07/19/2022, 09:48 AM
I feel very stupid... I'm breaking my brain into it... I'm realizing that I have some weak point in truly understanding that implicit function thing... I believed it was 80% clear to me but it's not.
Let \(f_\phi(s,y):=x[s]_\phi y\) for some fixed suitable \(x\). Starting from the assumption of the existence of a function \(\varphi(s,y)=\phi\) that satisfies
\[\alpha_{\phi}(s+1,f_\phi(s,y))= \alpha_\phi(s+1,y) + 1\]
how you derive, step-by-step, the f.eq.
\[\varphi(s,f_{\varphi(s,y)}(s,y)) = \varphi(s,y)\\\]
Pls, can you explain it very slowly, step by step, is if you were an algorithm, or as if I am an high school student that is learning how to solve 2nd degree polynomial equations.
Because if I start from the assumption I derive the following
\[\alpha_{\varphi(s+1,f_{\varphi(s,y)}(s,y))}(s+1,f_{\varphi(s,y)}(s,y) )=\alpha_{\varphi(s+1,y)}(s+1,y)+1\]
but from here I'm lost...
About finding creative ways... I'm not sure I can help you but when you said you wanted to switch to the Abel presentation... I was thinking you would consider what I can inverse goodstein equation, something that give rise not to an hyperoperations sequence but to an hyperlogarithms sequence.
\[a_0\circ a_{s+1}=a_{s+1}\circ a_s\]
Assume the seed is decrementation by one, assume \(\alpha_{\varphi,s}\) is the bi-indexed family of abel functions you defined.
Why don't we just ask for parameters \(\varphi_i\) for \(i=0,1,2\) that solve
\[S^{-1}\circ \alpha_{\varphi_0,s+1}=\alpha_{\varphi_1,s+1}\circ \alpha_{\varphi_2,s}\]
i.e. a function \(\varphi(s)\) that solves
\[\forall y.\,\, \alpha_{\varphi(s+1),s+1}(y)-1=\alpha_{\varphi(s+1),s+1}( \alpha_{\varphi(s),s}(y))\]
this will take care of all the \(y\) at the same time... even if probably this means that such triples can not exist, i.e. such triples exist locally only for some \(y\in U\), but we should prove this.
This probably speaks for my ignorance of implicit functions... so let's stick to your story of the parameter depending on two variables... why don't you consider the fully abel-like presentation?
\[\alpha_{\varphi_0}(s+1,y)-1=\alpha_{\varphi_1}(s+1, \alpha_{\varphi_2}(s,y))\]
I feel that to go deep inside this matter we need to develop further the theory of goodstein maps... we need to compare and manipulate them as we do with superfunctions... maybe like introducing the analogous of theta mapping but for measuring the difference between solutions to goodstein f.eq.
Your discussion about normal families should be part of it but I'm slow and I still have to absorb the idea.
Let \(f_\phi(s,y):=x[s]_\phi y\) for some fixed suitable \(x\). Starting from the assumption of the existence of a function \(\varphi(s,y)=\phi\) that satisfies
\[\alpha_{\phi}(s+1,f_\phi(s,y))= \alpha_\phi(s+1,y) + 1\]
how you derive, step-by-step, the f.eq.
\[\varphi(s,f_{\varphi(s,y)}(s,y)) = \varphi(s,y)\\\]
Pls, can you explain it very slowly, step by step, is if you were an algorithm, or as if I am an high school student that is learning how to solve 2nd degree polynomial equations.
Because if I start from the assumption I derive the following
\[\alpha_{\varphi(s+1,f_{\varphi(s,y)}(s,y))}(s+1,f_{\varphi(s,y)}(s,y) )=\alpha_{\varphi(s+1,y)}(s+1,y)+1\]
but from here I'm lost...
About finding creative ways... I'm not sure I can help you but when you said you wanted to switch to the Abel presentation... I was thinking you would consider what I can inverse goodstein equation, something that give rise not to an hyperoperations sequence but to an hyperlogarithms sequence.
\[a_0\circ a_{s+1}=a_{s+1}\circ a_s\]
Assume the seed is decrementation by one, assume \(\alpha_{\varphi,s}\) is the bi-indexed family of abel functions you defined.
Why don't we just ask for parameters \(\varphi_i\) for \(i=0,1,2\) that solve
\[S^{-1}\circ \alpha_{\varphi_0,s+1}=\alpha_{\varphi_1,s+1}\circ \alpha_{\varphi_2,s}\]
i.e. a function \(\varphi(s)\) that solves
\[\forall y.\,\, \alpha_{\varphi(s+1),s+1}(y)-1=\alpha_{\varphi(s+1),s+1}( \alpha_{\varphi(s),s}(y))\]
this will take care of all the \(y\) at the same time... even if probably this means that such triples can not exist, i.e. such triples exist locally only for some \(y\in U\), but we should prove this.
This probably speaks for my ignorance of implicit functions... so let's stick to your story of the parameter depending on two variables... why don't you consider the fully abel-like presentation?
\[\alpha_{\varphi_0}(s+1,y)-1=\alpha_{\varphi_1}(s+1, \alpha_{\varphi_2}(s,y))\]
I feel that to go deep inside this matter we need to develop further the theory of goodstein maps... we need to compare and manipulate them as we do with superfunctions... maybe like introducing the analogous of theta mapping but for measuring the difference between solutions to goodstein f.eq.
Your discussion about normal families should be part of it but I'm slow and I still have to absorb the idea.
Mother Law \(\sigma^+\circ 0=\sigma \circ \sigma^+ \)
\({\rm Grp}_{\rm pt} ({\rm RK}J,G)\cong \mathbb N{\rm Set}_{\rm pt} (J, \Sigma^G)\)