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July 12

Two questions regarding extremely large numbers

Is the small Veblen ordinal a number, and (at least in principle) quantifiable? I notice that ordinal numbers are not mentioned in Orders of magnitude (numbers), nor are cardinal numbers.
One could define a function $f(m,n)$ so that $f(m,1)=G(m,m,m)$ (using Goodstein's notation of the hyperoperation function), and the parameter n is the number of times the function is nested, so that $f(m,2)=G(G(m,m,m),G(m,m,m),G(m,m,m))$ . Of course, this function grows ridiculously fast and is a nightmare of recursion. This is evidently in the spirit of Black Hat. Has a function like this been suggested before? --Florian Blaschke (talk) 11:10, 12 July 2017 (UTC)[reply]

Ordinals are not the sort of numbers you assign an order of magnitude to. Rather they generalise the notion of a natural number by counting up to and beyond infinity. So you would start the ordinal numbers by counting {0, 1, 2, ...}, and each ordinal corresponds to the set of all ordinals below it, so that 3 for example corresponds to {0, 1, 2} for obvious reasons. This works because all these sets admit a rather obvious ordering.

The "infinite" ordinals then start by defining ω as corresponding to {0, 1, 2, ...} going on forever. Then you can keep going and adding the usual order definitions, and start with ω + 1 {0, 1, 2, ..., ω}, and then later ω * 2 {0, 1, 2, ..., ω, ω + 1, ...}, ω^2, ω^ω, ω^ω^ω, and so on. Then you get to a fixed point (as you always will going up this ladder), such that ω^x = x, and then we define x = ε₀ and keep going. The small Veblen ordinal comes long after all of these more obvious ones, of course. If you think these are numbers, then yes, the small Veblen ordinal is one, but it is not really quantifiable as it isn't even finite. (The cardinals are similar, but count one-to-one correspondences without regard to order.) Double sharp (talk) 11:38, 12 July 2017 (UTC)[reply]

Ah, I understand a little better now (I have read the introductions at Ordinal number and Cardinal number and it helped a lot, but I was still not completely sure regarding the quantifiability issue). Ordinals and cardinals are transfinite (quasi-infinite) numbers, right? Ah, Infinity § Set theory treats this. The background for my question was a remark I had encountered that said that Graham's number is still modest in comparison to monsters such as the small (!) Veblen ordinal, but the remark was evidently misleading. --Florian Blaschke (talk) 12:32, 12 July 2017 (UTC)[reply]

"Transfinite" and "infinite" mean basically the same thing, the distinction is mostly historical.

Most ordinal and cardinal numbers are infinite, but of course some are finite as well (the natural numbers).

A comparison between a finite and an infinite number isn't meaningful, but there are also some very big finite numbers. Graham's number is much, much bigger than everyday numbers, yet much, much smaller than some other big numbers.

See for example Conway chained arrow notation, which can be used for numbers much larger than Graham's. I think it can also be used for a much faster growing function than the one you've described.

There's also the Fast-growing hierarchy, which actually makes use of infinite ordinal numbers to specify very large finite natural numbers. If I'm not mistaken, they can be used for much faster-growing function than is made possible by even chained arrow notation. -- Meni Rosenfeld (talk) 12:42, 12 July 2017 (UTC)[reply]

Thanks, I'd read about Conway chains before but didn't think of them. My understanding of mathematics is too limited to completely understand even them, at least without a lot of work tracing the description and examples – and the fast-growing hierarchy uses a lot of advanced concepts where I'm clearly out of my depth ... --Florian Blaschke (talk) 13:20, 12 July 2017 (UTC)[reply]

Since you've looked at his notation, have you looked at Goodstein's theorem? it is an application of the fact a function grows extremely rapidly. Dmcq (talk) 17:46, 12 July 2017 (UTC)[reply]

I have now, thank you! Very interesting! --Florian Blaschke (talk) 00:28, 18 July 2017 (UTC)[reply]

All numbers are small because most numbers are bigger. Bo Jacoby (talk) 18:30, 12 July 2017 (UTC).[reply]

"Most" is a question of measure. For some measures it is true that all numbers are smaller than most. For others, not. For example, if you define a measure on natural numbers based on how frequently the number was used in real-world applications, then by this measure, Graham's number is decidedly bigger than most numbers. -- Meni Rosenfeld (talk) 13:32, 13 July 2017 (UTC)[reply]

Triangles in 3D space

Are there any theorems relating a reference triangle to a point, line, circle, or triangle (or anything else) not in the plane of the reference triangle? Loraof (talk) 22:26, 12 July 2017 (UTC)[reply]

There's a number in projective geometry, you might like Desargues's theorem. Dmcq (talk) 09:21, 13 July 2017 (UTC)[reply]

Thanks. I'm looking for something like "a specific function of the distances from an arbitrary point P in R³ to the vertices/sides/centroid/whatever is independent of the location of point P. Loraof (talk) 16:21, 13 July 2017 (UTC)[reply]