The consequences of this little harmless looking statement are deep enough that I worry that the majority will think that it is provably wrong.
Simple statement:
- "You can pick a real number at random between 0 and 1, so that any number is as likely as any other."
more colloquially, (Freiling):
- "You can throw a dart at the unit square."
more computer-scienc-y:
- "The process of coin flipping to determine the binary digits of a real number converges to a unique well defined real number answer in the limit of infinitely many throws".
The interpretation of this statement is not that the probability distribution of the result is a well defined function, nor that statements about whether this number is in this or that Borel set can be assigned a probability--- both these assertions are true and boring. The assertion above is that a real number "x" produced in this way actually exists as an element of the mathematical universe, and every question you can ask of it, including "does x belong to this arbitrary subset S of [0,1]" gets a well defined yes or no answer in the limit. If you believe this assertion is self-evidently true, as I do, beware the implications!
- The continuum hypothesis is false. (Sierpinsky,Freiling)
For contradiction, well order [0,1] with order type aleph-1, then choose two numbers x,y at random in [0,1]. What is the probability that $yle x$ in the well-ordering? Since the set ${ z|zle y} $ is countable for any y, the answer is 0. The same thing works whenever sets of cardinality less than the continuum always have zero Lebesgue measure.
- Every subset of [0,1] has well-defined Lebesgue measure. (Solovay, more or less)
Make a countable list of independent numbers $x_i$, and ask for each one whether the number is in the set S or not. The fraction of random picks which land in S will define the Lebesgue measure of S. In more detail, if you write down a "1" every time $x_i$ is in S, and write down a zero when $x_i$ is not in S, then the number of ones divided by the number of throws converges to a unique real number, which defines the Lebesgue measure of S.
In this forum, somewhere or other, someone had the idea that this process will not converge for sets S which are not measurable, alternating between long strings of "0"s and long strings of "1"s in such a way that it will not have an average frequency of 1's. This is impossible, because the picks are independent. That means that any permutation of the 0's and 1's is as likely as any other. If you have a long string of N zeros and ones, the only permutation invariant of these bits is the number of ones. Any segregation of zeros or ones that has oscillating mean has less than epsilon probability whenever the mean number of ones after M throws, deviates by more than a few times $sqrt{ln epsilon}/sqrt{N}$ from the mean established by the first N throws.
It is astonishing to me that someone here simultaneously holds in their head the two ideas: "there exists a non measurable subset of [0,1]" and "you can choose a real number at random between [0,1]". The negation of the first statement is the precise statement of the second.
(Solovay defined this stuff precisely, but did not accept the resulting model as true. Others take the axiom of determinacy, thereby establishing that all subsets of R are measurable and that choice fails for the continuum, but determinacy is a stronger statement than "you can pick in [0,1]".)
- The axiom of choice fails, already for sets of size the continuum.
Since the axiom of choice easily gives a non-measurable set.
- The continuum has no well order.
This is because you could then do choice on the reals. So the first bullet on this list should really be rephrased as "the continuum hypothesis is just a stupid question".
- Sorry, you can NOT cut up a grape and rearrange the pieces so that it is bigger than the sun.
Simply because if you put a grape next to the sun, and pick a random point in a big box that surrounds both, the probability that the random point lands in the grape is less than the probability that it lands in the sun. The Lebesgue measure of the pieces is well defined, and invariant under translations and rotations, so it never amounts to more than the measure of the grape.
- The reals which are in "L", the Godel constructable universe, have measure zero.
When the axiom of Choice holds for all elements of the powerset of Z (i.e. R), then the pea can be split up and rearranged to make the sun. The axiom of choice holds in L, so that the Godel constructible L-points in the pea can be cut up and rotated and translated to fit over the L-points of the sun. This means that these points make a measure zero set, both in the pea and in the sun, when considered as a sub-collection of the real numbers which admit random picks.
