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My mathematical prejudices: 1. I don't accept choice on the continuum 2. I accept choice on countable collections, as well as countable infinity. 3. I don't think of the continuum as small enough to be a set, 4. I think every subclass (subset) of the real line is measurable, 5. I prefer constructions to abstractions. In this, I hope I am following Paul Cohen in spirit.

my dream theorem is a proof of the consistency of ZFC from a countable computable ordinal, as explicitly describable as possible, which plays the same role for ZFC as $\epsilon_0$ does for PA. This would complete Hilbert's program, giving what I would consider a finitary proof of the consistency of set theory. If you think this is impossible because of Godel's theorem, you haven't understood Godel's theorem fully. The problem is that ordinal naming schemes crap out much too early for this to work with an explicit construction with today's methods.


Apr
17
comment Foundation for analysis without axiom of choice?
@Did: Yes, you are right, filtrations are an exception to the rule, but in probability filtrations are an abomination. The process by which you define a stochastic process in a measurable universe is simply by giving a convergent sequence of random approximations to the process, and this natural definition is butchered by the infernal nonmeasurables, so that you end up introducing filtrations. The filtrations are useless in this context, and positively harmful, they are simply barrier to entry, even here the probabilist secretly thinks that all path sets in path-space are measurable.
Apr
17
comment Why is the axiom of choice separated from the other axioms?
@Hurkyl: The "presupposition" is that there are any nonmeasurable sets in the first place! It is obvious that these make it impossible to speak about "randomly choosing a real number", and this is something that makes natural arguments onerous and convoluted, because you need to distinguish between "random variables" (which are not real numbers and cannot be when there is uncountable choice) and "real numbers" which are something else. It's an artificial and stifling distinction.
Apr
17
comment Foundation for analysis without axiom of choice?
@AsafKaragila: Downvoted, because of the statement that "in Solovay's model you can decompose R into more parts than it has elements". This is using the ridiculous idea that just because you can't inject aleph-1 into R that aleph-1 somehow has "more elements" than R. Not true, it doesn't, it is simply incomparable. The ordinal tower is separated conceptually from the powersets in Solovay model, and you can't compare the two in size at all. The reals are enormously large, and the ordinals are much smaller, but to demonstrate this by embedding you have to first take the ordinals to be countable.
Apr
17
comment Implication and Interpretation of Banach Tarski
@AsafKaragila: Borel sets are not dependent on the kind of axiom of choice one is talking about here. They have nothing to do with uncountable continuum choice.
Apr
17
comment Implication and Interpretation of Banach Tarski
Of course the Banach Tarski pieces behave non-paradoxically, because the axiom of choice is consistent! But this is not the same as saying that they are not paradoxical, because they contradict the concept of "randomly chosen real number", and require you to formulate the concept of "random real" differently, as a "random variable", for which you cannot do certain operations. For example, you cannot speak about the probability that a random chosen real lands in a non-measurable set, the concept is incoherent. So this means random reals are forbidden in your universe.
Apr
17
comment Implication and Interpretation of Banach Tarski
This particular "naive argument" does not break down, if you allow random picking, in other words, if you can choose real numbers at random in the interval [0,1]. The concept of picking real numbers at random is equivalent to saying all sets are measurable, and losing the ability to speak about random real numbers is not acceptable. This is a case where the mathematicians are stupid and brainwashed, not the critics.
Apr
17
revised Foundation for analysis without axiom of choice?
explain the gaussian argument better
Apr
17
comment Foundation for analysis without axiom of choice?
@AsafKaragila: You are forgetting the obvious paradox that choice is incompatible with the statement "consider this randomly chosen real number". This is a paradox which makes it forbidden to talk about random numbers with specific values! Instead, you have to talk about randomly chosen things as second-class objects in the universe, where certain operations cannot be defined.
Apr
17
comment Foundation for analysis without axiom of choice?
@MichaelGreinecker: No, no, no. Making all the sets measurable will make the probabalists ecstatic, because they operate as if every set is measurable anyway, you just never read their arguments, so you don't know this. The probabilists automatically speak about randomly chosen real numbers, random configurations, and only when they have a set-theorists gun against their head do they stop and say "well, what I really mean is this and such Lebesgue measurable set". There is no probabilist that wouldn't be happy in a measurable universe, it is simply you that is wrong.
Apr
17
comment Foundation for analysis without axiom of choice?
