# A method for counting real numbers? **URL:** \ **Category:** Factual Questions\ **Created:** [June 14, 2011, 11:38am UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412 "2011-06-14T11:38:23Z")\ **Posts on this page:** 10\ **Page:** 3
**Author:** ![Lance\_Turbo](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lance_turbo/32/6156_2.png) [@Lance\_Turbo](https://boards.straightdope.com/u/Lance_Turbo)\ **Post date:** [June 15, 2011, 2:44pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/41 "2011-06-15T14:44:54Z")
> [@brocks](#): > > No, I got it wrong, to my utter lack of surprise. The correct statement should have been, “We assumed that every nonempty subset of N contains a least element.” I think you’re misunderstanding what Ross is saying. Essential he means that you shouldn’t try to justify the peano axioms with the well ordering principle, the well ordering principle can be derived from the peano axioms. The well ordering principle is definitely true, but it follows from his N5. There is zero chance that someday someone will map the reals to the naturals injectively. Proofs other than Cantor’s exist and there is no way all of them had errors that lasted centuries. The same goes for trisecting the angle. Math is not like science in this regard. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [June 15, 2011, 4:28pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/42 "2011-06-15T16:28:41Z")
> [@brocks](#): > > If you google the quoted statement verbatim (including the quotes), you will get one hit from Google Books, which will show you the context, if you’re interested. The all important context which was left out is this: “The discussion may be plausible, but we emphasize that we have not proved axiom N5 _using the successor notion and axioms N1 through N4_, because we implicitly used two unproven facts. We assumed that every nonempty subset of N contains a least element and we assumed that if n0 != 1 then n0 is the successor to some number in N” (my emphasis). What is being pointed out is only that, within the particular formal system of the particular axioms N1 through N4, the work had not yet been done, within the book at that point, to derive the two “unproven” facts (about the abstract, uninterpreted term N); that is, if one wanted to use those facts in the process of showing that some statement was derivable _from N1 through N4 alone_, then one would have to first derive those facts themselves from N1 through N4 alone. [Which as it happens could not be done; however, those facts could be derived from N1 through N4 with the addition of N5] The facts themselves aren’t in question, for the ordinary interpretation of what “N” means; in general mathematics, no one would make a fuss were you to call upon them. It’s just in this particular formal game of working from scratch up in this particular formal system of N1 through N4 that those facts, interpreted as referring to the abstract unspecified N of that formal system and not some other externally defined concept with which you may have already acquired prior familiarity, demanded justification. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [June 15, 2011, 5:11pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/43 "2011-06-15T17:11:42Z")
> [@brocks](#): > > I’m just wondering if more sophisticated mathematicians can be similarly tripped up by more subtle assumptions, and that _very_ subtle errors can resist the attempts of all mathematicians to find them for centuries I suppose it’s possible. There are logical gaps and hidden assumptions in Euclid (long considered the gold standard of deductive reasoning) that were only noticed many centuries later. **However**. Modern standards of mathematical rigor and logical subtlety are much higher than they were during most of mathematical history, so it’s a lot harder to “get away with” things like that today. So it’s highly, highly unlikely that a relatively simple proof of a well-known theorem would be in any way incorrect or incomplete without anyone noticing. (It’d be a bit more likely in the case of a long, complicated proof, especially one that only a relatively few mathematicians are qualified to read and understand; but that certainly doesn’t describe Cantor’s diagonalization argument.) And it’s even more likely that such a theorem would turn out not to be true, after all, as opposed to the proof just needing some refinement. I don’t know of any examples at all of an untrue mathematical statement for which the long-held consensus was that it had been validly proved. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [June 15, 2011, 5:44pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/44 "2011-06-15T17:44:57Z")
What does happen, though, is that the manner in which preformal mathematical concepts are to be standardly interpreted or formalized can change, and with this context shift, previously valid proofs can fail. For example, the particular formalization of “continuous function” to be settled on changed from author to author and over time between a number of non-equivalent definitions at one point in history, results being validly proven about “continuous functions” and then just as validly disproven, only, of course, in different contexts. Darboux may have legitimately considered himself to have proven that every derivative is continuous; today, we consider this statement false, but only because we have standardized on a different definition of continuity than provided by the intermediate value property. Perhaps with time, the standard definition of “continuous” will change so that this statement may again be viewed as true. What is invariantly valid is the substance of Darboux’s work, of course; what changes is only how one chooses to describe what it establishes. Yet how one describes a result is important as well, and here we are rarely in a position to predict unchanging finality. As an example more in line with the thread, someone once proved that the unit interval injects into the countably-infinite digit strings and Cantor proved that if X injects into Y and Y injects into X then there is a bijection between X and Y. Brouwer would have denied both of these. Indeed, Brouwer proved that every total function from a real interval to a real interval is continuous; thus every total function from the unit interval to digit-strings is constant, while also (0, 1) and (0, 1) U (2, 3) both inject into each other without being in bijection. And all of these conflicting proofs are correct; they just work within different frameworks and understandings of what “reals”, “total functions”, etc., are. As the winds of fashion change, we may come to take Brouwer’s view as the more mainstream one, finding it more useful or clarifying in some way. Or maybe not. Again, the substance of the arguments is invariant, regardless of how one describes it, yet all the same, how one describes it is important and subject to change. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [June 15, 2011, 5:58pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/45 "2011-06-15T17:58:47Z")
