# Upward closure and amalgamation in the generic multiverse of a countable model of set theory

• J. D. Hamkins, “Upward closure and amalgamation in the generic multiverse of a countable model of set theory,” RIMS Kyôkyûroku, pp. 17-31, 2016. (also available as Newton Institute preprint ni15066)
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Abstract. I prove several theorems concerning upward closure and amalgamation in the generic multiverse of a countable transitive model of set theory. Every such model $W$ has forcing extensions $W[c]$ and $W[d]$ by adding a Cohen real, which cannot be amalgamated in any further extension, but some nontrivial forcing notions have all their extensions amalgamable. An increasing chain $W[G_0]\subseteq W[G_1]\subseteq\cdots$ has an upper bound $W[H]$ if and only if the forcing had uniformly bounded essential size in $W$. Every chain $W\subseteq W[c_0]\subseteq W[c_1]\subseteq \cdots$ of extensions adding Cohen reals is bounded above by $W[d]$ for some $W$-generic Cohen real $d$.

This article is based upon I talk I gave at the conference on Recent Developments in Axiomatic Set Theory at the Research Institute for Mathematical Sciences (RIMS) at Kyoto University, Japan in September, 2015, and I am extremely grateful to my Japanese hosts, especially Toshimichi Usuba, for supporting my research visit there and also at the CTFM conference at Tokyo Institute of Technology just preceding it. This article includes material adapted from section section 2 of Set-theoretic geology, joint with G. Fuchs, myself and J. Reitz, and also includes a theorem that was proved in a series of conversations I had with Giorgio Venturi at the Young Set Theory Workshop 2011 in Bonn and continuing at the London 2011 summer school on set theory at Birkbeck University London.

# A position in infinite chess with game value $\omega^4$

• C.~D.~A.~Evans, J. D. Hamkins, and N. L. Perlmutter, “A position in infinite chess with game value $\omega^4$,” to appear in Integers, vol. 17, 2017. (Newton Institute preprint ni15065)
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title = {A position in infinite chess with game value $\omega^4$},
journal = {to appear in Integers},
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Abstract.  We present a position in infinite chess exhibiting an ordinal game value of $\omega^4$, thereby improving on the previously largest-known values of $\omega^3$ and $\omega^3\cdot 4$.

This is a joint work with Cory Evans and Norman Perlmutter, continuing the research program of my previous article with Evans, Transfinite game values in infinite chess, namely, the research program of finding positions in infinite chess with large transfinite ordinal game values. In the previous article, Cory and I presented a position with game value $\omega^3$. In the current paper, with Norman Perlmutter now having joined us accompanied by some outstanding ideas, we present a new position having game value $\omega^4$, breaking the previous record.

A position in infinite chess with value $\omega^4$

In the new position, above, the kings sit facing each other in the throne room, an uneasy détente, while white makes steady progress in the rook towers. Meanwhile, at every step black, doomed, mounts increasingly desperate bouts of long forced play using the bishop cannon battery, with bishops flying with force out of the cannons, and then each making a long series of forced-reply moves in the terminal gateways. Ultimately, white wins with value omega^4, which exceeds the previously largest known values of omega^3.

In the throne room, if either black or white places a bishop on the corresponding diagonal entryway, then checkmate is very close. A key feature is that for white to place a white-square white bishop on the diagonal marked in red, it is immediate checkmate, whereas if black places a black-square black bishop on the blue diagonal, then checkmate comes three moves later.  The bishop cannon battery arrangement works because black threatens to release a bishop into the free region, and if white does not reply to those threats, then black will be three steps ahead, but otherwise, only two.

The throne room

The rook towers are similar to the corresponding part of the previous $\omega^3$ position, and this is where white undertakes most of his main line progress towards checkmate.  Black will move the key bishop out as far as he likes on the first move, past $n$ rook towers, and the resulting position will have value $\omega^3\cdot n$.  These towers are each activated in turn, leading to a long series of play for white, interrupted at every opportunity by black causing a dramatic spectacle of forced-reply moves down in the bishop cannon battery.

