In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 16 Jun., 03:05, "Mike Terry" > <news.dead.person.sto...@darjeeling.plus.com> wrote: > > "Virgil" <Vir...@home.esc> wrote in message > > > > news:Virgil-3B2F0B.email@example.com... > > > > > In article <87sk4ohwbt....@dialatheia.truth.invalid>, > > > Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote: > > > > > > Virgil <Vir...@home.esc> writes: > > > > > > > Note that it is possible to have an uncomputable number whose decimal > > > > > expansion has infinitely many known places, so long as it has at least > > > > > one unknown place. > > > > > > You need infinitely many unknown places. > > > > > If the value of some decimal digit of a number depends on the truth of > > > an undecidable proposition, can such a number be computable? > > > > Yes - e.g. imagine just the first digit of the following number depends on > > an undecidable proposition: > > > > 0.x000000000... > > This is not a real number. It restricts the set of numbers to 10 > differents numbers (in decimal).
If x is unambiguously defined as one of the decimal digits then 0.x000000000... IS a number, even if we do not know which one. > > > > There are only 10 possibilities for the number, and in each case it is > > obviously computable... > > But it is not clear which case will be chosen. Two real numbers a and > b satisfy a < b or a = b or a > b. > 0.y and 0.x do not.
That you cannot readily determine that trichotomy holds does not mean that it does not hold.
These symbols are number forms like x and y in > 3x + 5y = 0 > or > n in "let n be an even number". > Obviously n need not be an even number. > It is no number at all.
That n is still a number, even if which one is not known.
According to WM's philosophy, one cannot speak of a number in the abstract, but only in the concrete.