Thread Rating:
  • 1 Vote(s) - 5 Average
  • 1
  • 2
  • 3
  • 4
  • 5
[MO] Is there a tetration for infinite cardinalities? (Question in MO)
#9
(12/04/2014, 01:20 PM)tommy1729 Wrote: Doner and Tarski can not define exp^[1/2](w) and its cardinality , right ?
The problem is not if they are able.. probably they weren't but as far as I know the didn't even tried, at least not in their work about the hyperoperations over transfinite ordinals.

The reason is clear, it was about transfinite ordinals not cardinals or fractional iteration... so your problem about the half iterate of the ordinal exponentiation has nothing to do with their work.

When you ask something like that, it should be clear, we have to be precise and define our words.
-What exp means?
-What half iterate means?
-what means?

The answers to this questions involve a amount of non-trivial choiches like wich framework theory we should use, how to formalize the concept of infinity, potential or actual infinity? Cardinals or ordinals? How we define them? Axiom of choiche or not?

This mess of possible choices and interpretation makes your question blurry and ill defined even if I admit that most of the interpretations we can give to the question are really interesting.

Quote:Do Doner and Tarski give a different answer to 2^^w ?
Sure, lets define first the ordinal tetration for limit ordinals


So

About the first fixed point of the function
or in other words the solution of we have that is the first fixed point and is called epsilon zero



The next fixed points can be defined recursively as follows for all the ordinals


for limit ordinals we have


Is possible to see that for the recursive definition we do not iterate the ordinal superexponentiation but we use the iteration of the map so the ordinal pentation is not involved at all with the fixed points.
Quote:I still challenge to give exp^[1/2](w) even under the assumption that ZFC is good.
In my opinion the theory of surreal numbers can help us to extend the iteration of ordinal operations beyond the set theoretic ordinals (that are discrete, integer-like).

But is hard to think about the iteration of the Cardinal operations... really hard.
If I remember good the cardinals number are not even linear orderable without the axiom of choiche and the problem is that for infinite cardinals the cardinal exponentiation is linked more with the combinatorial constructions (like set of maps, powersets) rather than the recursive definitions.

So the key could be the combinatoral meaning of tetration (like the author of the questions seems to suggest).

Thinking about something simple I noteced that the beth numbers (I have to use the mathfrak symbol) are defined using iterated cardinal exponentiation.







the exponentiation is not the ordinal one, note that the cardinal exponentiation push us beyond the countable while the ordinal one (with the epsilon fixed point) can't go beyon the first uncontable

Anyways we can define a kind of tetration that is defined for the cardinal base and for transfinite ordinal superexponents







But about beth omega... that should be I'm not really sure it is also inaccessible... wikipedia says that it is the smallest strong limit cardinal but ZFC doesn't prove it (so I guess a fixed point of the cardinal exponentation) but there is the regularity requirement which I don't fully understand (it has to do with cofinalty but beth omega is the union of countable many smaller ordinal so i guess it is not inaccessible).

PS: I made i big mistake because strong limit doesn't mean that is a cardinal exp. fixed point but that for every smaller cardinal then . The fixed point statement is not compatible with the Cantor theorem.
MathStackExchange account:MphLee
Reply


Messages In This Thread
RE: [MO] Is there a tetration for infinite cardinalities? (Question in MO) - by MphLee - 12/06/2014, 02:26 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Kneser method question tommy1729 9 2,967 02/11/2020, 01:26 AM
Last Post: sheldonison
  A Notation Question (raising the highest value in pow-tower to a different power) Micah 8 5,162 02/18/2019, 10:34 PM
Last Post: Micah
  Math overflow question on fractional exponential iterations sheldonison 4 5,200 04/01/2018, 03:09 AM
Last Post: JmsNxn
  [repost] A nowhere analytic infinite sum for tetration. tommy1729 0 1,660 03/20/2018, 12:16 AM
Last Post: tommy1729
  Sexp redefined ? Exp^[a]( - 00 ). + question ( TPID 19 ??) tommy1729 0 2,031 09/06/2016, 04:23 PM
Last Post: tommy1729
  Remark on Gottfried's "problem with an infinite product" power tower variation tommy1729 4 6,297 05/06/2014, 09:47 PM
Last Post: tommy1729
  Another question! JmsNxn 4 5,205 08/27/2013, 06:57 PM
Last Post: JmsNxn
  Very curious question JmsNxn 3 4,629 08/20/2013, 08:56 PM
Last Post: JmsNxn
  Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... Gottfried 5 8,350 07/17/2013, 09:46 AM
Last Post: Gottfried
  Question about curvature tommy1729 0 2,030 12/15/2012, 11:38 PM
Last Post: tommy1729



Users browsing this thread: 1 Guest(s)