Comments on: Numbers that cannot be computed

By: Fern

Fern — Sat, 17 Apr 2010 20:14:56 +0000

Hi,

Thanks for this post! But I still don’t understand how you would use the diagnolisation method to prove that some numbers are non-computable?

Please help!

Many Thanks!

By: Matt Mahoney

Matt Mahoney — Fri, 09 Apr 2010 14:35:02 +0000

Most uncomputable numbers are hard to describe because if you could describe them, then you could turn that description into a program and they would not be uncomputable.

One exception is Chaitin’s constant Omega, the probability that a random program will halt. http://en.wikipedia.org/wiki/Chaitin‘s_constant

By: Aakash

Aakash — Wed, 03 Feb 2010 19:22:51 +0000

Simply put, some numbers are random.

PI and E are not at all random, but just the opposite, they have a very specific derivation.

A very nice article, this one.

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By: Igor Ostrovsky

Igor Ostrovsky — Thu, 02 Apr 2009 19:07:01 +0000

1) Programs are countable, real numbers aren’t. So, there are fewer programs than there are real numbers. It follows that some real numbers don’t have programs that generate them.

2) “Definable” numbers are also countable, so some real numbers are not definable say in English. However, there are some numbers can be defined, but cannot be computed on a Turing machine. I gave an example of one such number (see my response to 3 below).

3) I gave an example of a number that cannot be computed in the article (see the paragraph on diagonalization).

By the way, check out http://en.wikipedia.org/wiki/Definable_numbers. The page explains this stuff better than I can. Here is a short quote from the article:

While every computable number is definable, the converse is not true: the numeric representations of the Halting problem, Chaitin’s constant, the truth set of first order arithmetic, and 0# are examples of numbers that are definable but not computable. Many other such numbers are known.

By: Daniel

Daniel — Thu, 02 Apr 2009 10:17:58 +0000

1) Only that the number of programs is countable and the number of real floating point values isnt doesn’t actually prove that any of these values is NOT programmable. It only proves that it’s not possible to programm ALL of them.

2) The main problem with real floating point values is that not all are “Definable”. e.g.: Give me rule how the real number is defined and a program can be written to accomodate this rule.
Example: Calculation of Pi, e, Sqrt(2) etc.

3) The problem actually gets even worse: There is NO number for which you can “prove” that you cannot program it (assuming indefinite memory and processing power is provided).
-> To prove, you need a definition of the number to check the program against. But if such a definition exists, you then can create a program according to it.

By: Números no computables | FACIL TUTORIALES

Números no computables | FACIL TUTORIALES — Fri, 19 Dec 2008 22:23:27 +0000

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By: Gürkan Alkan

Gürkan Alkan — Thu, 18 Dec 2008 14:23:59 +0000

I think it is uncountable infinite. If it is countable the computer can.