BPFA and projective well-orderings of the reals

December 17, 2009

Sy Friedman and I recently submitted the paper {\sf BPFA} and projective well-orderings of the reals to The Journal of Symbolic Logic. The preprint is available at my papers page.

In a previous paper, Boban Velickovic and I showed that if {\sf BPFA}, the bounded version of the proper forcing axiom, holds, then one can define a well-ordering of the reals using what is in essence a subset C of \omega_1 as a parameter. The argument uses Justin Moore‘s technique of the Mapping Reflection Principle, and provides us with a \Delta_1(C) well-ordering. In this sense, the result is best possible. 

In earlier work, I had shown that {\sf BPFA} is consistent with a projective well-ordering of the reals. The result with Velickovic dramatically improves this; for example, if \omega_1=\omega_1^L and {\sf BPFA} holds, then there is a projective well-ordering of {\mathbb R}. Note that this is an implication rather than just a consistency result, and does not require that the universe is a forcing extension of L. The point here is that the parameter C can be chosen in L so that it is “projective in the codes,” and {\sf MA}_{\omega_1} then provides us with enough coding machinery to transform the \Delta_1(C) definition of the well-ordering into a projective one.

In the new paper, Friedman and I show that in fact under {\sf BPFA}+\omega_1=\omega_1^L, the projective definition is best possible, \Sigma^1_3. For this we need to combine the coding technique giving the \Delta_1(C) well-ordering with a powerful coding device in the absence of sharps, what Friedman calls David’s trick. The point now is that the forcing required to add the witnesses that make the coding work is proper, and {\sf BPFA} suffices to grant in the universe the existence of these objects.

As a technical problem, it would be interesting to see whether the appeal to the mapping reflection principle can be eliminated here. We only obtain a \Sigma^1_4 well-ordering in that case. Also, since {\sf BPFA} turns out, perhaps unexpectedly, to provides us with definable well-orderings, it would be interesting to see that {\sf MA}_{\omega_1} does not suffice for this.


December 17, 2009

Flashforward is a science-fiction novel by Canadian writer Robert Sawyer. I wanted to like the novel, I did. I first heard of it, of course, when ABC began to air commercials for their version of the story. The TV series has been kind of terrible so far. There is a lot of potential in the premise, I think, but neither the series nor the novel delivers it.

There are significant differences, too. In both cases, a global event occurs where everybody has a glimpse of their own future for a couple of minutes or so. Except for this, there is really not much in common in both cases. In the TV series, the `jump’ is six months or so. In the novel is several decades. One of the main characters finds out he is going to be killed, and when. Other than this, there is nothing much in common.

Sawyer has won several awards for his writing, but I do not understand it based on this novel. Perhaps his other works are better. In this one, what is clear is how little understanding he seems to have of the science he is using. His characters are actually well developed, and in a less plot driven story, I think his strengths would be evident. 

Here are three quotes:

Frau Drescher went as white as the snow cap on Mont Blanc. “Mein Gott,” she said. “Mein Gott.

I mean, seriously? Ok. Perhaps that is a bit unfair; his comparisons and metaphors usually work, and his characters tend to talk like people. Let’s forget that one.

“Of course,” said Michiko. “It’s not that three minutes passed during which planes and trains and cars and assembly lines operated without human intervention. Rather, three minutes passed during which nothing was resolved—all the possibilities existed, stacked into shimmering whiteness. But at the end of those three minutes, consciousness returned, and the world collapsed again into a single state. And, unfortunately but inevitably, it took the single state that made the most sense, given that there had been three minutes of no consciousness: it resolved itself into the world in which planes and cars had crashed. But the crashes didn’t occur during those three minutes; they never occurred at all. We simply went in one jump from the way things were before to the way they were after.”

“That’s… that’s crazy,” said Lloyd.


“No, it’s not. It’s quantum physics.”

No, it’s not. It’s crazy. It’s just absurd. It is what happens when rather than understanding the science, one takes the metaphors scientists use and erroneously assumes they are literal truths. The TV series quickly distanced itself from this nonsense. In the book, only humans see the future, since they are the only ones with “consciousness.” And no recording devices work during the `jump’, because nobody is observing them. Seriously. Sawyer seems to think that the tree actually makes no sound. Or, perhaps, there isn’t even a forest, if nobody is watching it. It gets worse, because of course it is not only quantum mechanics that is thoroughly misunderstood. 

Surely all consciousness everywhere had to agree on what constituted “now.”

This being what the main physicist of the story thinks. Everywhere being, of course, the entire universe. This in the same paragraph where the physicist explains some relativity theory to himself. 


502 – The Banach-Tarski paradox

December 17, 2009

1. Non-measurable sets

In these notes I want to present a proof of the Banach-Tarski paradox, a consequence of the axiom of choice that shows us that a naive understanding of the concept of volume can lead to contradictions. A good reference for this topic is the very nice book The Banach-Tarski paradox by Stan Wagon.

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