Implications

We have the following indirect implication of form equivalence classes:

20 \(\Rightarrow\) 243
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
20 \(\Rightarrow\) 101 Partition principles and infinite sums of cardinal numbers, Higasikawa, M. 1995, Notre Dame J. Formal Logic
101 \(\Rightarrow\) 40 On some weak forms of the axiom of choice in set theory, Pelc, A. 1978, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
40 \(\Rightarrow\) 43 Consistency results for $ZF$, Jensen, R.B. 1967, Notices Amer. Math. Soc.
On cardinals and their successors, Jech, T. 1966a, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
43 \(\Rightarrow\) 243 A First Course in Abstract Algebra (4th edition), Fraleigh, 1989, pages 333-336

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
20:

If \(\{A_{x}: x \in S \}\) and \(\{B_{x}: x \in  S\}\) are families  of pairwise disjoint sets and \( |A_{x}| = |B_{x}|\) for all \(x\in S\), then \(|\bigcup_{x\in S}A_{x}| = |\bigcup_{x\in S} B_{x}|\). Moore [1982] (1.4.12 and 1.7.8).

101:

Partition Principle:  If \(S\) is a partition of \(M\), then \(S \precsim M\).

40:

\(C(WO,\infty)\):  Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.

43:

\(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\)  is  a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\)  then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\).  See Tarski [1948], p 96, Levy [1964], p. 136.

243:

Every  principal ideal domain is a unique factorization domain.

Comment:

Back