How many solvable and unsolvable problems exist

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I am unable to quantify it as there are many problems which are in polynomial time and certain problems can be reduced to polynomial time.How exactly to quantify them?

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Are you talking about solvable versus unsolvable, or polynomial time versus not polynomial time?

Assuming we're restricted to problems that can be described by a finite expression in a given language having a finite alphabet, the number of problems is countably infinite.

There are countably many solvable problems (since if $A$ is one solvable problem, and $B$ is any problem, you can get a solvable problem of the form "do $A$ or $B$"), and countably many unsolvable problems (since if $C$ is one unsolvable problem, and $B$ is any problem, you can get an unsolvable problem of the form "do $B$ and $C$").