Question about the natural numbers.

[cragar]

Is the set of natural numbers the only infinite set that is not a power set of another set?

 

[chiro]

Is the set of natural numbers the only infinite set that is not a power set of another set?

Hey cragar.

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.

 

[cragar]

why can't we just start with 0 and 1 . and just add 1 to 1 as many times as we want

 

[Hurkyl]

Is the set of natural numbers the only infinite set that is not a power set of another set?

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 numbers​

or 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.

 

[phinds]

why can't we just start with 0 and 1 . and just add 1 to 1 as many times as we want

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