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 Pentation's definitional ambiguity bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 10/24/2009, 09:27 AM (This post was last modified: 10/24/2009, 09:47 AM by bo198214.) (10/22/2009, 09:45 PM)Base-Acid Tetration Wrote: Then I would think the number of branches is still countable, though i doubt it because a[4]b 's cardinality is that of the continuum. Potentially there uncountably many branches. Branches are usually determined by the non-homotopic curves that lead to a point. E.g. take the logarithm. There is one branching point at 0. Paths are homotopic iff they wind the same number around 0. log has a branch for each winding number which is countably many. Now consider a function with two branch points. For each branch point we have a winding number and so we have $\mathbb{Z}\times\mathbb{Z}$ branches which is still countable. For $n$ branch points we have $\mathbb{Z}^n$ branches. But for infinitely many branch points, which is the case for tetrations, they have a branch point at every integer $\le 2$, we have $\mathbb{Z}^\omega$ many branches. And we know that already $2^\omega$ is uncountable. On the other hand this is only under the assumption that branches dont coincide. E.g. consider $\sqrt{z}$. It has a branch point at 0. But only two branches. But even if we assume that each branch point gives only two branches we still have $2^\omega$, i.e. uncountably many branches. If however more branches coincide it can lessen the number of different branches. !!!EDIT!!!: After reading mike3's posting I see what is the error in the above reasoning: A path with finite length can not wind around infinitely many branchpoints of tetration. It always needs to choose a finite number out the infinitely many branchpoints to wind around. And thatswhy the number of branches must be countable. « Next Oldest | Next Newest »

 Messages In This Thread Pentation's definitional ambiguity - by Base-Acid Tetration - 10/17/2009, 09:54 PM RE: Pentation's definitional ambiguity - by andydude - 10/22/2009, 07:54 AM RE: Pentation's definitional ambiguity - by Base-Acid Tetration - 10/22/2009, 11:40 AM RE: Pentation's definitional ambiguity - by andydude - 10/22/2009, 08:27 PM RE: Pentation's definitional ambiguity - by Base-Acid Tetration - 10/22/2009, 09:45 PM Number of branches of tetration - by bo198214 - 10/24/2009, 09:27 AM RE: Number of branches of tetration - by Base-Acid Tetration - 10/24/2009, 01:02 PM RE: Number of branches of tetration - by bo198214 - 10/24/2009, 01:20 PM RE: Number of branches of tetration - by Base-Acid Tetration - 10/24/2009, 01:23 PM RE: Number of branches of tetration - by bo198214 - 10/24/2009, 02:01 PM RE: Pentation's definitional ambiguity - by Base-Acid Tetration - 12/15/2009, 08:52 PM RE: Pentation's definitional ambiguity - by bo198214 - 12/15/2009, 10:56 PM RE: Pentation's definitional ambiguity - by Base-Acid Tetration - 12/15/2009, 11:08 PM RE: Pentation's definitional ambiguity - by bo198214 - 12/15/2009, 11:15 PM RE: Pentation's definitional ambiguity - by Base-Acid Tetration - 12/15/2009, 11:23 PM

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