Not to you, no.
ETA: Sigh. Before that gets taken personally, I mean it’s clear you’re not the audience for that joke.
Same…I mean, I couldn’t make one, and I would have only the vaguest idea what they were if you were talking about them, but the name and the names of the variables in this one including the word ‘cost’ gave enough information about it for it to click.
Personally, I’m quite comfortable with taking the negation of the Banach-Tarski statement as an axiom, and thereby disproving the Axiom of Choice. I’ve never liked that axiom to begin with. “Given an infinite set, I can always choose an element from it.” “OK, which element did you choose?” “Um, I can’t answer that. But rest assured, I chose one of them. I just don’t know which one it was.”
Aw, but then you can’t make my favorite anagram of ‘Banach-Tarski’!
Banach-Tarski Banach-Tarski
LOL! I like this.
Are you equally comfortable with negating all of the equivalents to the Axiom of Choice?
The way I see it, set theory with the axiom of choice and set theory without it are just different abstractions. It’s not that one is true and the other isn’t; they describe different things, just as arithmetic with and without square roots of -1 describe different things. If what one intends to model is processes describable by some expressible rule or such things, then the abstraction which lacks the axiom of choice is the correct one [as well, this arises as the internal logic of most categories, as seen in topos theory; thus, we can make a theorem of it that, e.g., the computable universe, the continuous universe, etc., suitably formalized, lack the general axiom of choice]. If what one intends to model is some kind of idealized limit of unboundedly potentially many random choices, or such things, then the abstraction which incorporates the axiom of choice is the correct one (“If every islander has at least some hairs, and every islander has to pluck precisely a single hair from their head to avoid their extinction, then surely they’ll find a way to save themselves”). One may even find cause to use both abstractions at the same time, and the relations between them, depending on what one is doing.
Not sure why you put “infinite” in here, but ok. Note, though, that choosing an element from a single inhabited set at a time is easy; at least, it’s easy to prove that choice functions exist, in the sense that since there exist elements in the set, there exist functions which pick out single elements from that set [it’s a trivial tautology]. It can’t be denied that every set which has elements has ways of choosing elements from it, since the former and the latter mean the same thing.
Most of the discussion about the axiom of choice is about choosing elements from multiple sets simultaneously; that is, turning “for all x, there exists a y such that R(x, y)” into “there exists a function f such that for all x it’s the case that R(x, f(x))”.
Put another way, the axiom of choice states that every surjective function has a right inverse; if f : X -> Y is surjective, then there is some g : Y -> X such that fg : Y -> Y is the identity on Y. Think of the elements of Y as various questions which need to be answered, and the preimage of such an element under f as being the collection of possible correct answers to that question. The surjectivity of f is the condition that every question has at least one correct answer; finding a g, then, would amount to finding a way of simultaneously selecting a correct answer for every question.
In the particular case where one is only asked to select elements from a single set at a time, then Y is the 1 element set. But it’s not at all in doubt that every surjective function into the 1 element set has a right inverse [each element of X induces a corresponding right inverse, and because X will have at least some elements, there will therefore be at least some right inverses]. The only doubt is whether surjective functions into larger sets always have right inverses.
That having been said, one can draw a distinction between the so-called “internal” and “external” axioms of choice for a set theory. A set theory satisfies the internal axiom of choice if it proves that every surjection has a right inverse. It satisfies the external axiom of choice if for every definable function f which is provably surjective, there is a definable function g which is provably a right inverse to f. The concern about expressibly picking elements from a single set at a time would be a concern about the external axiom of choice, rather than the internal axiom of choice. However, the external axiom of choice is a question in the metatheory of a set theory, rather than in the set theory itself. Almost all discussion of the axiom of choice without a further qualifier is a discussion in some set theory about its internal axiom of choice. (Note that, for example, it is much, much easier (trivial, even) to construct a set theory in which the external axiom of choice is false than to construct one in which the internal axiom of choice is false)
(The reason being that one is more often doing set theory (talking about sets and functions) than doing metatheory of set theory (talking about definitions and proofs). In both cases, one of course is still using definitions and proofs, but in the former case, they’re definitions of and proofs about sets and functions, while in the latter case, they’re definitions of and proofs about definitions of and proofs about sets and functions.)
Apparantly today’s punchline is
“is it ‘soh cah toa’
or ‘coh sah toa’”
and I don’t understand what that means.
It means this.
And the joke is that since he can’t remember which way around it is Steve’s calculations may not be all that reliable and thus the Earth may be destroyed.
I’ve always thought it was a terrible mnemonic. What makes that string of gibberish any easier to memorize(/harder to get backwards) than what it represents?
From Explain xkcd:
Aaaand, beaten to the punch. Too bad, posting anyway.
Silly Old Harry Caught A Herring Trawling Off America.
Much easier to remember.
I learned it as:
Oscar Had A Hit Of Acid
…with O/H A/H and O/A representing sine, cosine and tangent. I guess the order of the functions had to be memorized, but it never occurred to me to think of them as “cosine, tangent, sine” or in any other order.
Soak a toe(uh) makes more sense than coe sah toe(uh).
By the way, quiet nerds burp only near school.
Gee, I never learned that mnemonic. I just memorized
sine = opposite over hypotenuse
cosine = adjacent over hypotenuse
tangent = opposite over adjacent
Zombie, etc.
We were given a transposed version, The Cat Sat On An Orange And Had Hysterics . But I’ve just realised what a poor mnemonic that is, too - it’s memorable, but you have to write the whole thing out before you can work out what Tangent, Cosine, and Sine are.