# First-Order Continuous Induction and a Logical Study of Real Closed Fields

## Abstract

Over the last century, the principle of “continuous induction” has been studied by different authors in different formats. All of these different forms are equivalent to one of the three versions that we isolate in this paper. We show that one of the three forms of continuous induction is weaker than the other two by proving that it is equivalent to the Archimedean property, while the other two stronger versions are equivalent to the completeness property (the supremum principle) of the real numbers. We study some equivalent axiomatizations for the first-order theory of real closed fields and show that some first-order formalization of continuous induction is able to completely axiomatize it (over the theory of ordered fields).

## Keywords

First-order logic Complete theories Axiomatizing the field of real numbers Continuous induction Real closed fields## Mathematics Subject Classification

03B25 03C35 03C10 12L05## Notes

### Acknowledgements

The authors are very much grateful to the anonymous referee of the Bulletin for carefully reading the article and for many helpful suggestions which improved the results and made the paper much more readable.

## References

- 1.Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry, 2nd edn. Springer, Berlin (2006). ISBN:9783540330981CrossRefzbMATHGoogle Scholar
- 2.Brown, R., Craven, T.C., Pelling, M.J.: Ordered fields satisfying Rolle’s theorem. Ill. J. Math.
**30**(1), 66–78 (1986). https://projecteuclid.org/euclid.ijm/1256044752 - 3.Chao, Y.R.: A note on “Continuous mathematical induction”. Bull. Am. Math. Soc.
**26**(1) 17–18 (1919). https://projecteuclid.org/euclid.bams/1183425067 - 4.Clark, P.L.: The instructor’s guide to real induction. Expository preprint (2017). Available on the net at: http://alpha.math.uga.edu/~pete/instructors_guide_2017.pdf. Accessed 15 May 2019
- 5.Deveau, M., Teismann, H.: 72+42: Characterizations of the completeness and Archimedean properties of ordered fields. Real Anal. Exch.
**39**(2), 261–304 (2014). https://doi.org/10.14321/realanalexch.39.2.0261 MathSciNetCrossRefzbMATHGoogle Scholar - 6.Epstein, R.L.: Classical Mathematical Logic: The Semantic Foundations of Logic. Princeton University Press, Princeton (2006). (corr. 2011). ISBN:9780691123004zbMATHGoogle Scholar
- 7.Fine, B., Rosenberger, G.: The Fundamental Theorem of Algebra. Springer, Berlin (1997). (softcover 2012). ISBN:9781461273431CrossRefzbMATHGoogle Scholar
- 8.Hathaway, D.: Using continuity induction. Coll. Math. J. (Classr. Capsul.)
**42**(3), 229–231 (2011). https://doi.org/10.4169/college.math.j.42.3.229 CrossRefGoogle Scholar - 9.Jingzhong, Z.: The induction on a continuous variable. IAEA-UNESCO, ICTP Report #IC/89/157 (1989). http://streaming.ictp.it/preprints/P/89/157.pdf. Accessed 15 May 2019
- 10.Kalantari, I.: Induction over the continuum. In: Friend, M., Goethe, N.B., Harizanov, V.S. (eds.) Induction, Algorithmic Learning Theory, and Philosophy, pp. 145–154. Springer, Berlin (2007). https://doi.org/10.1007/978-1-4020-6127-1_5 CrossRefGoogle Scholar
- 11.Kreisel, G., Krivine, J.-L.: Elements of Mathematical Logic: Model Theory. North-Holland, Amsterdam (1971). ISBN:9780720422658zbMATHGoogle Scholar
- 12.Mashkouri, M.: Sākhtāre A’dād, Mabāniye Riyāziāt. Structure of Numbers, Foundations of Mathematics. Isfahan University of Technology Publication Center, Isfahan (2004). ISBN:9648476012
**(in Farsi)**. https://publication.iut.ac.ir/book/89 - 13.Pelling, M.J.: Solution of advanced problem no. 5861 (proposed by Michael Slater). Am. Math. Mon.
**88**(2), 150–152 (1981). https://www.jstor.org/stable/2321145 - 14.Spivak, M.: Calculus, Publish or Perish, Houston, 4th edn (2008). ISBN:9780914098911Google Scholar