🔍 Do I have this consistency?

Explanation

This program tells you whether consistency of x < y for cardinal invariants x and y can be deduced or not using transitivity from known theorems of inequality and known consistency results.

I don't explain the syntax of the input. Imitate the preset inputs.

If you are not familiar with cardinal invariants of the continuum, see Wikipedia's article.

This program was created by Tatsuya Goto in 2025.

References

• Regarding the definitions of all cardinal invariants written in presets, see [Bla10].
• To know the proof of p<=h, see Proposition 6.8 of [Bla10].
• To know the proof of p<=e, see Theorem 10.4 of [Bla10].
• To know the proof of h<=b, see Theorem 6.9 of [Bla10].
• To know the proof of h<=g, see Proposition 6.27 of [Bla10].
• To know the proof of h<=s, see Theorem 6.9 of [Bla10].
• To know the proof of b<=a, see Proposition 8.4 of [Bla10].
• To know the proof of b<=r, see Theorem 3.8 of [Bla10].
• To know the proof of e<=r, see Theorem 10.4 of [Bla10].
• b<=d is clear.
• To know the proof of e<=r, see Theorem 10.4 of [Bla10].
• To know the proof of g<=d, see Proposition 6.27 of [Bla10].
• To know the proof of s<=d, see Theorem 3.3 of [Bla10].
• To know the proof of r<=u, see Proposition 9.7 of [Bla10].
• To know the proof of r<=i, see Proposition 8.12 of [Bla10].
• To know the proof of d<=i, see Theorem 8.13 of [Bla10].
• add(N)<=cov(N), non(N)<=cof(N) are trivial.
• To know the proof of add(N)<=add(M) and cof(M)<=cof(N), see Section 2.3 of [BJ95].
• To know the proof of add(M)<=b and d<=cof(M), see Corollary 5.4 of [Bla10].
• To know the proof of b<=non(M) and cov(M)<=d, see Proposition 5.5 of [Bla10].
• add(M)<=cov(M) and non(M)<=cof(M) are trivial.
• To know the proof of cov(M)<=non(N) and cov(N)<=non(M), see Theorem 5.11 of [Bla10].
• To know the proof of s<=non(M), s<=non(N), cov(M)<=r and cov(N)<=r see Theorem 5.19 of [Bla10].
• p<=add(M) can be derived from p<=b and p<=cov(M) using add(M)=min{b, cov(M)} (see Theorem 5.6 of [Bla10]).
• To know the proof of cof(M)<=i, see [BHH04].

Reference List (for the ZFC theorems)

• [Bla10] Blass, A. (2010). Combinatorial cardinal characteristics of the continuum. In Handbook of set theory (pp. 395–489). Springer.
• [BHH04] Balcar, B., Hernández-Hernández, F., & Hrušák, M. (2004). Combinatorics of dense subsets of the rationals. Fund. Math, 183(1), 59-80.
• [BJ95] Bartoszynski, T., & Judah, H. (1995). Set Theory: on the structure of the real line. CRC Press.

Reference List (for the consistency theorems)

• [Bre03] Brendle, J. (2003). The almost-disjointness number may have countable cofinality. Transactions of the American Mathematical Society, 2633-2649.
• [BCM21] Brendle, J., Cardona, M. A., & Mejía, D. A. (2021). Filter-linkedness and its effect on preservation of cardinal characteristics. Annals of Pure and Applied Logic, 172(1), 102856.
• [BF11] Brendle, J., & Fischer, V. (2011). Mad families, splitting families and large continuum. The Journal of Symbolic Logic, 76(1), 198-208.
• [BS87] Blass, A., & Shelah, S. (1987). There may be simple Pℵ1 and Pℵ2-points and the Rudin-Keisler ordering may be downward directed. Annals of pure and applied logic, 33, 213-243.
• [BM99] Blass, A., & Mildenberger, H. (1999). On the cofinality of ultrapowers. The Journal of Symbolic Logic, 64(2), 727-736.
  (This article mentions g is small in the Blass-Shelah model.)
• [GK21] Guzmán, O., & Kalajdzievski, D. (2021). The ultrafilter and almost disjointness numbers. Advances in Mathematics, 386, 107805.
  (This article removed the measurable cardinal assumption from [She04].)
• [GKMS21] Goldstern, M., Kellner, J., Mejía, D. A., & Shelah, S. (2021). Preservation of splitting families and cardinal characteristics of the continuum. Israel Journal of Mathematics, 246(1), 73-129.
• [GS90] Goldstern, M., & Shelah, S. (1990). Ramsey ultrafilters and the reaping number—Con (r< u). Annals of Pure and Applied Logic, 49(2), 121-142.
• [She84] Shelah, S. (1984). On cardinal invariants of the continuum. Axiomatic set theory (Boulder, Colo., 1983), Contemp. Math. 31, 183--207.
• [She92] Shelah, S. (1992). 𝐶𝑂𝑁 (𝔲> 𝔦). Arch. Math. Logic, 31(6), 433-443.
• [She04] Shelah, S. (2004). Two cardinal invariants of the continuum (∂< a) and FS linearly ordered iterated forcing. Acta Mathematica, 192(2), 187-223.