Szpilrajn’s Theorem states that any partial orderP=〈S,<p〉 has a linear extensionP=〈S,<L〉. This is a central result in the theory of partial orderings, allowing one to define, for instance, the dimension of a partial ordering. It is now natural to ask questions like “Does a well-partial ordering always have a well-ordered linear extension?” Variations of Szpilrajn’s Theorem state, for various (but not for all) linear order typesτ, that ifP does not contain a subchain of order typeτ, then we can chooseL so thatL also does not contain a subchain of order typeτ. In particular, a well-partial ordering always has a well-ordered extension.
We show that several effective versions of variations of Szpilrajn’s Theorem fail, and use this to narrow down their proof-theoretic strength in the spirit of reverse mathematics.
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This research was carried out while the first and second authors were visiting the third and fourth authors at the University of Wisconsin. The first and second authors’ research was partially supported by the Marsden Fund of New Zealand. The third author’s research is partially supported by NSF grant DMS-9732526. The fourth author is supported by an NSF postdoctoral fellowship.
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Downey, R.G., Hirschfeldt, D.R., Lempp, S. et al. Computability-theoretic and proof-theoretic aspects of partial and linear orderings. Isr. J. Math. 138, 271–289 (2003). https://doi.org/10.1007/BF02783429
- Linear Extension
- Algebraic Closure
- Order Type
- Peano Arithmetic
- Active Interval