Wikipedia talk:Errors in the Encyclopædia Britannica that have been corrected in Wikipedia/Archive 1

The EB article about "real numbers" claims that every set of real numbers with an upper bound has a least upper bound; this is false.

Is this false? can you give an example?. I'm under the impression that it is true. In fact, I just took it out of the page because the real number page says exactly the opposite.

Please contemplate the empty set and then put the comment back in. --AxelBoldt

I'm not sure you can consider the empty set "a set of real numbers"...It is clearly a subset of the real numbers, but I am not convinced it is the same. --AN

Allright, let's give the poor and abused editors of EB the benefit of the doubt :-) --AxelBoldt

I just restored the Big Oh and FFT paragraphs. The page is there to counter the common claim that Wikipedia can never be as accurate and complete as EB; the point of the page is that EB is neither as accurate nor as complete as people make it out to be. The first sentence of the article explains that goal.

Pointing out incompletenesses in EB is not hypocritical: nobody claims Wikipedia to be complete, but many people think EB is complete. --AxelBoldt

I suppose so, and apologize for simply making the remove instead of suggesting it first. Wikipedia is nowhere near as complete as Britannica, and omissions in the second don't serve to prove that it could be, so the above doesn't help with the goal stated. And I think it is a double standard to point out the Britannica is incomplete, and then turn around and say it doesn't matter that Wikipedia is or isn't. But hypocritical was definitely poor word choice.

But who says that it doesn't matter that Wikipedia is incomplete? I think everybody wants it to be more complete, and also everybody acknowledges that EB is vastly more accurate and complete than Wikipedia. So ultimately, this page is just propaganda, showing the tiny little specks where Wikipedia is better than EB and ignoring the vast ocean of inadequacies. I'll make that clearer in the first paragraph. --AxelBoldt

The entry about "transfinite number" in EB claims that aleph-one is the cardinality of the real numbers. This is in fact neither provable nor disprovable; see cardinality and continuum hypothesis for the full story.

Oh, they're just being up-to-date: see Chris Freiling, "Axioms of Symmetry: Throwing Darts at the Real Number Line," Journal of Symbolic Logic Vol. 51, Iss. 1, pp 190-200 for an argument against CH. Remember, the continuum hypothesis is independent of the ZFC axioms; but does that mean it has no truth-value, or just that the axiomatization is less than perfect?

What Freiling presents is an argument against CH, and a rather handwaving one at that. There are similar arguments against the axiom of choice. Nobody accepts them as proofs, and I'm sure Freiling himself wouldn't sell his argument as proof. For a proof, you need a system of axioms and a set of inference rules. Furthermore, EB can hardly be said to be "up to date" since Freiling argues against CH and EB claims that CH is true. --AxelBoldt

For a formal proof, yes, you need axioms & defined rules of inference; but mathematical proofs needn't be formal. It's true, of course, that Freiling gives an argument, and he makes sure to call it that; and you're right about EB--I was just poking fun (& misreading the page). --AF ("What, me pay attention?")