This is a rather disturbing flaw that I've come across with zeration that rather surprises me the authors didn't consider.
By the Ackermann function:
\( \text{(I)}\,\,a\,\,\bigtriangleup_\sigma\,\,(a\,\,\bigtriangleup_{\sigma +1 }\,\,b) = a\,\,\bigtriangleup_{\sigma + 1}\,\,b+1 \)
which is the core of how the function is defined.
I'm gonna put it right out there that zeration DOES NOT satisfy this property.
by definition:
\( a\,\,\circ\,\,b = max(a,b) + 1 \)
or
\( a\,\,\circ\,\,b = a + 2 = b + 2\,\,\text{if a = b} \)
Take the obvious example, a > 0:
\( a\,\,\circ\,\,(a + (-1)) = max(a, a-1) + 1 = a + 1 \)
however, by Ackermann's law (I):
\( a\,\,\circ\,\,(a + (-1)) = a + (-1) + 1 = a \)
Did anybody else notice this? How could they over look such a fatal flaw? I mean it took me fifteen minutes of research to see this?
It can actually be extended generally to all negative numbers
\( k \in \mathbb{R}\,\,;\,\,k<0 \)
\( a\,\,\circ\,\,(a - k) = a + 1 \)
however by (I) we know it must be
\( a\,\,\circ\,\,(a - k) = a - k + 1 \)
Clearly Zeration is not the operator below addition in the Ackermann function.
By the Ackermann function:
\( \text{(I)}\,\,a\,\,\bigtriangleup_\sigma\,\,(a\,\,\bigtriangleup_{\sigma +1 }\,\,b) = a\,\,\bigtriangleup_{\sigma + 1}\,\,b+1 \)
which is the core of how the function is defined.
I'm gonna put it right out there that zeration DOES NOT satisfy this property.
by definition:
\( a\,\,\circ\,\,b = max(a,b) + 1 \)
or
\( a\,\,\circ\,\,b = a + 2 = b + 2\,\,\text{if a = b} \)
Take the obvious example, a > 0:
\( a\,\,\circ\,\,(a + (-1)) = max(a, a-1) + 1 = a + 1 \)
however, by Ackermann's law (I):
\( a\,\,\circ\,\,(a + (-1)) = a + (-1) + 1 = a \)
Did anybody else notice this? How could they over look such a fatal flaw? I mean it took me fifteen minutes of research to see this?
It can actually be extended generally to all negative numbers
\( k \in \mathbb{R}\,\,;\,\,k<0 \)
\( a\,\,\circ\,\,(a - k) = a + 1 \)
however by (I) we know it must be
\( a\,\,\circ\,\,(a - k) = a - k + 1 \)
Clearly Zeration is not the operator below addition in the Ackermann function.