I will try to ask this question more correctly as it bothers me:
If we apply Mother Law of hyperoperations:
\( a[n-1](a[n]b) = a[n](b+1) \),
to infinite application of operations with \( a=I^{1/I}= {1/I}^I = e^{\pi/2} \) than:
\( I^{1/I}+I^{1/I}*{\infty} = I^{1/I}*(\infty+1) \)
\( I^{1/I}*(I^{1/I})^{\infty} = (I^{1/I})^{(\infty+1)} \)
Tetration is the first operation where something qualitatively new happens- I is returned as a result of infinite application of operation:
\( (I^{1/I})^{((I^{1/I})[4]\infty)} =I=(I^{1/I})[4]{(\infty+1)} \)
as \( (I^{1/I})^I=I=(I^{(1/I)})[4](\infty+1) \)
This allows(?) to apply formula recursively, so that by denoting \( \infty'=\infty+1 \):
\( (I^{1/I})^{(I^{1/I})[4]\infty')} = I=(I^{1/I})[4]{(\infty'+1)} \)
If this is continued infinitely, then in the end ( I have elaborated it here :Application of infinite hyperoperations to I^(1/I) .
\( (I^{1/I})^{((I^{1/I})[4](\infty[\infty]\infty-1))} =I= (I^{1/I})[4](\infty[\infty]\infty) \)
I wonder is this true as it got quite complex in the end.
If so, \( I \) is very strong attractor for infinite tetration of \( e^{\pi/2} \), as any type of infinity via operations above tetration returns I.
How I understand it, if we reverse the operations, any smallest, infinitesimal deviation from \( I \) will return a result infinitely away from \( e^{\pi/2} \).
If we had looked only at :
\( ((I^{1/I})[4]\infty)=I \)
we would not noticed how extremely, \( (\infty[\infty]\infty) \) sensitive is infinite tetration to deviations from \( I \) or \( e^{\pi/2} \) as input and output.
I my opinion, that allows to consider or define infinitesimal deviations from \( I -> dI \) and \( e^{\pi/2} -> d( e^{\pi/2}) \) however strange it may sound and look.
The basic property of such infinitesimal deviation would be divergence from \( I \) and \( e^{\pi/2} \) if all \( (\infty[\infty]\infty) \) operations are applied.
Howerever, as seen from mother law, if \( (\infty[\infty]\infty-1) \) is applied to the tetration of \( e^{\pi/2} \) than it will still converge to I if the deviation from \( e^{\pi/2} \) is the smallest possible special infinitesimal \( d(e^{\pi/2}) \) , however, it will diverge if the deviation is still infinitesimal ( as somehting infinitely small) , but any other (bigger than?) EXCEPT the smallest one.
Excuse me for strange ideas, but I believe both infinitesimals and infinities are scaled and has structure, and very exactly so.
I would love to hear if such ideas connect to some existing mathematical constructs.
Tetration seems the right operation, fast enough to start to discuss such ideas as structures and scales of infinities/infinitesimals.
Of course, it seems one can use any number of selfroot type in these formulas, \( a= x^{1/x} \) and that is true, except that \( I \) and \( I^{1/I} = {(1/I)}^I=e^{\pi/2} \) seems to be the only basic combination involving purely imaginary number \( I \) and purely real number \( e^{\pi/2} \).
Ivars
If we apply Mother Law of hyperoperations:
\( a[n-1](a[n]b) = a[n](b+1) \),
to infinite application of operations with \( a=I^{1/I}= {1/I}^I = e^{\pi/2} \) than:
\( I^{1/I}+I^{1/I}*{\infty} = I^{1/I}*(\infty+1) \)
\( I^{1/I}*(I^{1/I})^{\infty} = (I^{1/I})^{(\infty+1)} \)
Tetration is the first operation where something qualitatively new happens- I is returned as a result of infinite application of operation:
\( (I^{1/I})^{((I^{1/I})[4]\infty)} =I=(I^{1/I})[4]{(\infty+1)} \)
as \( (I^{1/I})^I=I=(I^{(1/I)})[4](\infty+1) \)
This allows(?) to apply formula recursively, so that by denoting \( \infty'=\infty+1 \):
\( (I^{1/I})^{(I^{1/I})[4]\infty')} = I=(I^{1/I})[4]{(\infty'+1)} \)
If this is continued infinitely, then in the end ( I have elaborated it here :Application of infinite hyperoperations to I^(1/I) .
\( (I^{1/I})^{((I^{1/I})[4](\infty[\infty]\infty-1))} =I= (I^{1/I})[4](\infty[\infty]\infty) \)
I wonder is this true as it got quite complex in the end.
If so, \( I \) is very strong attractor for infinite tetration of \( e^{\pi/2} \), as any type of infinity via operations above tetration returns I.
How I understand it, if we reverse the operations, any smallest, infinitesimal deviation from \( I \) will return a result infinitely away from \( e^{\pi/2} \).
If we had looked only at :
\( ((I^{1/I})[4]\infty)=I \)
we would not noticed how extremely, \( (\infty[\infty]\infty) \) sensitive is infinite tetration to deviations from \( I \) or \( e^{\pi/2} \) as input and output.
I my opinion, that allows to consider or define infinitesimal deviations from \( I -> dI \) and \( e^{\pi/2} -> d( e^{\pi/2}) \) however strange it may sound and look.
The basic property of such infinitesimal deviation would be divergence from \( I \) and \( e^{\pi/2} \) if all \( (\infty[\infty]\infty) \) operations are applied.
Howerever, as seen from mother law, if \( (\infty[\infty]\infty-1) \) is applied to the tetration of \( e^{\pi/2} \) than it will still converge to I if the deviation from \( e^{\pi/2} \) is the smallest possible special infinitesimal \( d(e^{\pi/2}) \) , however, it will diverge if the deviation is still infinitesimal ( as somehting infinitely small) , but any other (bigger than?) EXCEPT the smallest one.
Excuse me for strange ideas, but I believe both infinitesimals and infinities are scaled and has structure, and very exactly so.
I would love to hear if such ideas connect to some existing mathematical constructs.
Tetration seems the right operation, fast enough to start to discuss such ideas as structures and scales of infinities/infinitesimals.
Of course, it seems one can use any number of selfroot type in these formulas, \( a= x^{1/x} \) and that is true, except that \( I \) and \( I^{1/I} = {(1/I)}^I=e^{\pi/2} \) seems to be the only basic combination involving purely imaginary number \( I \) and purely real number \( e^{\pi/2} \).
Ivars
, I didn' mean that!) I think that Henryk is absolutely right. The entities that you mentioned are indeed constant. Where did you find the idea that they are only "symbols"? They are