# Is the languague L={, M accepts a finite amount of words} decdidable?

Is $$L=\{ | L(M) \ is \ finite\}$$ decidable ? M is a TM.

I think its relative simple to proof with the theorem of rice. But I am interested in a solution which does not use the Rice theorem.

This my try : Let f(<m,w>) be a function which works in the following way :

1. Run w on M
2. If M accepts Construct a TM M`which accepts only the word w and return M`
3. If M rejects Construct a TM M`which accepts everything. Return M`

So if m is in $$A_{TM}= \{|M \ accepts \ w\}$$ we know that f(<m,w>) is in L. If m is not in A then we know that f(<m,w>) does accept every word and therefore infinity words. So f(<m,w>) not in L.

Is this a correct mapping reduction ?