@AsafKaragila: I know the paper, it has nothing to do with my comment. I didn't say that "ZF+DC+LM" is less strong than "ZFC+I", I said that "ZFC+I" is not something one should worry about, and the reason you need it is just because of the fact that LM adds strength. "ZFC+I" is simply adding a set which serves as a universe, and then closing up the model using ZF operations, this cannot be controversial, it is simply used as propaganda against Solovay.
Apr
17
comment Foundation for analysis without axiom of choice?
@Did: The Gaussian random numbers argument is original to me, but it is very easy. You can see this mathoverflow answer: mathoverflow.net/questions/49351/… (there is a simple proof at the beginning of my answer). The key point is that it is impossible for the sum of two equidistributed positive integers to have the same probability distribution as the two summands.
Apr
17
comment Foundation for analysis without axiom of choice?
@Hurkyl: Obviously, I am speaking about admissable idealizations, not physical objects. But as admissible idealizations, there is definitely such a thing as a "randomly chosen number" in any reasonable mathematical universe, you need to speak about it to describe a specific state of an infinite 2d Ising model, a specific instance of a distribution random field, and so on and so on, and it is also completely consistent to speak about such things! It only conflicts with a stupid metaphysical principle you like, namely uncountable choice. Sorry, but your metaphysical principle should go.
Apr
17
comment Foundation for analysis without axiom of choice?
@Hurkyl: I can't make sense of measure spaces without specifying precisely what are the allowed set operations and the model first, so no, I don't.
Apr
17
comment Foundation for analysis without axiom of choice?
@Hurkyl: The most important thing is freedom. One must be free to work within whatever universe one chooses. The mathematicians have made a metaphysical barrier, and require one to take a ridiculous metaphysical stance as true, and reject the equal validity of all the different possible metaphysical stances which agree on the output of computation.
Apr
17
comment Foundation for analysis without axiom of choice?
@Hurkyl: In a different way, and arguably. Computation is absolute, everyone agrees what it means. For the real number system, it's about a lot of metaphysics and convenience. The non-measurable sets are inconsistent with the concept of a randomly selected real number, and random numbers are important. There is nothing gained from the admission of non-measurables, and you lose a whole world of straightforward probability arguments. The current conventions are simply unacceptable to a physicist, and will always be so. Measurability is acceptable.
Apr
17
comment Foundation for analysis without axiom of choice?
@Hurkyl: What you don't understand is that by simply admitting the concept of "unmeasurable set", you are disallowing the concept of "randomly chosen number". It is not an argument against choice, it is an argument against stupidity--- you can choose numbers at random, people do it all the time, and if your mathematical system does not allow this, it is time to readjust your mathematical system. There is NO SUCH THING as a nonmeasurable set, it is a figment of your imagination. You can easily see why such a figment would exist in a submodel, but don't force us to take this as the universe.
Apr
17
comment Foundation for analysis without axiom of choice?
@Hurkyl: It's not just the coefficient that is not measurable, it's the number of basis elements used in the decomposition. The decomposition itself is not measurable. You claim "every real number can be decomposed", and then "but no random variable can be decomposed". This means that you simply decided to never speak about the decomposition of randomly chosen numbers! As if there is no such thing. This is a ridiculous stance, you are simply declaring that random numbers don't exist, and blinding yourself like that is stupid, it is not acceptable to me, or to a lot of other people.
Apr
17
comment Foundation for analysis without axiom of choice?
@Hurkyl: There are operations that you can apply to real numbers in ZFC, like "decompose into a basis over Q" which cannot be applied to randomly selected numbers, to random variables. This makes for a completely mentally retarded distinction between "random variables" and "real numbers" which is useless and counterproductive, and makes developing analysis and physics a pain in the ass. Physicists use random numbers all the time, and they can't be forced to remember which operations are allowed. It is also ridiculous and anachronistic, because it is possible to just use a Solovay model.
Apr
17
answered Foundation for analysis without axiom of choice?
Apr
16
comment Foundation for analysis without axiom of choice?
This is not sufficient, as you need "ZF + ZF is consistent" is consistent, and so on, iterated over computable ordinals, and this is equivalent to the consistency of arbitrarily strong theories of the large cardinal type. The main issue is with the metaphysics of the continuum. For a physics application, you need that every subset is Lebesgue measurable, and one must never renounce this, as there is no gain from assuming a non-measurable set, only headaches.