My guess would be no. Specifically, I guess that 2^(2^(1/2)) + 3^(2^(1/2)) is not in Q^.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Dan Asimov Sent: Sunday, January 18, 2015 6:01 PM To: math-fun Subject: [math-fun] Least set of reals containing Q+ and closed under both multiplication and exponentiation
This may well be trivial, since I haven't thought about it more than a few seconds.
But let Q^ denote the least set of reals containing Q+ and closed under both multiplication and exponentiation.
I.e., Q^ is the union of all Q_n, n >= 0, where
* Q_0 := Q+
* Q_(n+1) = (Q_n)^(Q_n) * (Q_n)^(Q_n), n >= 1.
where for any subsets X and Y of R+,
X op Y := {x op y | x in X and Y in Y}
for op in {*,^}.
Question: Is Q^ closed under addition?
--Dan
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