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According to a result of Paul Cohen in a mathematical model without the axiom of choice there exists an infinite set of real numbers without a countable subset. The proof that every infinite set has a countable subset (http://www.proofwiki.org/wiki/Infinite_Set_has_Countably_Infinite_Subset) is dependant on the axiom of choice and therefore cannot be proven in a model without the axiom of choice. Given that it cannot be proven does it imply there MUST be atleast one infinite set without a countable subset?

I realise this may seem like a stupid questions because it seems intuitive that if something can't be proven then there must be a counter example but I have learnt to be skeptical about any naievely intuitive conclusions so I was hopping somebody could just clarify.

I realise this may seem like a stupid questions because it seems intuitive that if something can't be proven then there must be a counter example but I have learnt to be skeptical about any naievely intuitive conclusions so I was hopping somebody could just clarify.

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