Overall, this video provides a good intro to some important math ideas:
Here are some random thoughts on the video:
I don’t think principles like undecidability necessarily imply a “hole” or “flaw” in mathematics. It only appears to be a “hole” if you’re assuming something should be there, like decidability. But that’s based on your assumption that it should exist in the first place.
The video soon turns to Cantor, giving me the perfect opportunity to finally rant about how foolish he was! (I’ve been intending to do so for some time.)
Cantor’s infinities are a bit of a pet peeve of mine. His foolishness, and the foolishness of those who nod in bedazzled wonder and agreement with his nonsense, stem from a lack of understanding the implications of infinity. Infinite size means no size. That can be confusing, because in this instance “no size” does not mean a size of 0. So how can something have no size without the size being 0? By being infinity. One must stop thinking about infinity as a number, but as a concept parallel to that of number.
So when Cantor asks (at 4:30 in the video): “Are there more natural numbers or more real numbers between 0 and 1?”
Woah, back it up, back it up, beep, beep, beep!
That is a nonsense question. It’s like asking: “What is five divided by green?” By definition, there cannot be “more” or “less” of something that has no amount to begin with. There are infinite natural numbers. There is not an amount of natural numbers, because there are an infinite amount, which means there is no amount. So there cannot be more, or less, or the same. There can be no comparison whatsoever because what you’re trying to compare is the amount, which does not exist.
Does infinity equal infinity? If you’re intending to compare amounts, the question is again meaningless nonsense. Infinity cannot, in this sense, equal or not equal infinity because you cannot compare them like finite amounts.
As the video shows, Cantor goes on to (rather stupidly) compare two lists. His methods are meaningless because his premise (that infinities can be compared) is already flawed. He then finds with his “diagonalization proof” that you can’t logically define a pairing between every natural number and every real number between 0 and 1, and in the depths of his infinite stupidity thinks that this somehow proves that there are “more” real numbers than natural numbers.
Uh, no it doesn’t, Mr Georg without an “e”. All you’ve shown is that you didn’t actually succeed in defining a pairing. You haven’t proven anything about sizes because infinite sets do not have sizes. Whether or not you can rigorously define a pairing (a one-to-one corresponce) implies nothing at all about sizes. It only proves your definition of the pairing to be paradoxical nonsense. You can’t say “let’s assume we’ve paired all natural numbers to real numbers between 0 and 1” and then say “here’s a real number that can’t be in the list!” That just means we didn’t actually pair the sets to begin with!
The crux of the paradox doesn’t lie in the “sizes” of the sets anyway (which don’t exist). It lies in the inability to express all real numbers with finite decimal places in the decimal system. If we take for granted that we could instead express some otherwise undefined real number with an arbitrary symbol (like, gee I don’t know, a natural number), the paradox is completely resolved. There is nothing to “diagonalize” and the one-to-one corresponse is complete. Logic 1, Cantor 0.
At 6:45: “Cantor’s work was just the latest blow to mathematics…” Perhaps more of a blow to mathematical philosophies than to math itself. Aside from being complete nonsense, it had no implications aside from morons thinking “oh wow, different size infinities sure is amazing, derp!” which is about as meaningful as thinking, “oh wow, five sure is colorful, derp!”
At 7:27: “On the one side were the intuitionists who thought that Cantor’s work was nonsense. They were convinced that math was a pure creation of the human mind and that infinities like Cantor’s weren’t real.”
Perhaps, but whether or not an infinity can be “real” is really not the issue with Cantor’s illogic. Also, his lack of logic in this particular area does not necessarily imply inherent weakness with set theory in general.
The video goes on to speak of set theory’s self-reference paradox. It is indeed a paradox, but is by itself really no weakness of set theory anymore than the existence of paradox itself is somehow a weakness of the human mind that conceives of them. In fact, one could say the ability of a system to define a paradox is actually a strength. It’s like trying to make a programming language that doesn’t allow for infinite loops by taking out the ability to have any loops at all.
I really like the video’s explanation of Gödel’s work with using; actually, perhaps because it’s visual and tangible, I think it may be the best explanation I’ve seen!
At 31:34: Haha, what is this artsy-fartsy shot? “Look at my back as I gaze at the sky and ponder the deep thoughts of the world…”
The video ends by circling back to the “hole” in math, which is now defined as not being able to know everything with certainty, which seems a rather imprecise way of summing up undecidability and incompleteness as it takes for granted the meaning of “certainty”. I guess we could say: “Hey, Gödel, if math is incomplete, then your proof is incomplete and therefore not a proof! Hyuck hyuck!”