Restricted versions of the Tukey-Teichmüller theorem that are equivalent to the Boolean prime ideal theorem

Abstract

Dedicated to W.W. Comfort on the occasion of his seventieth birthday. AbstractWe formulate a restricted version of the Tukey-Teichmüller Theorem that we denote by (rTT). We then prove that (rTT) and (BPI) are equivalent in ZF and that (rTT) applies rather naturally to several equivalent forms of (BPI): Alexander Subbase Theorem, Stone Representation Theorem, Model Existence and Compactness Theorems for propositional and first-order logic. We also give two variations of (rTT) that we denote by (rTT) + and (rTT) ++ ; each is equivalent to (rTT) in ZF. The variation (rTT) ++ applies rather naturally to various Selection Lemmas due to Cowen, Engeler, and Rado. IntroductionLet ZF denote Zermelo-Fraenkel set theory. It is well known that the axiom of choice (AC) is independent of ZF. When we add AC to ZF we obtain the most widely used axiom system for classical mathematics, namely ZFC = ZF + AC.It is also well known that there are many theorems of ZFC that are in fact equivalent forms of AC. For example, we have the following short list (see [12], [13], and [19] for much longer lists): In this paper we are especially interested in (TT). Recall that a collection A of subsets of a set X has finite character if for every A ⊆ X, A ∈ A if and only if every finite subset of A is in A. Note that ∅ ∈ A (assuming that A is non-empty). There are two versions of the Tukey-Teichmüller Theorem; however, in ZF they are equivalent to each other and also to AC.TT (weak form) Let X be a set and let A be a non-empty collection of subsets of X with finite character. Then A has a maximal element (with respect to ⊆).TT (strong form) Let X be a set and let A be a non-empty collection of subsets of X with finite character. Then for each A ∈ A, there exists B ∈ A such that A ⊆ B and B is maximal.The Tukey-Teichmüller Theorem seems tailor-made for certain applications. For example, it can be used to prove that every vector space has a basis by simply noting that the property of being linearly independent has finite character. Another nice application is a proof of the Alexander Subbase Theorem (see [14] or [7]).The axiom of choice and its equivalent forms are non-constructive principles; they assert the existence of a set without giving instructions on how to construct the set. There are other such non-constructive principles that are perhaps not so well known as those given above. Here is a short list (see The logical status of these ideas is not trivial. The easy part is:The hard part states:• AC cannot be proved from ZF + BPI (Halpern and Lévy [10]);• BPI cannot be proved from ZF (S. Feferman [8]).In section 2 we formulate a version of the Tukey-Teichmüller Theorem (TT) that is equivalent to (BPI) rather than to (AC). We then use this new 2 non-constructive axiom, denoted (rTT) (for restricted Tukey-Teichmüller), to give direct proofs of the Alexander Subbase Theorem and the Stone Representation Theorem. In section 3 we show directly that (rTT) and (BPI) are equivalent in ZF. Finally, in section 4 we use (rTT) to derive various theorems...

Cut‐conditions on sets of multiple‐alternative inferences

“…Note rTT$$_{\mathit {po}}$$ is formulated in the second‐order language based on one 2‐place predicate‐constant and ‘is finite’ as a primitive second‐order predicate. It is a distant cousin of [3, Theorem 3.2], which is also a restricted version of the Tukey‐Teichmüller Lemma, there called rTT$$^{++}$$.…”

“…Note that rTT$$_{\mathit {fubs}}$$ is formulated in the second‐order language of fub‐selector structures, again taking ‘is finite’ as primitive. It is a cousin of rTT$$^{++}$$ from [3], closer to it than was rTT$$_{\mathit {po}}$$. Theorem UT entails rTT$$_{\mathit {fubs}}$$.…”

Liars and Circles: essays on truth, self-reference and paradoxes

“…Another problem then arises: the restrictions imposed by the hierarchy are perhaps too severe, and exclude from the language things that we would like to be able to affirm, and that we can in fact affirm in the ordinary use of language. Priest's example is as follows: 20 I would like to be able to claim that all the sentences written in this text (including this one) are true. But I cannot do such deed, for the sentences said to be true occur under the same truth predicate as the sentence which asserts that they are true.…”