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I'm not sure how to fix this, but I don't think it can stand as is. There is no modern "disagreement" on the question, not phrased this way. (I'm not sure whether intuitionists consider the proof constructive, because it might use excluded middle (?) but that's a bit of a different issue.) --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 07:05, 13 September 2018 (UTC)
Thank you for your feedback. I agree with you that "Sheppard correctly notes that Cantor's method can be applied to find a particular transcendental." However, I disagree that "Stewart correctly notes that Cantor did not in fact ''do'' that."
To understand what Cantor did, I'll quote from the article [http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gray819-832.pdf Georg Cantor and Transcendental Numbers ]. From the bottom of page 819 to page 820, it states:
:Cantor begins his article by defining the algebraic reals and introducing the notation: (ω) for the collection of all algebraic reals, and (ν) for the collection of all natural numbers. Next he states the property mentioned in the article's title; namely, that the collection (ω) can be placed into a one-to-one correspondence with the collection (ν), or equivalently:
::... the collection (ω) can be thought of in the form of an infinite sequence: (2.) ω<sub>1</sub>, ω<sub>2</sub>, ..., ω<sub>η</sub>, ... which is ordered by a law and in which all individuals of (ω) appear, each of them being located at a fixed place in (2.) that is given by the accompanying index.
:Cantor states that this property of the algebraic reals will be proved in Section 1 of his article, and then he outlines the rest of the article:
::To give an application of this property of the collection of all real algebraic numbers, I supplement Section I with Section 2, in which I show that when given an arbitrary sequence of real numbers of the form (2.), one can determine [Note: By this Cantor means that he can "construct"] in any given interval (α ··· β), numbers that are not contained in (2.). Combining the contents of both sections thus gives a new proof of the theorem first demonstrated by Liouville: In every given interval (α ··· β), there are infinitely many transcendentals, that is, numbers that are not algebraic reals. Furthermore, the theorem in Section 2 presents itself as the reason why collections of real numbers forming a so-called continuum (such as, all the real numbers which are ≥ 0 and ≤ 1), cannot correspond one-to-one with the collection (v); thus I have found the clear difference between a so-called continuum and a collection like the totality of all real algebraic numbers.
:To appreciate the structure of Cantor's article, we number his theorems and corollaries:
:Theorem 1. The collection of all algebraic reals can be written as an infinite sequence.
:Theorem 2. Given any sequence of real numbers and any interval [α, β], one can determine a number η in [α, β] that does not belong to the sequence. Hence, one can determine infinitely many such numbers η in [α, β]. (We have used the modern notation [α, β] rather than Cantor's notation (α ··· β).)
:Corollary 1. In any given interval [α, β], there are infinitely many transcendental reals.
:Corollary 2. The real numbers cannot be written as an infinite sequence. That is, they cannot be put into a one-to-one correspondence with the natural numbers.
:Observe the flow of reasoning: Cantor's second theorem holds for any sequence of reals. By applying his theorem to the sequence of algebraic reals, Cantor obtains transcendentals. By applying it to any sequence that allegedly enumerates the reals, he obtains a contradiction—so no such enumerating sequence can exist. Kac and Ulam reason differently [20, p. 12-13]. They prove Theorem 1 and then Corollary 2. By combining these results, they obtain a non-constructive proof of the existence of transcendentals.
So Cantor does give a method of constructing transcendental numbers. By the way, Kac and Ulam present the same non-constructive proof that Stewart uses.
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