This sort of thing just strikes me as so pointless. How could one possibly decide? Obviously, if we're talking about new axioms that aren't provable from the old ones, you can't use those old ones to settle the matter. The axioms under discussion don't really affect normal math so you can't appeal to their usefulness there (unlike with, say, the axiom of choice, which famously was controversial for a while). And I don't think people can really have much of a claim to have any good intuition for such things, or else we'd have had such an axiom a long time ago. My bet is neither of these will become widely accepted.
That said, if I had to bet on one of the two becoming accepted, I'd go with "V = ultimate L". Why? Because mathematics already has an axiom roughly analogous, that doesn't actually affect normal mathematics but prunes away sets we don't need so people don't waste their time with them. I'm talking, of course, about the axiom of foundation. It's like, why is that even in ZFC? Ordinary math doesn't actually care about the material content of sets; it really affects nothing. The only thing it does, is that, well, without it, non-well-founded sets are something of a problem, in that they could exist, or they could not, and you can't really prove much about them either way. There are non-well-founded set theories that replace the axiom of foundation with an axiom of antifoundation, allowing you to actually prove things about non-well-founded sets -- Peter Aczel wrote a good book on the subject -- but if you just take it out and don't put anything in to replace it, then, well, you just can't really say anything about the matter. The axiom of foundation prunes these irrelevant sets away and tells you to focus on actual math rather than this mystery that's not really much of a mystery at all. So I'd bet on another axiom along those lines becoming accepted, rather than one that does the opposite. After all, if we want to take the opposite perspective that anything that can exist ought to... well, then why not non-well-founded sets, too?
(Note: Not a logician, my understanding of the two axioms being discussed is based pretty much entirely on this article, so the analogy I claim may not actually work as well as I think it does.)
They speak of the Philsophy of Mathematics. I typed a lot out and chose to delete it. In short, it is considered the highest order among Mathematicians. However, us Applied Mathematicians just get on with the real work and have pretty much figured out that they are insane.
It's a bit like true random and infinity. It's best not to think about it to much and only useful for post doc bar conversations or impressing stoned Econ undergrads of your preferred gender. Even then, it's best not to think about it too much.
> They speak of the Philsophy of Mathematics. In short, it is considered the highest order among Mathematicians.
Speaking as a pure mathematician: It's really not.
When I say this stuff is pointless, that it doesn't affect ordinary mathematics, I'm not just talking applied math; I'm talking, well, all of ordinary mathematics, pretty much. Number theory, combinatorics... every now and then a question comes up whose answer depends on the continuum hypothesis admittedly but it's seriously rare (and on those few occasions when it has happened, I think often someone comes along and says that this means that you were really asking the wrong question in the first palce).
I mean, OK, there are people such as Harvey Friedman who work out arithmetic consequences of such assumptions. But the statements they come up with are seriously contrived, not the sort of thing ordinary mathematicians care about, even if the statements no longer involve infinities.
Hell I work with infinities sometimes and I don't care about this stuff! Of course, I'm not a proper logician or set theorist. But doing computations with ordinals and well partial orders is fun... and the answers you get never end up depending on this sort of stuff. :P
Infinity is a rubbish idea, by the way. I've given it much thought and concluded it is best to just not think about it. Cantor was a lunatic, and I'm not entirely certain that infinity isn't a causal factor.
Don't get me started. It doesn't end well. ;-)
Eventually I go off on the absurdity of 'true random' and our presumption to know the differences between it and unpredictability. It's hubris and madness, all the way down.
They really should stop teaching mathematics by rote. If you're familiar with my posting history, I very seldom say 'should.'
Eh, everyone but the set theorists ignores the set theorists at this point. I think some exciting things happened in the 60's with forcing, but a lot of the subsequent developments seem pretty contrived.
People who get confused about that equality are in actuality confused about what a real number is. In particular they probably think that real numbers are digit expansions, and they have probably never seen a construction of the reals. You can show this person a proof, and you might get them to shut up about it, but they’ll never believe it on an intuitive level until you show them what real numbers actually are and what their relationship to their (non-unique!) digital representation is.
I believe the confusion is spread across several layers. For those in the know, infinite decimal expansion is syntactic sugar for a limit over a particular infinite series of sums, but this convention sidesteps how people are taught syntactic manipulation and how it's intetpteted in a particular algebra.
0.999... is presented as a term, that can be syntactically replaced with an actual infinite decimal expansion, upon which the normal syntactic opetations can be applied, specifically "=" as in "0.999...=1".
This is ofcourse impossible, because even if the rules are valid you would be applying them forever. This is the apparent absudity of the equation. In defence of people who hold this intuition: it's a valid concern in many branches of math(discrete, constructive, infinitesimal etc.), just not in informal expositions of the properties of real numbers.
If you could step up to the board and actually expand the ... for us, we'd be much obliged.
You can keep writing 9 forever and you will never reach a whole number. But, it does reach a whole number and there is proof! So, they made a rule. The rule is, of course, silly. Infinity itself is silly. Not even atoms are infinite. Not even atoms can be infinity divisible.
It's a horrible idea, really. (I have karma to burn.)
By the way, it's a damned good thing infinity doesnt exist. If you had a tape measure that was infinitely long and folded it in half, we aren't really sure what would happen.
Hell, I wasn't using that karma for anything better.
While I'm here... When you're done finding an infinite tape measure, I'd like it if you could bring me -1 apple. No, not a slip of paper representing an owed apple, but -1 apple.
Math and reality are only loosely coupled. Mathematics is a language. It can tell lies. 1 + 1 = 74 See? That's pretty obviously a lie. It can tell stories, physicists are pretty fond of using it for that. It can be so very descriptive and elegant. It can be so deceptive, as we see with the 'well, let's just let that be represented by 9/10' silliness.
