Modal graph theory as a foundation of mathematics

This will be a talk for the Barcelona Set Theory Seminar, 17 March 2021 4 PM CET (3 PM UK, 4 PM Poland). I understand the talk will be held on Zoom; please contact Claudio Ternullo for access.

aerial view of city buildings

Abstract. One can consider the class of all graphs as a Kripke model of possible worlds, where a graph extends or accesses a larger graph just in case it is an induced subgraph thereof. In this way, we can introduce modal operators of possibility and necessity. A statement is possible at a graph if it is true in some extension of that graph, and it is necessary if it is true at all such extensions.  We can thus enlarge the first-order language of graphs by closing it under modal operators, Boolean connectives, and quantification. The resulting modal language of graph theory turns out to be rather fruitful—it can express finiteness, countability, size continuum, size \(\aleph_1\), \(\aleph_2\), \(\aleph_\omega\), first \(\beth\)-fixed-point, first \(\beth\)-hyper-fixed-point, and so on. Perhaps most remarkably, modal graph theory can interpret set-theoretic truth in \(V_\theta\) for quite a long way into the cumulative hierarchy. Does it run out of steam or can it interpret truth in the full set-theoretic universe \(V\), and serve as a foundation of mathematics?

This is joint work with Joel David Hamkins, Professor of Logic at Oxford University and the Sir Peter Strawson Fellow at University College Oxford.

  • J. D. Hamkins and W. A. W. l, “Modal model theory,” Mathematics arxiv, 2020.
    archiveprefix = {arXiv},
    author = {Joel David Hamkins and Wojciech Aleksander Wo{\l}oszyn},
    date-added = {2021-03-15 00:30:53 +0100},
    date-modified = {2021-03-15 00:30:53 +0100},
    eprint = {2009.09394},
    journal = {Mathematics arXiv},
    keywords = {under-review},
    note = {Under review},
    primaryclass = {math.LO},
    title = {Modal model theory},
    year = {2020}}

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