Here is the Knol referenced above, with my comments in [square brackets]. Everything not in square brackets is verbatim quotation. If you have any background in math, hopefully if you don’t, feel free to follow along:
The short answer to the above [Are the real numbers well defined] question is no. [OK, stating a "fact" with nothing to back it up.]
Why? Although all numbers are well-defined up to and including the rational numbers, the irrational numbers have never been well-defined. [Restating the same "fact" again, as evidence?.]
What does it mean to talk about anything being well-defined? An object or a noun that has clearly distinguishable limits or boundaries is well-defined. [I thought it was something that has a rigorous definition.]
In the theory of real analysis, real numbers are defined as limits of equivalent convergent sequences (a.k.a Cauchy sequences). [Not exactly. The are several definitions, but the closest one to what he describes is "real numbers of equivalence classes of Cauchy sequences". The AKA is certainly not true -- over many metric spaces, there are Cauchy sequences that do not converge.] Herein [in the incorrect statement of what the real numbers are defined as] lies the core problem with the modern theory of real numbers [or rather, with the theory as stated by John Gabriel]: it is impossible to know the limit or boundary of any irrational number given a convergent sequence that represents the same.[Irrational numbers do not have a "limit" -- they are numbers, not sequences. If we do use the 'Cauchy sequences of rationals' definition for real numbers, irrational numbers would be the equivalence classes containing non-convergent sequences (it's an easy lemma to see that an equivalence class either all converges or all diverges) But regardless of the specific definition, irrational numbers do not have a "limit" or "boundary", as they are numbers. Whether we can know stuff about them is irrelevant to their definition. "All Cauchy sequences up to the equivalence relationship defined as 'the difference converges to zero" is a definition of the real numbers which is as clear cut as any.]
So how then, is it sensible to claim that such a definition is well-defined? [By not giving a strawman definition, I guess.] It fails in the very first requirement to match the attribute for being well-defined, that is, its limit or boundary is unknown. [The strawman definition fails in that it begs the question: if irrational numbers are, indeed, limits, then they would have to be defined before they are used. I guess that this is a roundup way of going about that.] For example, consider the well-known irrational number called pi. [That "irrational number" can be represented as a real number easily enough: one example is through the Taylor expansion for arc-tan, and taking the arc-tan of something we can construct Pi from via additions and multiplications with rationals.] Although we can calculate the approximate representation of pi in a radix system to almost any degree of accuracy [In other word, for any real number, there is a unique Cachy sequence in its equivalence class such that a_n is b_n/10^n with b_n being natural, and such that the sequence is not eventually constant.], it is incorrect to state that we know the limit of a convergent sequence that represents pi. [No it isn't. After defining the real numbers as such Cauchy sequences, we can identify the rationals with the equivalence class of constant sequences. It is then trivial to prove that if r is a real number, and q_n is a member of it, then q_n seen as a Cauchy sequence of real numbers converges to r. It is subtle to understand that we identify two different representation of the rational numbers: one as the regular ol' rationals, and one as the subset of the real numbers. However, in general, any two structures isomorphic up to a unique isomorphism can be identified is a usual thing, and brings no ill effect. A different way of saying it is "the rationals are the smallest field of characteristic 0", and take any field satisfying that as the field of rationals.] To say that 3.1415926 as a value for pi is accurate to seven decimal places means nothing. [Ummm....it needs to be well-defined, which is not to say it means nothing. Here's a stab at a definition: a finite decimal expansion is accurate to n digits if it differs from the number it represents by less than 10^n.] It is only an approximation. [Yes, it is an approximation, accurate to 7 digits. What's the problem with that?] Furthermore, this approximation is not sufficient for all calculations involving pi. [Right. If it was, it would be pi, not an approximation of pi.] Therefore it can be confidently stated that pi is not well-defined, [Umm...no, it can be confidently stated that pi is distinct from its finite expansions. At least confidently if you prove it, but the details of this proof are immaterial to his thesis, which could have used some other number.] except to say that it is a ratio of a circle’s circumference length to its diameter length. [That's certainly one way to define pi.] The components of such a ratio are neither rational numbers [indeed, though not trivial to prove], nor limits of sequences composed of rational numbers that supposedly represent these irrational numbers [Ummm...yes, they are. I just constructed such a sequence above in my definition of pi, and it converges to pi by the lemma I alluded to earlier.] . Reason? Such limits are indeterminate. [No, such limits are determinate. Here is a sketch of the proof: assume r is a real number, and q_n is a member of it. Let p_n be another member of it. By Cauchy propery and equivalence, for a given epsilon, for sufficiently large m, n, q_m-p_n will be less than epsilon. Therefore, q_m-r will be less than epsilon. The details are beyond the scope of this text, but are not hard to fill in.]