To understand the Godel constructible universe, and choice, I will pretend that the phrase "Godel constructible" simply means "computable." This is a bald-faced lie. the Godel constructible universe contains many non-computable numbers, but they all resemble computable numbers, in that they are defined by a process which takes an ordinal number of steps and at each step uses only text sentences of ZF acting on previously defined objects. If you replace ordinal by "omega" and "text sentences acting on previously defined elements" by "arithmetical operations defined on previously defined memory", you get computable as opposed to Godel-constructible. To well order the Godel universe, you just order the objects constructed at each ordinal step by alphabetical order and ordinal birthday. To well order the computable reals, you just order their shortest program alphabetically (like the well-ordering of the Godel-universe, this ordering is explicitly definable, but not computable).
- That stupid hat trick doesn't work in the random-pick real numbers
There is a recently popularized puzzle: A demon puts a hat, either red or green, on the head of a countably infinite number of people. Each person sees everyone else's hat, and is told to simultaneously guess the color on their heads. If infinitely many get this wrong, everybody loses. If only finitely many people get the answer wrong, everybody wins.
When the demon picks the hat color randomly on each person's head, they lose. Each person has 50% chance of getting their hat right. End of story. Nothing more to say. Really. This is why set theory has nothing to do with weather prediction.
- The stupid hat trick does work over the computable reals, but is intuitive.
If the demon is forced to place hats according to a fixed definite computer program, there are only countably many different programs, the demon must pick a program, and stick with it. Then it is reasonable that each person can figure out the program used from the infinite answers at their disposal, up to a finite number of errors.
Supposing the people are provided just with some halting oracles and a prearranged agreement regarding computer programs. They do not need a choice function on the continuum. The people see the other hats, and they test the computer programs one by one, in lexicographic order, until they find the shortest program consistent with what they see. They then go through all the programs again, until they find the shortest program on integers which will give be only different from what they see in finitely many places (this requires a stronger oracle, but it still doesn't require a choice function). Then they answer with the value of this program at their own position.
(more precisely, to see everyone else's hat means that the demon provides a program which will give the value of everyone elses hat. You use the halting oracle to test whether each program successively will answer correctly on everyone else's hat, until you find the shortest program that does so.)
This version also has application to weather prediction: by studying the weather long enough, you can guess that it is obeying the Navier Stokes equations. Then you can simulate these equations to predict the weather. Come to think of it, this is exactly what we humans did.
- The stupid hat trick is also intuitive in L, so long as you always think inside a countable model of ZF(C).
The demon again is constrained to definable reals below omega one, which is now secretly a countable ordinal (but ZF doesn't know it). So there is very little difference between the conceptual method to guess the definable real, except that now it isn't so easy to interpret things in terms of oracles.
- There is no problem with "$R_L$, the L version of R, coexisting inside $R_R$, the actual version of R, in your mental model of the universe.
The axiom of choice is true in L, which includes a particular model for the real numbers (and all powersets). This model is fine for interpreting all the counterintuitive statements of ZFC, since they are just plain true in L. When you read a choicy theorem, you just imagine little "L" subscripts on the theorem, and then it is true (this is called relativizing to L in logic). But you always keep in mind that L is measure zero. Then that's it. There are no more intuitive paradoxes.
Your mental axiom system for no-intuitive-paradox mathematics than can be this
ZF (but you interpret powerset as "L-powerset")
V=L (and therefore Choice for all sets in your universe, and the continuum hypothesis for the constructible reals, and for the constructible powersets)
For each set S in the universe L, (which is well-ordered by V=L), there is a non-well-orderable proper class of subsets of S, the true power-class. Every subclass of a powerclass has a real valued Lebesgue measure, and every subset (meaning well-orderable sub-SET, not a non-well-orderable subCLASS) of a powerclass has measure zero. All powerclasses are the same size, since they are not powerclasses of previous powerclasses, just powerclasses of dinky little sets. The measure of the proper-class completion of the dinky little measure-zero L-Borel sets is the same as the measure assigned to these Borel sets in L.
This system does nothing but shuffle the intuition around. There is no new real mathematics here (no new arithmetic theorems). But with this in your head, you banish all the choice paradoxes to the dustbin of history. No more puzzles, no more paradoxes, no more nothing. This has been a public service announcement.
No comments:
Post a Comment