> somebody creates a good invention but deny it from the public

for about 15 years.. After 15 years the public can make use of the invention without paying a penny. How it is not in publics interest?

Because even if removing the patent system meant that the invention was kept secret from the public, it would most likely be reinvented many times in that 15 year time. The worst part of patents is that they extend even to others who do the legwork themselves, and frequently stifle an entire area (for example, 3D printing exploded in popularity and accessability right after some key patents held by stratasys expired, and this was in the best case of a company actually developing the technology they had patented).

We are talking about an exceptional case here. The patent owner is a wealthy and extentric agent who disregarts potential profit from selling/licensing/using the patent. And even in such rare and unexpected cases the public eventually gains access to inventions - which is not neccessarely the case for inventions kept secret. Basically, the society trades speed of technological development for certaininity of achieving its goals.

For 15 years nobody can use a invention, even if willing to pay. How is that in public interest?

I'm guessing the practical meaning is that the inventor is not required to license the invention "at a fair and reasonable" price but is free to ask whatever he/she feels is appropriate. I could imagine a case where the industry the inventor wishes to market to is controlled by a cartel which colludes to offer a low price for the invention. In this case, the inventor may wish not to sell or license in the hope that the cartel breaks up or gives in.

The public also benefits from people inventing alternatives to patented inventions.

That's debatable. Your point is that patent would encourage diversity which I'm sure it doesn't change much. Worst it sometimes prevents advances

How’s that good for the public?

More alternatives to choose from.