Comments on: Numbers that cannot be computed

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

[...] de Microsoft que trabaja en computación en paralelo; escribió una interesante anotación titulada Numbers that cannot be computed donde describe un fenómeno curioso, los números que no son [...]

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.

By: Sergio

Sergio — Tue, 02 Dec 2008 18:25:54 +0000

@Pablo: as Igor has said, you can only write finite programs, but most of real numbers are infinite/uncountable, so you would never finish to write your program. This is different from integer numbers, because no matter how big is your number, you can always represent it with log2(number) bytes.
As an example, try to write the pi number here. You would NEVER finish to write it.

Sergio

By: Números no computables « ¡La Trapaleta es lo de Hoy!

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