- #1
bedi
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Let X be a set and L a signature; write k(X,L) for the number of distinct L-structures which have domain X. Show that if X is a finite set then k(X,L) is either finite or at least 2^w (w stands for omega).
Attempt: as L is fixed, all L-structures have the same n-ary relation and function symbols and the same set of constants. Although the symbols can be different, they represent the same function or relation or set so I can't use that fact to construct distinct L-structures. How can I even find 2 distinct L-structures when they have the same set of constants, relations and functions? Perhaps I should consider the complement of the set of constants. But their domain is X...
Attempt: as L is fixed, all L-structures have the same n-ary relation and function symbols and the same set of constants. Although the symbols can be different, they represent the same function or relation or set so I can't use that fact to construct distinct L-structures. How can I even find 2 distinct L-structures when they have the same set of constants, relations and functions? Perhaps I should consider the complement of the set of constants. But their domain is X...