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 [MO] Is there a tetration for infinite cardinalities? (Question in MO) tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 12/07/2014, 05:32 PM What I wrote is in perfect harmony with sup and beth numbers. Maybe I should apologize for my tricky question. Tricky because trying to answer might be a trap. Long ago , I was challenged on sci.math to answer my own question. There cannot be a bijection from exp^[1/2](w) to w nor 2^w. Here is why : Let f(x) be exp^[1/2](x) + o(1). If card f(w) = card w then card f(f((w)) = card f( card f(w) ) = card f(w) = card(w) But f(f(w)) = 2^w Contradiction If card f(w) = card 2^w then card f(f(w)) = card f ( card f(w) ) = card f ( 2^w ) = card(2^(2^w)) But f(f(w)) = 2^w Contradiction Likewise for exp^[a] with noninteger a. SO exp^[1/2] does not exist in set theory. SO set theory is not suitable for functions since only the power function 2^ is a function that exists and makes a "difference". ( Polynomials do not make a "difference" , they do not change cardinality ) Considering that infinite set theory is not suitable for functions , it makes more sense to be skeptical about its use in other branches of math such as number theory , algebra , calculus and dynamical systems. Also notice that even substraction is not well defined (for ordinals) since w-1 does not even exist. That should shed some light on my skeptisism. ( it is JUST the tip of the iceberg , there is way more reason to be skeptical ) Since half-iterates (functions) do usually not exist in set theory , I do not see how one can continue to combine set theory and dynamics. regards tommy1729 " Truth is what does not go away when you stop believing in it " « Next Oldest | Next Newest »

 Messages In This Thread [MO] Is there a tetration for infinite cardinalities? (Question in MO) - by Gottfried - 11/30/2014, 10:37 AM RE: [MO] Is there a tetration for infinite cardinalities? (Question in MO) - by MphLee - 11/30/2014, 09:38 PM RE: [MO] Is there a tetration for infinite cardinalities? (Question in MO) - by tommy1729 - 12/01/2014, 11:55 PM RE: [MO] Is there a tetration for infinite cardinalities? (Question in MO) - by sheldonison - 12/02/2014, 02:35 PM RE: [MO] Is there a tetration for infinite cardinalities? (Question in MO) - by MphLee - 12/02/2014, 02:40 PM RE: [MO] Is there a tetration for infinite cardinalities? (Question in MO) - by tommy1729 - 12/04/2014, 12:08 AM RE: [MO] Is there a tetration for infinite cardinalities? (Question in MO) - by MphLee - 12/04/2014, 09:31 AM RE: [MO] Is there a tetration for infinite cardinalities? (Question in MO) - by tommy1729 - 12/04/2014, 01:20 PM RE: [MO] Is there a tetration for infinite cardinalities? (Question in MO) - by MphLee - 12/06/2014, 02:26 PM RE: [MO] Is there a tetration for infinite cardinalities? (Question in MO) - by tommy1729 - 12/07/2014, 05:32 PM RE: [MO] Is there a tetration for infinite cardinalities? (Question in MO) - by MphLee - 12/28/2014, 10:22 PM

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