The crankery in question is David McGoveran's paper Interval Arguments: Two Refutations of Cantor's 1874 and 1878 1 Arguments found here or here (archived pdf).

Just from the title, it shouldn't be hard to guess that there's badmath inside. Let's take a look. First, the bad definitions and obviously incorrect theorems.

The reals, rationals, and integers each satisfy Dedekind’s definition of infinite set: they each contain infinite proper subsets.

That's not quite the definition of Dedekind infinite. It'd be a bit odd if his definition of infinite required proving something else is infinite, though that might work as a sort of coinductive definition. The correct definition of a Dedekind infinite set is that it has a proper subset that is in bijection with the whole set. The reals, rationals and integers are, of course, infinite by this definition.

Deny the Hypothetical: From the contradiction and law of the excluded middle, conclude the hypothetical is false: There exists an η not in L and so L cannot, as was assumed, include all the members of I₀.

More of a common misconception, but this isn't the law of excluded middle. It's instead either the law of non-contradiction plus the principle of explosion (in classical logic) or simply the definition of negation in logics without the law of excluded middle.

Since every continuous subinterval of the positive reals [0,∞] has the same cardinality as the reals by definition, the argument’s conclusions will apply to the entirety of the reals.

Definitely not true by definition. Any open interval has any easy bijection with the positive reals (0, ∞), but closed intervals and half-open intervals necessitate something trickier to get a bijection with (0, ∞) like Cantor-Schoeder-Bernstein. It may be just a notation thing, but I also find it odd that the positive real numbers are denoted by "[0, ∞]", which usually denotes an interval containing (among other things) 0 (not positive unless tu comprennes ça) and ∞ (not a real number).

Dasgupta’s real construction procedure depends on the Nested Interval Theorem [6, p. 61], which states that every infinite sequence of nested intervals identifies a unique real η.

This is missing two very-necessary conditions regarding the size of the intervals and whether they include endpoints. The citation Dasgupta, A. Set Theory: With an Introduction to Real Point Sets has the correct statement.

Next up, let's look at the substantial mistakes that lead to an incorrect conclusion.

Note that both 𝔸 and ℚ are countable, which guarantees they can be included in list L.

[𝔸 refers to the set of real algebraic numbers]. While this isn't too bad on the face of it, this does betray a line of thinking that other Cantor cranks like to follow: that the purported list of real numbers can be modified after the fact. The author makes this mistake more explicitly later on.

if need be, Kronecker can add to his list any specific real η that Cantor specifically identifies, which Cantor will then have to exclude by defining a next nested interval.

In this game that Cantor and Kronecker are playing, Cantor doesn't specify a real not on the list until after the game is over. By then, it's too late to change anything.

Even as n → ∞, there is no finite cardinality k < ∞ such that |Rₙ| → k: The sequence of cardinalities of |Rₙ| is diverges for as n → ∞, since |Rₙ| = ∞ for all n. Treating ℓₙ ∉ Iₙ as having any relevance to the entirety of L is erroneous. It can be meaningful only if one treats the interval sequence I as a completed infinite set, a rather dubious enterprise since it entails showing that Rn is empty as n → ∞.

[L is the purported list of real numbers, Rₙ here is the remainder of the list that hasn't been inspected after n steps] Here the author makes the mistake of conflating a limit of cardinalities of nested sets with the cardinality of the intersection of the sets. The intersection of Rₙ is the set of elements of L that are never inspected for any finite n. But there are no such elements, so the cardinality of the intersection is 0.

A simpler case is the intersection of the sets of integers Kₙ = [n, ∞). Claiming that there might still be something in one of these sets in the "limit" n → ∞ is the same as claiming that there is an integer that is larger than every finite integer.

Rational Interval Theorem: Given any infinite sequence L of distinct rational numbers (all belonging to interval I₀ over the rationals ℚ) and progressing according to some law, there exists a subinterval of I₀ containing at least one rational number η such that η ∈ I₀ and η ∉ L.

To be fair, this is supposed to be an absurd theorem. The bad part is in the proof. Everything is spelled out nicely, but it uses (essentially) our old friend the nested interval theorem from before. But remember that even in the author's incomplete statement of that theorem we have a unique real number in the intersection of the intervals. But the theorem above says rational η. How does the author deal with that?

