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I was able to find the set theory definitions of addition and multiplication, but not of tetration. I wondered if somebody could define tetration in terms of set theory, and hopefully provide some (not too formally written) sources (books) which contain the definition.

Edit: Is this doable with n-fold Cartesian products? If $|A|^3$ is $|A × A × A|$ (cf. this source), what is $|A|^{|A|}$ (etc.)?

Edit 2: This has a segment on cardinal exponentiation, but is highly technical.

Also, is there a source which describes the infinite power tower of 1 in a few sentences (as a special case)? Usually it's merely said that it's trivial and then the square root of 2 is considered. I'm interested in the difference between the infinite power tower of 1, and 1 to the power of infinity, which difference seems to exist (see this answer).

string_knot
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    Unless someone has offered/introduced such a construction recently, I'm pretty sure there is no such set-theoretic formulation for tetration. (Comments to this MSE question are relevant to my background on this topic.) Note that for finite sets $A^A$ (all functions from $A$ to $A)$ has $n^n = {}^2n$ many elements, where $n$ is the cardinality of $A,$ and thus only represents $n$ with "tetration exponent" of $2.$ BTW, I'm sure this question has been asked at least once before in MSE, but I don't think anything significant was ever said. – Dave L. Renfro Oct 29 '23 at 16:09
  • There is a problem with combining set theory and tetration in a naive way ; if the powerset is $2^x$ then tetration says there is a semi-powerset : $f(f(x)) = 2^x$, thereby contradicting the GCH and more. – mick Nov 20 '23 at 22:14

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