(Actually, as one minor technical point, The Right Thing to do is to be a little more general and allow sequences which may have some undefined components, as long as they’re fully defined from some point on [since that tail-end “germ” is all that matters for anything in the end]; but that’s no huge change)
So, basically, the curly braces have to be inside the angle brackets, not the other way around. OK, I think I’ve got that, then.
Is this also how one shows the indeterminability of the Continuum Hypothesis? Like, even if the “real” world had no number between Aleph_0 and 2^Aleph_0, in the Fréchet world constructed from that “real” world, there is one? And since the Fréchet world is indistinguishable, from within, from the “real” world, we don’t know whether we’re in the world where the continuum hypothesis is false or the one where it’s true?
Not by this exact same technique, but, at least in part, by a somewhat related refinement:
We can’t use this exact same technique because Fréchet-world is too indistinguishable from Standard-world. Any definable property will hold of a Standard object in Fréchet-world just in case it holds of it in Standard-world (since <E, E, E, …> satisfies P from some point on just in case E satisfies property P). As a special case of this, the continuum hypothesis holds in Fréchet-world just in case it holds in Standard-world. (At least, if you interpret everything Fréchetly. If you mix and match in how you phrase the question (e.g., “Does there exist a Standard-world subset of what Fréchet-world considers to be 2^N such that Standard-world sees a mapping such that…”), then things can change, but this isn’t what we are concerned with when asking whether CH follows from certain axiom systems; our concern is with the possibilities where everything is interpreted in terms of the same world, whatever world that is).
Instead of using Fréchet-world, we’ll use Borel-world instead (obviously, none of this is standard terminology, but standard terminology is boring). There are analogies between these two constructions, but also important differences. To wit:
[ul]
[li]Fréchet-world is generated by adjoining a “random” natural number to Standard-world, in such a way as that you can no longer remember what the borders of Standard-world were. By the “random” natural number, I mean <0, 1, 2, 3, …>; it satisfies every (Standard) property that holds for “almost all” natural numbers [i.e., all but finitely many]. In a suitable sense, everything else about Fréchet-world follows from this.[/li][li]Borel-world, on the other hand, is generated by adjoining a “random” bunch of reals to Standard-world, but in such a way as that you can still see the outlines of Standard-world. By a “random” bunch of reals, I mean they have every (Standard) property that holds for “almost all” bunches of reals [i.e., with probability 100%]. In a suitable sense, everything else about Borel-world follows from this.[/li][/ul]
How exactly do you adjoin these random reals without destroying the ability to see Standard-world? I’ll leave the details for another thread, another time. For now, I’ll just outline how this lets us show that you can’t prove CH from certain axiom systems:
Not all axioms which hold true in Standard-world will still hold true in Borel-world (and, indeed, this is the point, since we want to at least be able to change CH’s truth value), but lots of foundational theories of various nice types will carry over successfully (that is, their holding in Standard-world will guarantee that they still hold in Borel-world); e.g., this holds for ZFC, as well as for topos-theoretic foundations. So if we assume that some such a theory holds of Standard-world, and can prove from this that CH fails in Borel-world, then we’ll have shown that said theory cannot consistently prove CH. I’ll not be very specific about what base theory we’re using, since the independence result here applies in some generality; the only notable point is that we’ll take it to include Choice.
Let N be a countable set, A be an uncountable set and let B be one of even larger cardinality (all in Standard-world). I didn’t say above how many random reals to adjoin for Borel-world; use at least B many. Now, all B of these random reals are distinct (since the probability that any two independent reals are distinct is 100%, which we’ve taken as a guarantee), so in Borel-world, there are at least B many reals [for Borel-world to agree to this, it’s important that it can actually see the Standard map from B to the terms representing the corresponding random reals and how it must be injective]. And since N <= A <= B in Standard-world, we’ll have N <= A <= B <= Reals in Borel-world.
It’s pretty easy to show that, since Standard-world thinks N is countable, so does Borel-world. So all that’s left for us to do is to show that, since A is in fact strictly inbetween N and B in Standard-world, it remains strictly inbetween them in Borel-world. For this, we invoke the fact that any set of disjoint events can include only countably many of non-zero probability, and use it in some simple combinatorial fiddling to show that strict cardinal inequalities in Standard-world are preserved by Borel-world. And that finishes the proof.
Next, let me tell you about Lava-world…
[That just demonstrates how to show that some theory doesn’t prove CH. To show that it doesn’t prove the negation of CH either, one generally goes another way; instead of augmenting the universe of Standard-world, one diminishes it, tossing out everything which can’t be generated in a certain restricted way. Again, for suitable starting theories, you can show that what you’re left with still satisfies the theory, plus CH will hold in it. But the details of this are far afield and not particularly analogous with Fréchet-world]
Actually, no, that doesn’t follow. It proves it inductively, since “finite” has a technical definition that we are using (and which forms the basis for our assumptions. Rudy Rucker is being stupid, for some reason. The act of counting 7 steps is inherently finite, since it is not infinite in any relevant scale.
But which definition of “finite”? As Hari Seldon pointed out, there are several definitions of “finite” which are not identical.
Coy, probably; stupid, I don’t think so. Remember, this was a “proof” in a popular text for introducing the masses to transfinite cardinals. If Rucky had to submit the proof to a peer-reviewed journal, you can bet it would look very different.
That is exactly what we are requested to prove.
If you are offering an inductive proof similar to Hari Seldon’s, then I agree, that is a valid proof and side-steps the issue of circularity.
Indistinguishable – thanks for taking the time to write up that explanation on Fréchet-world. Although I’m familiar with vectors/tensors etc., that was a little above my level. I’ve read through your explanation several times, and looked up the relevant stuff in Wikipedia, but I’m going to have to go through a little more to absorb what you wrote. Fascinating stuff, though; makes me wish I were back at university, studying math again.
By the way, I got a response from Professor Rucker. About my proof, he merely said “that could work”.
Anyway…I’m now satisfied that 7 is finite (what a relief!), that mine and SaintCad’s proof proves it for a particular set-theoretic construction of the naturals, and that Hari Seldon’s inductive proof based on the Peano axioms applies to any reasonable construction of the naturals.
It’s interesting that an apparently obvious and even silly question has led to a lengthy and informative (for me, at least) debate.
My proof is the damn dictionary. Frankly, and I’m sure you mean well, but your OP is the least intellectual form of sophistry imagimable. No one here has proven anything, except how to parse and read a grammatically correct sentence. I do not criticize such much as make a complaint of obtuseness: you are obviously way overthinking this.
smiling bandit, please quote the context – the words you removed from my quote changes the meaning:
[emphasis added]
The dictionary doesn’t prove or disprove anything about mathematics. Are you saying that “7 is finite because the dictionary says so”?
This is GQ; I politely request that you refrain from personal attacks.
Some mathematical big hitters have posted in this thread, with (what I considered) interesting points of view on the OP. No, I’m not one of them – I can barely swing a bat by comparison, and that’s what I love about the Dope. Are you really willing to dismiss what they have written? Isn’t it possible that these people have seen subtleties to the question that you haven’t?
Perhaps by your standards; some of us enjoy pondering fundamental questions of mathematics, and I often find that even if they are “obvious”, just discussing and pondering them can lead you to interesting places. The Axiom of Choice is “obvious”, yet accepting or rejecting it has deep implications that I am nowhere near fathoming.
If you are as frustrated as you appear to be about this discussion, perhaps a thread entitled “How would you prove 7 is finite?” is not for you.