03/23/2017, 01:27 PM

(03/12/2017, 03:08 PM)Xorter Wrote: I am interested in iterates (and uniterates) of logical operators, like nand, xor ... etc. .

Let's investigate the nand op:

x↑y = ¬(x&y)

Iterating x O y means this: x O x O ... O x (y-times)

Thus

x ↑ x ↑ ... ↑ x := x ↑↑ y

x ↑↑ 1 = x

x ↑↑ 2 = x ↑ x = ¬x

x ↑↑ 3 = x ↑ ¬x = ¬(x & ¬x) = x v ¬x

x ↑↑ 4 = x ↑ (x v ¬x) = ¬(x & (x v ¬x)) = x & ¬x

x ↑↑ 5 = x ↑ (x & ¬x) = ¬(x & (x & ¬x)) = x v ¬x

x ↑↑ 6 = x & ¬x

x ↑↑ 7 = x v ¬x

... etc.

So

x ↑↑ 2k = x & ¬x

x ↑↑ 2k-1 = x v ¬x

where k is bigger integer than 2

My question: Can k be any real or complex number?

(In my view to do this, we should know what "between" & and v is.)

---

I like the basic idea to do dynamics in set theory , Logic and the alike.

However i see many issues.

First you write alot " x and not x " and "x or not x" and they are ( in boolean ) either trivial or paradoxical !

Like " this sentense is false ".

or " this is true and false " , " this is true or false.

Secondly

suppose we let times -1 mean not.

Than iterations of not give the unit circle in the complex plane.

but what does it mean " i " ? What does it mean to half-iterate NOT ?

0r the pi th iteration of x OR y ???

you only have a few things like and or not true etc.

but you want continue iterations ??

or if you introduce new things like " i " above you need to Well define it !!

so as of now im very skeptical for continue iterations.

as for integer iterates that might work.

i think one then needs to associate it with groups or rings.

maybe modular arithmetic too.

Regards

tommy1729

the master