Probably has been answered many times,but I have to ask:
If cardinality of Natural numbers is Aleph_0, and their powerset P(N) has cardinality of continuum, |P(N)|= c = 2^Aleph_0 which is also the cardinality of R, then:
x^x has cardinality of R^R = c^c= cardinality of power set of Reals=|P(R )|=2^c=2^2^Aleph_0 then for each step of tetration we have to add 2, so via recursion:
Cardinality
|x[4]1|=|P(N)|= |R|=2^Aleph_0
|x[4]n|= |P(x[4]n-1|= 2^| x[4]n-1| and generally:
|x[4]n|= |P(x[4]n-1|= (2[4]n)^| x[4]1|
What would be cardinality for infinite tetration , known to converge to real values or give complex values by analytic continuation? And what is the cardinality of a number obtained as a result of tetration, even for finite n?
Some references here:
Cardinality of continuum
Another question is , what is than the cardinality of:
x[4]y, x[4]I? From previous considerations, can it be (for y>1):
|x[4]y|= |P(x[4]y-1|=2^?
What is cardinality of I^I?
And what would such cardinalities mean?
Similarly the same question arises when trying to understand what is fractional or real Cartesian product of sets? May be the answer is in tetration.
Ivars
If cardinality of Natural numbers is Aleph_0, and their powerset P(N) has cardinality of continuum, |P(N)|= c = 2^Aleph_0 which is also the cardinality of R, then:
x^x has cardinality of R^R = c^c= cardinality of power set of Reals=|P(R )|=2^c=2^2^Aleph_0 then for each step of tetration we have to add 2, so via recursion:
Cardinality
|x[4]1|=|P(N)|= |R|=2^Aleph_0
|x[4]n|= |P(x[4]n-1|= 2^| x[4]n-1| and generally:
|x[4]n|= |P(x[4]n-1|= (2[4]n)^| x[4]1|
What would be cardinality for infinite tetration , known to converge to real values or give complex values by analytic continuation? And what is the cardinality of a number obtained as a result of tetration, even for finite n?
Some references here:
Cardinality of continuum
Another question is , what is than the cardinality of:
x[4]y, x[4]I? From previous considerations, can it be (for y>1):
|x[4]y|= |P(x[4]y-1|=2^?
What is cardinality of I^I?
And what would such cardinalities mean?
Similarly the same question arises when trying to understand what is fractional or real Cartesian product of sets? May be the answer is in tetration.
Ivars