you're over
10^100, because your options with that much of a canvas have already shot over
400 digits.

You can get many more pixels if you assume your constants are a word, but I
think Rayo's constraints would start counting subscripts and word-like
constants as multiple "symbols".

Then again, we usually don't just flip a single pixel to switch symbols
either, so I'm being too generous there. I'm not sure anyone's done a study on
how many potential single-width characters might be reasonably
distinguishable...

Studies on human perception seem like a much needed prerequisite for further
classification of large numbers!

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pavel_lishin
That seems a little bit like cheating, in that you can calculate the number in
the third turn - e.g., 111! - but as I understand it, to calculate Rayo's
number, you basically have to calculate all the numbers that it's not.

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stouset
If you restrict yourself to calculable numbers, you're capped at the busy
beaver number[1] for the number of rules you're willing to allow in your
Turing machine.

[1]:
[http://www.scottaaronson.com/writings/bignumbers.html](http://www.scottaaronson.com/writings/bignumbers.html)

~~~
zeroer
> And in Go, which has a 19-by-19 board and over 10^150 possible positions,
> even an amateur human can still rout the world’s top-ranked computer
> programs.

History

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cobbzilla
It seems composition would yield ever larger numbers. If Rayo's number is
F(10^100), then wouldn't F(F(10^100)) be even larger?

OK, now repeat that nesting F(10^100) times (maybe use Knuth's up-arrow
notation?) and now you have something even bigger. Now call that whole thing
G, and start over from the top.

There is no end to how many times you can do this.

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psyc
In Googology, these kinds of operations are viewed almost the same as
multiplying by a large constant. The result of everything you just did would
be considered still in the neighborhood of Rayo's number, even if you did it
rn^^^rn times. These people have a whole other notion of what "order of
magnitude" means.

~~~
cobbzilla
I think I get it, the trick is to find a wholly "new dimension" of hugeness.
My solution just continues going bigger in a known direction/dimension.

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cafebabbe
No graham's number ? that one is so big it's scary.

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contravariant
It's tiny compared to the numbers described in the article.

