Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Cardinality of Infinite tetration
#1
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
Reply


Messages In This Thread
Cardinality of Infinite tetration - by Ivars - 06/17/2008, 01:02 PM
RE: Cardinality of Infinite tetration - by Ivars - 06/17/2008, 07:59 PM
RE: Cardinality of Infinite tetration - by Ivars - 06/18/2008, 08:12 AM
RE: Cardinality of Infinite tetration - by Ivars - 06/20/2008, 04:28 PM
RE: Cardinality of Infinite tetration - by Ivars - 06/21/2008, 07:27 PM
RE: Cardinality of Infinite tetration - by Ivars - 07/13/2008, 08:07 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  [repost] A nowhere analytic infinite sum for tetration. tommy1729 0 1,426 03/20/2018, 12:16 AM
Last Post: tommy1729
  [MO] Is there a tetration for infinite cardinalities? (Question in MO) Gottfried 10 13,178 12/28/2014, 10:22 PM
Last Post: MphLee
  Remark on Gottfried's "problem with an infinite product" power tower variation tommy1729 4 5,630 05/06/2014, 09:47 PM
Last Post: tommy1729
  Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... Gottfried 5 7,526 07/17/2013, 09:46 AM
Last Post: Gottfried
  Wonderful new form of infinite series; easy solve tetration JmsNxn 1 4,669 09/06/2012, 02:01 AM
Last Post: JmsNxn
  the infinite operator, is there any research into this? JmsNxn 2 5,849 07/15/2011, 02:23 AM
Last Post: JmsNxn
  Infinite tetration of the imaginary unit GFR 40 59,444 06/26/2011, 08:06 AM
Last Post: bo198214
  Infinite Pentation (and x-srt-x) andydude 20 28,129 05/31/2011, 10:29 PM
Last Post: bo198214
  Infinite tetration fractal pictures bo198214 15 24,707 07/02/2010, 07:22 AM
Last Post: bo198214
  Infinite towers & solutions to Lambert W-function brangelito 1 4,089 06/16/2010, 02:50 PM
Last Post: bo198214



Users browsing this thread: 1 Guest(s)