> [@](#): > > I suppose it’s possible. There are logical gaps and hidden assumptions in Euclid (long considered the gold standard of deductive reasoning) that were only noticed many centuries later. I remember back in high school, when I was reading Euclid for fun (what?), I was very excited to see his proposition that the intersection of two planes is a straight line, since I knew of counterexamples. Turns out that what he actually did was assume that the intersection was a line, and then proved that that line was straight. Similarly, his proof that there are only five Platonic solids also fails, since he was insufficiently rigorous in his definition of Platonic solids. By his definition, there are at least 7, even without counting the non-convex ones. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [June 15, 2011, 6:14pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/46 "2011-06-15T18:14:20Z")
> [@Chronos](#): > > I remember back in high school, when I was reading Euclid for fun (what?), I was very excited to see his proposition that the intersection of two planes is a straight line, since I knew of counterexamples. Turns out that what he actually did was assume that the intersection was a line, and then proved that that line was straight. Heh. This is a bit confusing if one does not realize (as I did not till just now) that the term “line”, as Euclid (or I guess his translators) uses it, encompasses curves as well as straight lines. ---
**Author:** ![brocks](https://avatars.discourse-cdn.com/v4/letter/b/7feea3/32.png) [@brocks](https://boards.straightdope.com/u/brocks)\ **Post date:** [June 15, 2011, 10:50pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/47 "2011-06-15T22:50:35Z")
> [@Lance\_Turbo](#): > > I think you’re misunderstanding what Ross is saying. Essential he means that you shouldn’t try to justify the peano axioms with the well ordering principle, the well ordering principle can be derived from the peano axioms. The well ordering principle is definitely true, but it follows from his N5. Yes, I got that. N5 is an axiom, after all, so there was no need to prove it, and I didn’t think Ross was saying that it, or well ordering, was false; he was just saying that the motivation he gave for N5 was not a proof. It just strikes me that it’s much easier to know that you have solved a numerical problem correctly, than to know that you have not overlooked any details in a proof. The proofs used as examples in this thread are simple enough that it’s almost inconceivable that they will one day be found invalid, but I still object to the blanket statement that something once proved need never be revisited. ---
**Author:** ![brocks](https://avatars.discourse-cdn.com/v4/letter/b/7feea3/32.png) [@brocks](https://boards.straightdope.com/u/brocks)\ **Post date:** [June 15, 2011, 10:58pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/48 "2011-06-15T22:58:00Z")
> [@Thudlow\_Boink](#): > > I don’t know of any examples at all of an untrue mathematical statement for which the long-held consensus was that it had been validly proved. Since a postulate does not require proof, Euclid’s “parallel postulate” does not refute your statement, and it was certainly not proven untrue in the context of Euclidean geometry. But it _is_ an example of something that was considered inviolable for over a thousand years before the discovery of non-Euclidean geometry. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [June 15, 2011, 11:02pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/49 "2011-06-15T23:02:20Z")
Right. An example of how a proven result is neither the final word, nor is it generally lost in revolution; it just gets reinterpreted. What once was the one-and-only way to look at geometry was turned over, but not lost; it became a small part of a much larger space of possibilities, acknowledged as still sometimes but not always the most apt way of analyzing a situation. We don’t consider the Pythagorean theorem to be false, as such, but we recognize that it is not always true, either, depending on the context. ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [June 15, 2011, 11:09pm UTC](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412/50 "2011-06-15T23:09:30Z")
(And so it would be for Cantor’s diagonalization arguments… we can be certain (as certain of anything) that they will continue to tell us something valid about such domains of interpretation as bless the formal reasoning involved, but our extra-formal views on what exactly those domains are may fluctuate over time.) [This is all rather separate from an out-and-out _error_, where an element of an argument is discovered to have been made in haste, in clear contradiction to the implicit rules of the proof-game the argument was intended to be made within and such that no one has any interest in investigating alternative proof-games which would still accommodate it; of course, these happen (go multiply large numbers by hand till you hit a brainfart), but the frequency with which they happen to short, highly-publicized and commonly-formalized arguments is essentially zero; at any rate, one cannot be skeptical of these without being skeptical of _all_ reasoning, and thus falling into such thoroughgoing Cartesian skepticism as to refuse to be certain of anything or worth talking to] [Previous page](https://boards.straightdope.com/t/a-method-for-counting-real-numbers/585412.md?page=2)