Talk:Constructive proof

message: Alan Kay and other computer scientists use the term "Existence Proof" to mean proof by example, e.g. demonstrating a red car is proof that some cars are red. I think this article should add this second connotation. --Mcandre
It seems to me that this, existence theorem and nonconstructive proof all cover the same ground. It would be better to merge all three, I think, and get one substantive article.
Charles Matthews 17:37, 10 Nov 2003 (UTC)
thread_title: Untitled

message: The statement that the set of algebraic numbers is countable
is equivalent to the statement that there exists an algorithm
that enumerates them all with increasing precision, i.e. it outputs
the decimal approximation of each consecutive number with precision
growing by 1 in each turn.  We can now use diagonalization
to construct a decimal expansion of a real number where each digit
is different from the respective digit of the respective algebraic
number in the enumerating sequence.  This procedure produces
a transitive number and it is well-defined: it can produce any digit
of the decimal expansion upon request.  That allows us to compare the
number to any rational number and to create a Dedekind section
out of it.  What is nonconstructive here?
:Well, sure, but that's a diagonal argument, and it is not the same as the proof based on uncountability. As the article says:
::We may distinguish three different problems. The first is that of proving the existence of transcendental numbers (without necessarily providing a specimen). The second is that of giving an example of a transcendental number by a construction specially designed for the purpose. The third, which is much more difficult, is that of proving that some number given independently ... is transcendental.
:So "algebraics are countable, reals are uncountable, therefore transcendentals exist" is a (non-constructive) solution to the first, your proof is a solution to the second (we could include it in the article as such), and a proof that pi is transcendental is a solution to the third. On the other hand, perhaps we should just find a better example. 192.75.48.150 15:12, 24 July 2006 (UTC)
thread_title:  Why is the proof based on uncountability nonconstructive? 

message: The article should be rewritten completely because the approach used in this article is academic. The concept of non-constructive proof is different from constructive proof and it (non constructive proof) should have a separate article. arsalan... (talk) 19:20, 22 December 2009 (UTC)
replies: text: I completely agree Habiliscus (talk) 18:21, 4 May 2024 (UTC)
replies: text: I disagree, unless if there is an evidence that a non-constructive proof is not the same as a proof thatis not constructive. D.Lazard (talk) 19:38, 4 May 2024 (UTC)
thread_title: Non Constructive Proofs

message: The page states that "It turns out that \sqrt{2}^{\sqrt2} is irrational because of the Gelfond–Schneider theorem, but this fact is irrelevant to the correctness of the non-constructive proof".  Does anyone know whether the theorem is valid in a constructive framework?  Tkuvho (talk) 15:06, 27 February 2012 (UTC)
thread_title:  Is Gelfond constructive? 

message: I think the part 'all of its prime factors are greater than n' should instead be 'some of its prime factors are greater than n'.
Am I mistaken? Notrium (talk) 20:54, 11 April 2015 (UTC)
replies: text: The statement holds as written.  However, I fail to see why this proof is inconstructive, given that you can take the smallest factor as before.  Yecril (talk) 21:48, 26 April 2015 (UTC)
thread_title:  Mistake in section Examples/Non-constructive proofs ? 

message: Substituted at 01:55, 5 May 2016 (UTC)
thread_title: Assessment comment

message: Right now, the article says:
"Define a function f of a natural number x as follows:
:f(x) = \begin{cases} 0 & \mbox{if Goldbach's conjecture is false} \\ 1 & \mbox{if Goldbach's conjecture is true}\end{cases}
Although this is a definition by cases, it is still an admissible definition in constructive mathematics."
Is it? I think usually this sort of example is given by something like f(n)=0 if 2n is the sum of two primes or n is 0 or 1 and f(n)=1 otherwise. This second definition by cases uses a decidable criterion and so demonstrably yields a computable function. If definitions by cases are generally allowed to take the form the article suggests, however, that would seem to imply the existence of noncomputable functions (at least if there are no restrictions on what sorts of cases we can use), which at least some constructive theories outright deny. 
I suppose we could, in a constructive theory, postulate the existence of a function such that f(x)=0 for all x if Golbach's conjecture is false and f(x)=1 for all x if it is true and reason about such a function's properties, but it's not even obvious to me that most constructive theories would be able to prove that two such functions are equal (without either proving or disproving Goldbach's conjecture)107.77.211.108 (talk) 19:52, 4 January 2017 (UTC)
replies: text: That example was indeed unclear - thank you for pointing it out. Although something was defined, it would not be possible to prove constructively that f was even a function. I changed the example to a more classical one involving the limit of a Cauchy sequence, with a reference to the SEP. — Carl (CBM · talk) 21:15, 4 January 2017 (UTC)
replies: text:  I'm glad to have helped. I also edited your definition to include the condition that the counterexample is even in the second case, which I believe is necessary unless I'm having a really bad day for thinking clearly. 107.77.212.225 (talk) 18:17, 6 January 2017 (UTC)
thread_title:  Definition by cases 

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thread_title:  External links modified 

message: "Some non-constructive proofs show that if a certain proposition is false, a contradiction ensues; consequently the proposition must be true (proof by contradiction). However, the principle of explosion (ex falso quodlibet) has been accepted in some varieties of constructive mathematics, including intuitionism."
These two statements are not related to each other. This appears to be an act of vandalism or a mistaken edit, removing some important context relating the two sentences. Comiscuous (talk) 01:31, 11 January 2022 (UTC)
thread_title:  Unrelated sentences