It's pretty horrible, really. Like I said, it's best not to think about it too much. It's kind of contrived in a few areas.
> If you could step up to the board and actually expand the ... for us, we'd be much obliged.
It seems you don't know what a sequence is. A sequence is just a function whose codomain is the natural numbers, i.e. you give any natural number you choose, and I give you the corresponding digit. I don't need to 'expand' the ellipses to define that function, any more than I have to write down all perfect squares in order to define what a perfect square is.
Seriously, this is basic mathematics. You should study it before publicly embarrassing yourself by calling mathematicians 'liars'.
I do understand it. In fact, I understand it quite well. I can give you the textbook definitions. I've studied them completely. I hold a Ph.D., you know. You are accepting rote still, as opposed to concerning yourself with the Philosophy of Mathematics.
The ... represents an infinitely repeating numeral, does it not? It is a shorter way of writing 0.999999999999 (into infinity).
Now, you can do that forever and you will never reach a whole number. Ever... Go ahead, try it. I'll wait.
So, we've all happily agreed on some very contrived rules that you'll quote verbatim and gleefully declare to be basic mathematics and I the embarrassment. You'll commit no more thought to the process and, as I've already stated, that's a very good choice.
I, on the other hand, would bet KGIII's Ph.D. is in math. I have a Ph.D. in math, too, and I've met, shall we say, a wide variety of fellow Ph.D. holders.
You have given me one of the best chuckles I've had in weeks. Thanks! Err... If you really wanted, you could probably use a search engine and figure out who I am. I actually am a mathematician. But, you can keep your wager.
Of course I do. It's rather contrived, now isn't it?
We've effectively just said, 'Well, bugger it. We don't have to expand that.' We've all agreed it equals one. That's kind of silly.
I can quote the rules, the reasons, and even helped to author the early versions of the Wikipedia article on the subject. I understand the textbooks, and they are silly.
I haven't found the history of when this was discovered, so I have to speculate. I think it went a bit like this...
Bob, that's his name, was smoking opium one day and pondering mathematics. On a hunch, he sat down at his desk and, between nods, he discovered that 0.999... = 1.
This, of course, is absurd. After all, anyone would notice that the deception lies in the conversion to a fraction. But, we can't have that.
Bob chose to keep it a secret.
Except Bob was quite an opiate addict and only made it about a week before he started telling anyone who would listen.
Unfortunately, so many people heard about it that the higher order of mathematicians decided they couldn't just assassinate Bob and, after all, he was right - and he could prove it!
So, they thought long and hard. How could this be made true? How could this still be right and the order still be maintained?
Eventually, a bright young neophyte named Hank piped up, 'Wait! I have it! Let's just imply a limit!'
They all cheered and headed to the bar to buy Hank a round of drinks.
Something like that. ;-)
In an earlier comment you said you didn't have to expand it. You're right. You don't have to, but you can. There is no rule that prevents you from stepping up to the board and writing 9s until the heat death of the universe, only an imposition made to keep order. Kinda silly, isn't it?
As I said above, mathematics is a language. You can say anything you want. Some of it is trivially proven false, though it is still useful.
Regarding -1 apple, you can argue in the same way that any concept "doesn't exist" if you refuse to recognize a correspondence between it and its referent. What makes "-1 apple" less real than "1 apple"? I could argue, for example, that the conceptual boundary between an apple and the tree it came from cannot be guaranteed to be the same as its actual physical point of separation, and thus any real apple is more like 1.000001 or 0.99999 apples, so there is no such thing as "1 apple" in reality.
Oh! I like that. I will counter that I can hold what I believe to be an apple, or close enough to being a single apple. I can not do so with a -1 apple. Horrible, really. ;-)
Oh, well then could you point me to a formal definition of the syntax and semantics for the unary "..." operator then? What is then domain and codomain, for example?
Rightfully, they just voted me down and refuse to think about it. I can't blame them, really. They were probably taught mathematics by rote and maybe stopped at arithmetic.
Non-well-founded sets seem to be grammars from a CS point of view. As soon as you admit the Kleene-star '*' operator you have potential infinity. If you allow circular grammar definitions, then you are specifying non-well-founded-sets.
When parsing finite structures, it just so happens that the parsing terminates, but it's easy to extend to open-ended (potentially infinite) streams. There are even type system constructs and language expressions for them, but only executable in lazy languages, or systems open to external input.
CS has also made theoretical progress, perhaps the most surprising is differentiating regular expressions and other regular datatypes:
That said, if I had to bet on one of the two becoming accepted, I'd go with "V = ultimate L". Why? Because mathematics already has an axiom roughly analogous, that doesn't actually affect normal mathematics but prunes away sets we don't need so people don't waste their time with them. I'm talking, of course, about the axiom of foundation. It's like, why is that even in ZFC? Ordinary math doesn't actually care about the material content of sets; it really affects nothing. The only thing it does, is that, well, without it, non-well-founded sets are something of a problem, in that they could exist, or they could not, and you can't really prove much about them either way. There are non-well-founded set theories that replace the axiom of foundation with an axiom of antifoundation, allowing you to actually prove things about non-well-founded sets -- Peter Aczel wrote a good book on the subject -- but if you just take it out and don't put anything in to replace it, then, well, you just can't really say anything about the matter. The axiom of foundation prunes these irrelevant sets away and tells you to focus on actual math rather than this mystery that's not really much of a mystery at all. So I'd bet on another axiom along those lines becoming accepted, rather than one that does the opposite. After all, if we want to take the opposite perspective that anything that can exist ought to... well, then why not non-well-founded sets, too?
(Note: Not a logician, my understanding of the two axioms being discussed is based pretty much entirely on this article, so the analogy I claim may not actually work as well as I think it does.)