Is the set of natural numbers the only infinite set that is not a power set of another set?
Well you need to have a building block for numbers in the simplest manner and the natural numbers are that block.
Do you have any thoughts about a building block that is a subset or rather something simpler than the natural numbers?
It is a very good question to ask, because these kinds of things get people thinking and understanding and that's always good.
why cant we just start with 0 and 1 . and just add 1 to 1 as many times as we want
Nearly every set is not the power set of another set.
Did you instead mean
the only infinite sets for which there is not a bijection from X to a power set of another set are sets for which there is a bijection to the natural numbersor maybe
the cardinality of the natural numbers is the only infinite cardinal number that is not the cardinality of a power set?
If so, then your question is essentially the Generalized Continuum Hypothesis.
You might just as well start with 0 and add 1,000,000 and keep adding 1,000,000. You'll end up with a set that is the same size as if you had added 1 each time
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