The rook towers

At every opportunity, black mounts a long distraction down in the bishop cannon battery.  Shown here is one bishop cannon. The cannonballs fire out of the cannon with force, in the sense that when each green bishop fires out, then white must reply by moving the guard pawns into place.

Bishop cannon

Upon firing, each bishop will position itself so as to attack the entrance diagonal of a long bishop gateway terminal wing.  This wing is arranged so that black can make a series of forced-reply threats successively, by moving to the attack squares (marked with the blue squares). Black is threatening to exit through the gateway doorway (in brown), but white can answer the threat by moving the white bishop guards (red) into position. Thus, each bishop coming out of a cannon (with force) can position itself at a gateway terminal of length $g$, making $g$ forced-reply moves in succession.  Since black can initiate firing with an arbitrarily large cannon, this means that at any moment, black can cause a forced-reply delay with game value $\omega^2$. Since the rook tower also has value $\omega^2$ by itself, the overall position has value $\omega^4=\omega^2\cdot\omega^2$.

Bishop gateway terminal wing

With future developments in mind, we found that one can make a more compact arrangement of the bishop cannon battery, freeing up a quarter board for perhaps another arrangement that might lead to a higher ordinal values.

Alternative compact version of bishop cannon battery

Read more about it in the article, which is available at the arxiv (pdf).

# Open determinacy for class games

• V. Gitman and J. D. Hamkins, “Open determinacy for class games,” in Foundations of Mathematics, Logic at Harvard, Essays in Honor of Hugh Woodin’s 60th Birthday, A. E. Caicedo, J. Cummings, P. Koellner, and P. Larson, Eds., American Mathematical Society, 2016. (also available as Newton Institute preprint ni15064)
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title = {Open determinacy for class games},
booktitle = {Foundations of Mathematics, Logic at Harvard, Essays in Honor of Hugh Woodin's 60th Birthday},
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Abstract. The principle of open determinacy for class games — two-player games of perfect information with plays of length $\omega$, where the moves are chosen from a possibly proper class, such as games on the ordinals — is not provable in Zermelo-Fraenkel set theory ZFC or Godel-Bernays set theory GBC, if these theories are consistent, because provably in ZFC there is a definable open proper class game with no definable winning strategy. In fact, the principle of open determinacy and even merely clopen determinacy for class games implies Con(ZFC) and iterated instances Con(Con(ZFC)) and more, because it implies that there is a satisfaction class for first-order truth, and indeed a transfinite tower of truth predicates $\text{Tr}_\alpha$ for iterated truth-about-truth, relative to any class parameter. This is perhaps explained, in light of the Tarskian recursive definition of truth, by the more general fact that the principle of clopen determinacy is exactly equivalent over GBC to the principle of transfinite recursion over well-founded class relations. Meanwhile, the principle of open determinacy for class games is provable in the stronger theory GBC$+\Pi^1_1$-comprehension, a proper fragment of Kelley-Morse set theory KM.

See my earlier posts on part of this material:

# A mathematician’s year in Japan

• J. D. Hamkins, A Mathematician’s Year in Japan, author-published, via Amazon Kindle Direct Publishing, 2015. (ASIN:B00U618LM2, 156 pages, http://www.amazon.com/dp/B00U618LM2)
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Years ago, when I was still a junior professor, I had the pleasure to live for a year in Japan, working as a research fellow at Kobe University. During that formative year, I recorded brief moments of my Japanese experience, and every two weeks or so—this was well before the current blogging era—I sent my descriptive missives by email to friends back home. I have now collected together those vignettes of my life in Japan, each a morsel of my experience. The book is now out!

A Mathematician’s Year in Japan
Joel David Hamkins

Glimpse into the life of a professor of logic as he fumbles his way through Japan.

A Mathematician’s Year in Japan is a lighthearted, though at times emotional account of how one mathematician finds himself in a place where everything seems unfamiliar, except his beloved research on the nature of infinity, yet even with that he experiences a crisis.