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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 $C$").

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  • $\begingroup$ Thanks got it I was talking about solvable versus unsolvable $\endgroup$ – ten do May 15 at 1:11
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    $\begingroup$ There's apparent tension here with the idea that for any real number $x$, there should be problems like "Is $x$ computable?" or "Is the function $f$ defined by $f(z)=xz$ differentiable?", so there should be uncountably many "problems". I suppose the resolution lies in saying that almost all real numbers cannot be named—which is pretty weird really; this supposedly familiar object which we introduce to grade schoolers consists mostly of things forever beyond our ken. $\endgroup$ – Kundor May 15 at 2:51
  • $\begingroup$ @Kundor thanks got it $\endgroup$ – ten do May 15 at 23:53

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