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By the way, you bring up an important point—would intuitionists accept Cantor's proof? I believe the answer is no because his proof of his Theorem 2 uses the fact that an increasing or decreasing bounded sequence of reals has a limit. Intuitionists like limiting procedures that are coming from below and above. However, the footnote on page 821 in [http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gray819-832.pdf Georg Cantor and Transcendental Numbers] says that page 27 in Bishop and Bridges ''Constructive Analysis'' (1985) has a proof of Theorem 2 that meets the demands of constructive mathematicians (and probably also intuitionists).
Concerning the word "disagree", I chose it to replace the word "controversy" in my first rewrite of the Wikipedia article. The word "controversy" has two problems: It's a "peacock" term and it's inaccurate because controversy implies that the mathematicians who are stating the proof is constructive or non-constructive are aware of the choice they are making. Disagreement simply means that what an article or book says disagrees with at least one source in the literature. I'm not particularly attached to the word "disagree". However, changing it could take some time (depending of course on the particular term chosen) and the GA review had no trouble with "disagree".
Most sources seem to say that Cantor's proof is non-constructive. Ivor Grattan-Guinness in his two-sentence 1995 review of "Georg Cantor and Transcendental Numbers" stated: "It is commonly believed that Cantor's proof of the existence of transcendental numbers, published in 1874, merely proves an existence theorem. The author refutes this view by using a computer program to determine such a number". Working on this Cantor article rewrite, I still found more sources stating his proof is non-constructive, but it was nice to find a few saying the proof is constructive.
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