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I am trying to prove that two definitions of a finite set are equivalent.
1.) A set ##A## is finite if and only if it is equipollent to a natural number ##n##. ( natural number as the set containing all the previous natural numbers including ##0## )
2.) A set ##A## is finite if and only if every collection of subsets has a minimal element with respect to the ##\subset## relation. ( this means there is at least one element of every collection that contains no others as subsets ).
I have figured out how to prove (1) ##\rightarrow## (2) but am struggling with (2) ##\rightarrow## (1).
Please don't post a full proof. Just a hint or basic sketch would be most appreciated.
Also, I'd like to prove it without using the axiom of choice. Induction on the natural numbers is okay ( that's how I proved (1) ##\rightarrow## (2) ).
1.) A set ##A## is finite if and only if it is equipollent to a natural number ##n##. ( natural number as the set containing all the previous natural numbers including ##0## )
2.) A set ##A## is finite if and only if every collection of subsets has a minimal element with respect to the ##\subset## relation. ( this means there is at least one element of every collection that contains no others as subsets ).
I have figured out how to prove (1) ##\rightarrow## (2) but am struggling with (2) ##\rightarrow## (1).
Please don't post a full proof. Just a hint or basic sketch would be most appreciated.
Also, I'd like to prove it without using the axiom of choice. Induction on the natural numbers is okay ( that's how I proved (1) ##\rightarrow## (2) ).
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