#
*“On the shoulders of giant numbers”*

http://www.iteror.org/big/book/ch2/ch2_7.html
big**Ψ**

http://www.iteror.org/big/book/ch2/ch2_7.html

## 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

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 `+`

is the _{0}*initial case*
of a whole family of countable unary operators `+`

culminating in the operatorial _{c}*bigA*.

The start of `+`

as `1`

is familiar,
for there's this given number `a`

and we count one more.

The next step, operations with subscripts
`+`

or _{+}`+`

results in doubles, of doubles, of…_{1}

So
`a`

expresses integers in a
random
binary system._{i}+_{1;}^{.. }.:.
`{a _{i}:1? +_{1;}#i≥0 #n}`

_{+;}= a+

_{1;}= a+

^{...}

`{+#a}`

`= aa = a*11`

a+

_{1;}

^{...}

`{+`= (.a+

_{1;}#b}_{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…

*3*, 2^

*11*, 2^

*2059*, 2^2^

*2070.*, 2^...

*2070.*

`{2^#`}

*n-3*}
The basic formula for left associative operators `+`

has the same footprint._{c}

_{c;}

^{...}

`{+`= a

_{c;}#0}a+

_{c1;}= a+

_{c;}

^{...}

`{+`

_{c;}#a}a+

_{c;}

^{...}

`{+`= (a+

_{c;}#b1}_{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._{-}

_{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+

`= 2`

2+

_{2;}+

_{2;}= 2+

_{1;}+

_{1;}+

_{2;}= 8+

_{2;}= 8+

_{1;}

^{..}

`{+`

_{1;}#8}`= 2^`*11*

v+

_{v;}

^{..}

`{+`~ 2+

_{v;}#v v:2^*11*}_{2049;}

f

_{m1;}= v+

_{v;}

^{..}

`{+`~ 2+

_{v;}#v v:f_{m;}}_{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.

“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!”