User blog comment:Simplicityaboveall/Insanely Fast-Growing Functions/@comment-25912386-20171103024156/@comment-30754445-20171103152150

Joe, I am considering your work seriously.

My analysis was based directly on the definitions that you've given for your functions. These definitions are crystal clear and unambigious, which is precisely how I could calculate the value of your numbers. And yes, I'm fully aware of the nestings-within-nestings that is going on in your functions.

The thing is, ordinal-based nesting is far deeper and much more insane than anything you've done here.

Let's try to look at the situation from a different angle. I think the whole ordinal thing is just confusing the issue here, so let me introduce a simple way to write and compare numbers of this size. Here are the rules:

(0,0)|n = 10n

(a,b+1)|n = (a,b)|(a,b)|(a,b)|...|(a,b)|1 with n (a,b)'s

(a+1,0)|n = (a,n)|10

Let me also add a few shorthands to facilitate writing these numbers:

Instead of (1,1)|n we will write Kn

Instead of (2,0)|n we will write Mn.

Simple, right? And also: no ordinals. It's just a general way to write numbers, similar to scientific notation.

So here is a question for you: Which number do you think is bigger? Your H(22) or fω×2+1(2)?

I'm claiming that fω×2+1(2) is much much larger. And armed with the above notation, you can find out for yourself that it is true. Indeed, it can be shown that:

(i) H(22) = (2,0)|12 = M12 (exactly!)

(ii) fω×2+1(2) ~ (2,0)|(1,1)|(10↑↑↑↑↑↑↑8) =M[K(10↑↑↑↑↑↑↑8)]

And since K(10↑↑↑↑↑↑↑8) is much larger than 12, it is clear which of the two numbers is bigger.

Now, if you think I'm wrong with this statement, then show me my error. Is equation (i) wrong? Is equation (ii) wrong? We're now talking about actual specific numbers rather than some vague ordinal progressions, so perhaps we're going to make some progress now.