Skip to main content Accessibility help
×
Home

A reflection principle and its applications to nonstandard models

  • James H. Schmerl (a1)

Extract

Some methods of constructing nonstandard models work only for particular theories, such as ZFC, or CA + AC (which is second order number theory with the choice scheme). The examples of this which motivated the results of this paper occur in the main theorems of [5], which state that if T is any consistent extension of either ZFC0 (which is ZFC but with only countable replacement) or CA + AC and if κ and λ are suitably chosen cardinals, then T has a model which is κ-saturated and has the λ-Bolzano-Weierstrass property. (Compare with Theorem 3.5.) Another example is a result from [12] which states that if T is any consistent extension of CA + AC and cf (λ) > ℵ0, then T has a natural λ-Archimedean model. (Compare with Theorem 3.1 and the comments following it.) Still another example is a result in [6] in which it is shown that if a model of Peano arithmetic is expandable to a model of ZF or of CA, then so is any cofinal extension of . (Compare with Theorem 3.10.) Related types of constructions can also be found in [10] and [11].

A reflection principle will be proved here, allowing these constructions to be extended to models of many other theories, among which are some exceedingly weak theories and also all of their completions.

Copyright

References

Hide All
[1]Beller, A. and Litman, A., A strengthening of Jensen's □ principles, this Journal, vol. 45 (1980), pp. 251264.
[2]Chang, C. C. and Keisler, H. J., Model theory, 3rd ed., North-Holland, Amsterdam, 1990.
[3]Devlin, K. J., The combinatorial principle ◊*, this Journal, vol. 47 (1982), pp. 888899.
[4]Hajnal, A., Kanamori, A., and Shelah, S., Regressive partition relations for infinite cardinals, Transactions of the American Mathematical Society, vol. 299 (1987), pp. 145154.
[5]Keisler, H. J. and Schmerl, J., Making the hyperreal line both saturated and complete, this Journal, vol. 56 (1991), pp. 10161025.
[6]Kotlarski, H., On cofinal extensions of models of arithmetic, this Journal, vol. 48 (1983), pp. 253262.
[7]Schmerl, J. H., Peano models with many generic classes, Pacific Journal of Mathematics, vol. 46 (1973), pp. 523536.
[7a]Schmerl, J. H., Correction to "Peano models with many generic classes", Pacific Journal of Mathematics, vol. 92 (1981), pp. 195198.
[8]Schmerl, J. H., A partition property characterizing cardinals hyperinaccessible of finite type, Transactions of the American Mathematical Society, vol. 188 (1974), pp. 281291.
[9]Schmerl, J. H., Generalizing special Aronszajn trees, this Journal, vol. 39 (1974), pp. 732740.
[10]Schmerl, J. H., Recursively saturated, rather classless models of Peano arithmetic, Logic year 1979–1980 (Lerman, M., editor), Lecture Notes in Mathematics, vol. 859, Springer-Verlag, Berlin, 1981, pp. 268282.
[11]Schmerl, J. H., Models of Peano arithmetic and a question of Sikorski on ordered fields, Israel Journal of Mathematics, vol. 50 (1985), pp. 145159.
[12]Schmerl, J. H., Peano arithmetic and hyper-Ramsey logic, Transactions of the American Mathematical Society, vol. 296 (1986), pp. 481505.
[13]Velleman, D. J., Morasses, diamond, and forcing, Annals of Mathematical Logic, vol. 23 (1982), pp. 199281.

A reflection principle and its applications to nonstandard models

  • James H. Schmerl (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed