2013/05/01 by Samuel Reid, Reid, Samuel
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · Physics and Astronomy · #03B44 #05A16 #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Logic (math.LO) #Logic in Computer Science (cs.LO) #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Receptor Mechanisms and Signaling #cs.LO #math.CO #math.LO #msc:03B44 #msc:05A16
paper · pdf · doi:10.48550/arxiv.1305.0064
8 pages and 3 figures
arxiv created 2013/05/01 · openalex publication_date 2013/05/01 · arxiv updated 2013/05/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A temporal-theoretic formalism for understanding game theory is described where a strict ordering relation on a set of time points T defines a game on T. Using this formalism, a proof of Zermelo's Theorem, which states that every finite 2-player zero-sum game is determined, is given and an exhaustive analysis of the game of Nim is presented. Furthermore, a combinatorial analysis of games on a set of arbitrary time points is given; in particular, it is proved that the number of distinct games on a set T with cardinality n is the number of partial orders on a set of n elements. By generalizing this theorem from temporal modal frames to S5 modal frames, it is proved that the number of isomorphism classes of S5 modal frames F = < W, R > with |W|=n is equal to the partition function p(n). As a corollary of the fact that the partition function is asymptotic to the Hardy-Ramanujan number (1)/(4√(3)n)eπ√(2n/3) the number of isomorphism classes of S5 modal frames F = < W, R > with |W|=n is asymptotically the Hardy-Ramanujan number. Lastly, we use these results to prove that an arbitrary modal frame is an S5 modal frame with probability zero.