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text: 'A [-3jz] $X$ is *uncountable* if there is no [499 bijection] between $X$ and [45h $\\mathbb{N}$]. Equivalently, there is no [4b7 injection] from $X$ to $\\mathbb{N}$.\n\n## Foundational Considerations ##\n\nIn set theories without the [69b axiom of choice], such as [ZF Zermelo Frankel set theory] without choice (ZF), it can be [5km consistent] that there is a [-cardinal_number] $\\kappa$ that is incomparable to $\\aleph_0$. That is, there is no injection from $\\kappa$ to $\\aleph_0$ nor from $\\aleph_0$ to $\\kappa$. In this case, cardinality is not a [540 total order], so it doesn't make sense to think of uncountability as "larger" than $\\aleph_0$. In the presence of choice, [5sh cardinality is a total order], so an uncountable set can be thought of as "larger" than a countable set.\n\nCountability in one [-model] is not necessarily countability in another. By [skolems_paradox Skolem's Paradox], there is a model of set theory $M$ where its [6gl power set] of the naturals, denoted $2^\\mathbb N_M \\in M$ is countable when considered outside the model. Of course, it is a [6fk theorem] that $2^\\mathbb N _M$ is uncountable, but that is within the model. That is, there is a bijection $f : \\mathbb N \\to 2^\\mathbb N_M$ that is not inside the model $M$ (when $f$ is considered as a set, its graph), and there is no such bijection inside $M$. This means that (un)countability is not [absoluteness absolute].\n\n## See also\n\nIf you enjoyed this explanation, consider exploring some of [3d Arbital's] other [6gg featured content]!\n\nArbital is made by people like you, if you think you can explain a mathematical concept then consider [-4d6]!',
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