Abstract
This paper surveys some results on combinatorial aspects of infinite Ramsey-type problems inspired by finite properties, and intends to explain the relevance of an alternative set-theoretical principle formulated in the language of colors, the so-called Principle of Ariadne. This principle, intended to be a rival of the Axiom of Choice, can be consistently added to the usual axiom stock of ZF set theory under certain conditions. Such a new axiom, which preserves all the finite contents of mathematics but deviates from the standard in the infinite contents, may help us to understand the finite-infinite divide in mathematics, making clear that there is more than one way to generalize from finite principles of order (or choice) to the infinite. In other words, several infinite principles are possible starting from the same finitary content.
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Notes
- 1.
We’re considering here only undirected graphs. A directed graph consists of a collection of vertices and a collections of arcs (ordered pairs of vertices). Finite Ramsey’s Theorems also extend to directed graphs, but are slightly more complicated.
- 2.
Paul Erdős is reported to have said: “If the demon asked us to tell him the value of R(6, 6) we should devote all our resources to finding a way to kill the demon”. Erdős was well aware that it would be easier to kill a demon than to compute R(6, 6).
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Acknowledgements
Some results in this paper have been written years ago during a visit of the first named author to the Instituto Venezolano de Investigaciones Científicas (IVIC) in Caracas, Venezuela. This author also acknowledges support from FAPESP Thematic Project LogCons 2010/51038-0, Brazil, and from a research grant from the National Council for Scientific and Technological Development (CNPq), Brazil.
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Carnielli, W., Prisco, C.d. (2017). The Wonder of Colors and the Principle of Ariadne. In: Silva, M. (eds) How Colours Matter to Philosophy. Synthese Library, vol 388. Springer, Cham. https://doi.org/10.1007/978-3-319-67398-1_18
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