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2 years ago in Logic , Mathematics By Usha K

Can an injective map always be defined from a countable set into an arbitrary set?

As I'm working on foundational problems in set theory and model construction, this precise point about cardinality and injective mappings keeps coming up. I need a clear, expert take on whether the definition is always possible, regardless of the cardinality of the target set, to clarify my arguments.

SetTheory Cardinality CountableSet

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By Sonam Bhatia Answered 2 years ago

Yes, absolutely. From my work in foundational mathematics, I can confirm that an injective map from a countable set into any arbitrary set always exists. The key is to recognize that "countable" means you can list the elements as a sequence. You simply map the first element of your countable set to some element in the target set, the second to a different element, and so on. Even if the target set is huge (like the real numbers), you're only using a countable subset of it. The Axiom of Choice ensures this selection process is valid. It's a fundamental result for comparing infinite sizes.

 

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