Ψ.2. Up to the Ackermann numbers

chapter 2.7, edit 0.2.9
published: 2010-11-05
updated: 2011-12-30

# 2.7. Unary plus of bigA

Floating on smoke high above a letter A island in the Atlantic After addition a+b is shown to be the void operation ab, we can re-employ the plus sign as a unary operator + acting as a single count on a left operand.
This new +0 is the initial case of a whole family of countable unary operators +c culminating in the operatorial bigA.

The start of + as 1 is familiar, for there's this given number a and we count one more.

a+ = a+0; = a1 = a+1
a+... {+#b} = (.a+).. {(#b#+)}
            = a1... {1#b} = ab

The next step, operations with subscripts ++ or +1 results in doubles, of doubles, of…
So ai+1;.. .:. {ai:1? +1;#i≥0 #n} expresses integers in a random binary system.

a++; = a+1; = a+... {+#a} = aa = a*11
a+1;... {+1;#b} = (.a+1;).. {(#b#+1;)} = a*11**b

Now if we'd initially take 1 for the two variables a,b and feed back the result in the above formula, and repeat this feedback for n times, then the results are approximately…

{ 2, 2^3, 2^11, 2^2059, 2^2^2070., 2^...2070. {2^#n-3}}

The basic formula for left associative operators +c has the same footprint.

a+c;... {+c;#0} = a
a+c1; = a+c;... {+c;#a}
a+c;... {+c;#b1} = (a+c;)+c;... {+c;#b}

In a bigA context with subscripts the inverse plus - is a strange beast. Entities +- do not exist (or perhaps inexist) as/if individual numbers exist. This might not lead to contradiction when we assume no number is ever finished.

a+0; = a+-;... {+-;#a} => +-;... {#n≥0} = + = 1
                     & +-;... {#0} = 1 <= +-;+-; = +-; = 0

Feedback of results into the three variables a,b,c delivers the biggest numbers of this chapter.
Start from 1 and take a few steps, then try to fathom the magnitude of that number – these are quickly getting Bigger than those in our previous feedback of only two variables.

1+1; = 1+ = 2
2+2;+2; = 2+1;+1;+2; = 8+2; = 8+1;.. {+1;#8} = 2^11
v+v;.. {+v;#v v:2^11} ~ 2+2049;
fm1; = v+v;.. {+v;#v v:fm;} ~ 2+fm; ~ 2+..2049.; {2+#m--#;}

The feedback formula for 3 parameters can be estimated by a downward progression of pluses. The nested subscript depth delimiter ; is displayed here, which is usually hidden from view.
More on subscripted pluses in chapter 3. The main story line continues with superstar operators in chapter 2.

# Addictive counting

There's more to counting than meets the eye. In the real world for instance, it's not obvious when we've counted all the objects we have – people become tired, quit, come back, find what's lost under the table…
Maybe later more items can be added. Counting is a process, it never stops (but doesn't actually reach infinity). We'll use the name addictive for all operations that have no official end (Latin addictio = assignment).

a ≤ a+... {+#n n≥0}

Therefore a philosophically inclined mathematician would consider all natural numbers to be lower bounds. And he would raise our arithmetical system up from the practical experience that counting is addictive.

Our ability to keep book and the inspiration to get the work done are often more important than the figures, which are only assumed to be exact at an approximate moment in time. When time rolls on (and time in principle can be rolled back), what used to be known as a fact may miraculously reappear as a mirage in this fantastic desert of Heisenberg uncertainty that is our universe.

Workers rolling up the sea like a carpet, whipped by horsemen

“But what do these people do?
 They are rolling up the sea!”
– Philémon

“One two! One two! One two! One two! One two!”
“Roll on!  Show some spirit, men!”