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Hello,

I have read that Σ* is countably infinite and power set of Σ* (ie. 2^ Σ*) is uncountably infinite.

So by Cantor’s theorem, power set of any countably infinite set is uncountably infinite.

Then what can be said about 0^any countably infinite set or 3^any countably infinite set? Do these things have any significance?

Thank you.

edited | 141 views
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What do you mean by "0^any countably infinite set"? What is "0" here exactly?
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Oops, I meant 0 ^ any countably infinite ( zero raised to the power any countably infinite set). I am not even sure if it has any significance but since 2 ^ any countably infinite is uncountably infinite, I had a doubt what if we replace 2 by 0 or 3 (or 4,5 etc).
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$2^{anyset}$ denotes the power set of a set.

According to the above source

In set theory, given two sets A,B the notation $A^B$ is often used to denote the set of all functions f:B→A

So, as you say, if we take 2={0,1} then $2^S$ is the set of functions f:S→{0,1}

Hence if you want to say A is a set of a finite number of elements and B is a countable infinite set then yes the resultant set will be uncountably infinite.