I meant to say that
\( \forall x,y > 2\,\,\forall s \in \mathbb{R} \,\,s\ge 0 \)
\( x\,\bigtriangleup_{s_1}\,y = x\,\bigtriangleup_{s_2}\,y\,\,\,\,\Leftrightarrow\,\,\,s_1 = s_2 \)
\( \forall n \in \mathbb{N}\,\,n \ge x+y\,\,\,\,\exists s_0\,\,x\,\,\bigtriangleup_{s_0}\,\,y = n \)
Now the distribution of the set \( \mathbb{I}_{x,y} \) (which depends on x and y) is pivotal insofar as, if we extend hyper operators for all natural arguments and real operators they have to obey the recursion on these points and these points alone. The recursive Identity is written:
\( x\,\,\bigtriangleup_{s_0-1}\,\,(x\,\,\bigtriangleup_{s_0}\,\,y) = x\,\,\bigtriangleup_{s_0}\,\,(y+1) \)
And we need to only worry when \( x\,\,\bigtriangleup_{s_0}\,\,y \) is an integer, which is when \( s_0 \in \mathbb{I}_{x,y} \)
I was trying to think of ways to talk only about recursion on this discrete set rather than the whole plane. What I was thinking, which I am no longer is that there maybe binary operations on this set \( \mathbb{I}_{x,y} \) which allows us to talk about the operators more freely. I haven't really had much luck in uncovering much of anything.
By isomorphism I meant one to one and onto insofar as \( x\,\bigtriangleup_s\,y \) obeys the rules I laid out at the beginning.
\( \forall x,y > 2\,\,\forall s \in \mathbb{R} \,\,s\ge 0 \)
\( x\,\bigtriangleup_{s_1}\,y = x\,\bigtriangleup_{s_2}\,y\,\,\,\,\Leftrightarrow\,\,\,s_1 = s_2 \)
\( \forall n \in \mathbb{N}\,\,n \ge x+y\,\,\,\,\exists s_0\,\,x\,\,\bigtriangleup_{s_0}\,\,y = n \)
Now the distribution of the set \( \mathbb{I}_{x,y} \) (which depends on x and y) is pivotal insofar as, if we extend hyper operators for all natural arguments and real operators they have to obey the recursion on these points and these points alone. The recursive Identity is written:
\( x\,\,\bigtriangleup_{s_0-1}\,\,(x\,\,\bigtriangleup_{s_0}\,\,y) = x\,\,\bigtriangleup_{s_0}\,\,(y+1) \)
And we need to only worry when \( x\,\,\bigtriangleup_{s_0}\,\,y \) is an integer, which is when \( s_0 \in \mathbb{I}_{x,y} \)
I was trying to think of ways to talk only about recursion on this discrete set rather than the whole plane. What I was thinking, which I am no longer is that there maybe binary operations on this set \( \mathbb{I}_{x,y} \) which allows us to talk about the operators more freely. I haven't really had much luck in uncovering much of anything.
By isomorphism I meant one to one and onto insofar as \( x\,\bigtriangleup_s\,y \) obeys the rules I laid out at the beginning.
