A Turing machine can't pick random numbers of any kind.

Once you accept that you have an entropy source in the physical world, you can easily be injecting random real numbers (from some range) and in fact, usually are, which are then being binned into integers by ADCs.

What about PI? We can represent it in terms of "we know what we are talking about" and we can distinguish it from other numbers.

You can only have countable number of first or second order logic statements each defining a specific real number.

Those things sound equivalent, but I don't want to be the one to try to prove it.