I firmly believe that "alpha" is a position, and not a mindset. The "alpha mindset" as described in forums like this is simply a "masculine mindset" IMO. This is the mindset you should have if you are a man, and it can help give you a chance of reaching an alpha position.
The alpha male is literally the top, most dominant male. Women want the top male, the best male they can get. This gives them bragging rights and they can share in the fruits of his power and dominance. Women literally want the top male, not just the male who thinks he is the top male.
PUAs try to teach you to think like alphas, to mimic alpha behaviors. This can be effective, if it's congruent with your personality, your life, or your actual situation. Thing is, I believe there are a lot of different types of males who are in alpha positions, so the PUAs' attempt to isolate their mindset and boil them down into one personality is a bit of an approximation.
