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 ???.
Reference List
- [BJ95] Bartoszynski, T., & Judah, H. (1995). Set Theory: on the structure of the real line. CRC Press.
- [Bla10] Blass, A. (2010). Combinatorial cardinal characteristics of the continuum. In Handbook of set theory (pp. 395–489). Springer.