Has this been proven? ... 10^n + 1

[xilman]

Originally Posted by R.D. Silverman

Another example: We know that almost all real numbers are Transcendental.
The subset that is algebraic is countable, while the reals are uncountable.
Thus, the algebraic numbers have density 0 in the reals. But knowing that
almost all numbers are transcendental does not help us prove that e.g.
The Euler-Mascheroni constant is.

Even worse: almost all numbers are not computable by a Turing machine but, on the C-T hypothesis, we can't give even a single example!

Paul

[R.D. Silverman]

Originally Posted by xilman

Even worse: almost all numbers are not computable by a Turing machine but, on the C-T hypothesis, we can't give even a single example!

Paul

Sure we can. One such example is:

For a given finite state machine, what is the probability that a random input
tape will halt?

[ewmayer]

Originally Posted by R.D. Silverman

For a given finite state machine, what is the probability that a random input
tape will halt?

Your Turing machine still uses a tape drive?!! You should consider upgrading.

[Fry's has specials on TMs nearly every week, you just need to be careful to read the fine print, send any rebate form in promptly, and make sure to save a copy of the latter in case it gets "lost" in the mail, as rebate forms appear to be distressingly prone to doing, similarly to insurance claim forms.]

[xilman]

Originally Posted by R.D. Silverman

Sure we can. One such example is:

For a given finite state machine, what is the probability that a random input
tape will halt?

Fair enough --- my statement was too sloppily phrased and should not have used "give". A more accurate verb would be "compute".

Even this statement needs tightening up a little. For instance, e is transcendental but we can compute it to an arbitrary but fixed precision and do so efficiently. Non-computable numbers are, by definition, non-computable.

Paul

[R.D. Silverman]

Originally Posted by ewmayer

Your Turing machine still uses a tape drive?!! You should consider upgrading.

[Fry's has specials on TMs nearly every week, you just need to be careful to read the fine print, send any rebate form in promptly, and make sure to save a copy of the latter in case it gets "lost" in the mail, as rebate forms appear to be distressingly prone to doing, similarly to insurance claim forms.]

This is terrific! What is the price on a standard NDTM??? I'd like to buy one.

[Zeta-Flux]

Originally Posted by xilman

Even worse: almost all numbers are not computable by a Turing machine but, on the C-T hypothesis, we can't give even a single example!

Paul

Even weirder is the fact that if we work outside of the system, there can be an 'equal' number of non-computable numbers as there are computable ones. In other words, there is a countable model for the (standard) reals. And, of course, one could just work with the "computable" reals in the first place depending on which model one favors.