08/30/2012, 05:24 PM
Well. The reason I ask is because I was structuring my semi operators around the distribution of the set:
\( \mathbb{I}_{y} = \{ s_0 | s_0 \in \mathbb{R}\,\,;\,\,x\,\,\bigtriangleup_{s_0}\,\,y\,\,\in \mathbb{N}} \)
claim that there are operators unique to x and y which allow us to perform operations on elements of \( \mathbb{I}_y \) instead of operations on \( \mathbb{N} \). We then say that \( x\,\,\bigtriangleup_s\,\,y \) is an isomorphism from \( \mathbb{I}_y \to \mathbb{N} \)
Then I found out I only needed to prove the recursive identity for primitive elements of \( \mathbb{I}_y \); (i.e elements that return primes in N); and then do the rest by induction and breaking up the real argument into a product of primitive elements.
However; this all and all sounded plausible but I hit some huge wall. Which is proving the recursive identity for primitive elements; mostly.
I have a new technique now. It may or may not work.
But having more information about how \( x\,\,\bigtriangleup_n\,\,y \) behaves for naturals would really help.
\( \mathbb{I}_{y} = \{ s_0 | s_0 \in \mathbb{R}\,\,;\,\,x\,\,\bigtriangleup_{s_0}\,\,y\,\,\in \mathbb{N}} \)
claim that there are operators unique to x and y which allow us to perform operations on elements of \( \mathbb{I}_y \) instead of operations on \( \mathbb{N} \). We then say that \( x\,\,\bigtriangleup_s\,\,y \) is an isomorphism from \( \mathbb{I}_y \to \mathbb{N} \)
Then I found out I only needed to prove the recursive identity for primitive elements of \( \mathbb{I}_y \); (i.e elements that return primes in N); and then do the rest by induction and breaking up the real argument into a product of primitive elements.
However; this all and all sounded plausible but I hit some huge wall. Which is proving the recursive identity for primitive elements; mostly.
I have a new technique now. It may or may not work.
But having more information about how \( x\,\,\bigtriangleup_n\,\,y \) behaves for naturals would really help.