[I'm skipping over some discussion of a flaw in Cauchy's argument. I don't know enough history, perhaps Cauchy's arguments were wrong. That does not matter. The definitions still work. Above I alluded to proofs that work. You can pick up any advanced calc. book and get the details, and the proofs there work -- I did check the proofs in my copy. That's the nice thing about math -- even if Cauchy was wrong, it doesn't impact any of modern math. Calling the sequences Cauchy as an honor is still appropriate -- he showed the way, even if he got stuff wrong. Real mathematicians are happy to fix the problems and move on.]
An irrational number exists as a magnitude in reality but does not exist in theory. [Here we veer off math into philosophy. For all I know, this guy's philosophy is sound. His math, in any case, isn't. What does "exist" mean in this context? For me, "the real numbers exist" is a theorem in ZFC which states "there exists a complete ordered field". I can prove this theory, and every decent calc book proves this by constructing the real numbers. There are many different constructions. Some of the easiest use Choice, but it's not endemic -- there's a famous construction from almost-homomorphism which uses only ZF. See wikipedia for more details. Whether it exists in "reality" or "theory" is a different question entirely. It is possible that ZF is inconsistent, and thus proves everything including the existence of real numbers in uninteresting ways.] For example, in an isosceles right-angled triangle whose equal sides are joined by a theoretically immeasurable [what does "theoretically immeasurable" mean?] quantity called a hypotenuse, the length of the hypotenuse is real in every respect. [Things can be real in more than one respect? Clearly, this is pure philosophy.] It can be formed from one of the equal sides and part of the other equal side. The part of the other equal side can be constructed to complete the length of the hypotenuse:
[A drawing follows]
Now it is easy to see how an irrational ‘line segment’ exists in practice. Theoretically, an irrational number cannot be defined (neither as a Cauchy sequence, nor in any other way). [I have no idea what he means by "theory" or "practice". Mathematically, irrational numbers can be defined. I just gave a definition above: equivalence classes of Cauchy sequences that do not converge.]
The successor of zero is an indeterminate quantity (not an infinitesimal because an infinitesimal is an ill-defined concept) [As opposed to an indeterminate quantitiy, which is a well-defined quantity? I've never seen a definition for that, and certainly not in this text. In any case, an infinitesimal can be well defined (a member of a an ordered field that contains R which is smaller than every positive real), and is irrelevant as it is still not "the sucessor of zero" which I have no idea what he means by. Infinitesimals in classical non-standard analysis are still members of a field, so the "smallest infinitesimal" still does not exist -- if p is infinitesimal, so is p/2.] that arises in the measure of all irrational numbers because it cannot be represented as a/b. [It does not arise in the measure of all irrational numbers, even though they can be represented by a/b. His conclusion fails to follow from the premises. It, in fact, fails to be clear enough for me to understand...] So when one tries to measure any irrational number [what does "trying to measure an irrational number" mean?] using sums of averages (a Cauchy sequence for example) [How is a Cauchy sequence a sum of averages?], there is no such thing as ‘limit’. [Cauchy sequences over the real numbers always have a limit.] It does not matter that a value for the limit can be fixed for any given partial sum as in the epsilon-delta arguments. Why? [Why, indeed!] Provided more terms are summed in a partial sum, the limit continues to change even though it (limit) converges [I don't understand. The limit is a number. It does not converge to anything. The sequence converges to the limit.] (To what? No one knows for certain…the limit cannot be a number because a number is defined as a ratio!) [What! The! Hell! Where was a number defined as a ratio? He pulled this out of his hat in the middle of the discussion. I assume he means "an integral ratio". Well, if that's his definition of a number, clearly the real numbers are not numbers. But it's just using a different definition. Fine, call a complete ordered field the real "lumbers". Now every Cauchy sequence over the rationals is a lumber. Real numbers are well defined -- they just differ from his definitions. While theorems are a matter of truth, definitions are a matter of convenience and thus convention. Flauting convention is just not useful here. Is the point maybe to say "I want to have an unorthodox definition of numbers"? Because I could say that too. A number is 5. 5 is the only number. This definition is just as good as his -- equally unpopular, unhelpful and yet valid.]. The successor of zero [which he didn't define] is the reason a limit does not exist for any irrational number. [An irrational number has no limit because it's a number. Sequences have limits. Numbers...are just numbers.]
Posted by moshez 
Posted by moshez 
Posted by moshez 