Whereas Cantor 1874 and 1878 both rely on the usual formulation of the Bolzano-Weierstrass Theorem (as it pertains to the continuum) to ensure that the limits α_limit and β_limit exist, the everywhere dense and countable properties of the rationals ensure that these monotonic and bounded sequences have a limit over the rationals.

Hoo boy. So the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, ... has a rational limit? Monotonic: check. Bounded: check. Rational limit: no check.

There's also a lot of confusion about the limit interval, with sentences like

If η = α_limit = β_limit then I_limit, = [α_limit, β_limit] is empty and closed.

Again, this might be a notation thing, but here the interval is explicitly described as closed, meaning including its endpoints. I mean, I guess the empty set is closed, so there's still a loophole here.

The concluding remarks go into some points regarding computability and some things that are handled by the axiom of dependent choice. While rejecting such axioms is perfectly valid, any assumptions should be mentioned up front. Cantor was not working in a system where dependent choice is explicitly rejected, so it would be unfair to criticize his proof on those grounds. This paper raises some interesting questions, but doesn't really make any progress on them. It's obvious to constructivists where Cantor's proof uses LEM and dependent choice, so if that's the only criticism, why write a paper? Why not write about how to avoid them, or prove that you can't?

Yesterday I stumbled over a post in /r/confidentlyincorrect.

I should have known better than trying to engage in the comments but I did and now my hair is white. While it was frustrating to argue with people who cannot be convinced no matter how rigorous your proof is, the more infuriating thing was this article on medium that was cited multiple times in the comments:

https://medium.com/@kenahlstrom/proof-that-99999-is-not-equal-to-1-5672e7dd58ce

This thing is so full of bad math (starting with the claim that 1/3 does not equal 0.333...) it really made me mad. What's even more annoying is that if you google something like "why does 0.999 equal 1" this medium post is among the top results.

I wish we were able to purge bad math posts from the internet because it's really frustrating when people spread this bullshit everywhere.

This isn't interesting bad math. After all, P vs NP isn't proven, yet most computer scientists assume P≠NP when doing their work or teaching classes. I wouldn't post a computer science lecture here because the instructor said something like "this is NP-complete so it's hard to solve". I think that Pi not being proven to be normal falls in the same category. Data from its digits suggests that it is normal, and almost all numbers are normal, so it isn't unreasonable to assume Pi is normal, and it isn't bad math to assume that, just like how it isn't bad math to assume P≠NP.

https://www.youtube.com/watch?v=Merc32fl_Rs&t=559s&ab_channel=150yearsofdelusionsinmathematics

R4: Dr Beomseok Jeon, PhD and professor of neurology at Seoul National University has started a youtube channel called "150 years of delusions in mathematics". So far he has made 4 videos (hopefully more to come soon) where he claims he will prove modern mathematics is inconsistent, using limits and set theory.

In the 2nd video of the series (linked above), he attempts to prove lim(1/3^n) > 0. He first assumes lim(1/3^n) = 0, and says "if we were not to doublespeak, this indicates a natural number n such that 1/3^n = 0". But this is a contradiction, so he concludes lim(1/3^n) > 0, and therefore lim(1/n) > 0.

This is not correct, lim(1/3^n) = 0 only indicates for any ε > 0 there exists an N such that for any n > N: 1/3^n < ε.

Edit: Title should be: The √4=±2 saga

Recently on r/mathmemes a meme was posted about how√4=±2 is wrong. And the comments were flooded with people not knowing the difference between a square root and the principle square root (i.e. √x)

Then the meme was posted on r/PeterExplainsTheJoke. And reposted again on r/mathmemes. More memes were posted about how ridiculous the comments got in these posts [1] [2] [3] [4] [5] (this is just a few of them, there are more).

The comments are filled with people claiming √4=±2 using reasons such as "multivalued functions exists" (without justification how they work), "something, something complex analysis", "x ↦ √x doesn't have to be a function", "math teachers are liars", "it's arbitrary that the principle root is positive", and a lot more technical jargon being used in bad arguments.

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