This will be a talk I’ll give at the CUNY Graduate Center Graduate Student Colloquium on Monday, April 11 (new date!), 2016, 4-4:45 pm. The talk will be aimed at a general audience of mathematics graduate students.
![By PeterPan23 [Public domain], via Wikimedia Commons](https://upload.wikimedia.org/wikipedia/commons/f/fb/Darts_in_a_dartboard.jpg)
Abstract. I shall give an elementary presentation of Freiling’s axiom of symmetry, which is the principle asserting that if $x\mapsto A_x$ is a function mapping every real $x\in[0,1]$ in the unit interval to a countable set of such reals $A_x\subset[0,1]$, then there are two reals $x$ and $y$ for which $x\notin A_y$ and $y\notin A_x$. To argue for the truth of this principle, Freiling imagined throwing two darts at the real number line, landing at $x$ and $y$ respectively: almost surely, the location $y$ of the second dart is not in the set $A_x$ arising from that of the first dart, since that set is countable; and by symmetry, it shouldn’t matter which dart we imagine as being first. So it may seem that almost every pair must fulfill the principle. Nevertheless, the principle is independent of the axioms of ZFC and in fact it is provably equivalent to the failure of the continuum hypothesis. I’ll introduce the continuum hypothesis in a general way and discuss these foundational matters, before providing a proof of the equivalence of $\neg$CH with the axiom of symmetry. The axiom of symmetry admits natural higher dimensional analogues, such as the case of maps $(x,y)\mapsto A_{x,y}$, where one seeks a triple $(x,y,z)$ for which no member is in the set arising from the other two, and these principles also have an equivalent formulation in terms of the size of the continuum.
Freiling axiom of symmetry on MathOverflow | On Wikipedia | Graduate Student Colloquium
I just assumed that the higher dimensional analogues were equivalent to the one-dimensional case. Is there a good paper on it?
Yes, the higher-dimensional arguments are actually in Chris Freiling’s original paper. The original $1$-dimensional version (that is, the axiom of symmetry itself) is equivalent to the negation of CH, which is to say, to the assertion that the continuum is strictly larger than $\aleph_1$. Meanwhile, the $2$-dimensional case is equivalent to the assertion that the continuum is strictly larger than $\aleph_2$, which is a strictly stronger statement, and the $n$-dimensional case is equivalent to $2^\omega>\aleph_n$.
I hope you’re not planning to throw darts at the pizza.
Well, that would be the higher-dimensional case, which won’t come until the end of the talk, but I expect that the pizza will be all eaten up by then.
I have just recently found out that, due to budget cuts, the Pizza Seminar no longer always offers free pizza, and has been renamed the Graduate Student Colloquium. What a pity!
This could be possibly a question for MathOverflow but I really don’t know how to make it more precise because in the current form it is not more than an intuitive suggestion for formulating new axioms in set theory which are originated from “our probabilistic intuition in the real world” and then using such axioms to settle the independent sentences like CH!
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Freiling’s axiom of symmetry makes me to think about professional gamblers’ intuitive arguments when they are playing cards.
For example if you had the chance to be in a game with such experts, you would see their pseudo-magical ability in predicting the game situations or the cards in another player’s hands with an acceptable degree of accuracy!
This is possibly because they “expect” a completely random shuffling and some other probabilistic phenomenon to happen in the game and then they build their chain of reasoning on such “expected” probabilistic facts which seem “intuitively true” to them.
Back to Freiling’s axiom of symmetry, it seems that the same thing happened in the process of formulating this axiom.
– There is a “real game” which depends on chance, namely throwing darts!
– There is an “expected situation” in this real game which is “intuitively justifiable”.
– The axiom formulates this “probabilistic expectation” formally.
The question is whether we can find more such “seemingly true” axioms through investigating “intuitively expected” probabilistic phenomenon in the other games, say in (an infinite form of) cards game?
In the better words if a professional gambler says that such and such things “intuitively” should happen during shuffling a deck of finitely many cards, and if we “carelessly” generalize this probabilistic intuition in the finite case to the infinite case while we are shuffling a deck of $\omega$ – many (or more) cards which are sorted in $\omega$ order type, then we find a statement in the language of set theory which is possibly true or false within ZF and if we are lucky it might be an independent axiom like Freiling’s axiom of symmetry which could be used for determining the truth or falsity of other independent statements in set theory.
Note that by shuffling a deck of $\omega$ – many cards, we are changing not only the position of single cards but also possibly the order type of the infinity which these cards are sorted in. A possibly interesting case could be also considering $2^{\aleph_0}$ – many cards each associated to a real number and sorted in $\mathbb{R}$ natural order and then shuffling them and applying gamblers’ intuition to predict the position or distribution of there cards or some expected situations in the game and finally deriving our intended formal axiom from these probabilistic intuitive arguments.
A related video on gamblers’ intuition about the best way of shuffling a deck of 52 cards between Riffle, smooshing or overhand shuffling, could be found in the below link. It is provided by mathematician and former professional magician, Persi Diaconis from Stanford University.
https://www.youtube.com/watch?v=AxJubaijQbI&feature=youtu.be
https://en.wikipedia.org/wiki/Persi_Diaconis
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