An engaging account of the origins of the modern mathematical theory of the infinite, his book is also a spirited defense against the attacks and misconceptions ...

How can the infinite, a subject so remote from our finite experience, be an everyday tool for the working mathematician? Blending history, philosophy, mathematics, and logic, Shaughan Lavine answers this question with exceptional clarity. Making use of the mathematical work of Jan Mycielski, he demonstrates that knowledge of the infinite is possible, even according to strict standards that require some intuitive basis for knowledge.

Quineans have taken the basic expression of ontological commitment to be an assertion of the form '' x '', assimilated to theEnglish ''there is something that is a ''. Here I take the existential quantifier to be introduced, not as an abbreviation for an expression of English, but via Tarskian semantics. I argue, contrary to the standard view, that Tarskian semantics in fact suggests a quite different picture: one in which quantification is of a substitutional type apparently first proposed by (...) Geach. The ontological burden is borne by constant symbols, and truth is defined separately from reference. (shrink)

If quantum mechanics (QM) is to be taken as an atomistic theory with the elementary particles as atoms (an ATEP), then the elementary particlcs must be individuals. There must then be, for each elementary particle a, a property being identical with a that a alone has. But according to QM, elementary particles of the same kind share all physical properties. Thus, if QM is an ATEP, identity is a metaphysical but not a physical property. That has unpalatable consequences. Dropping the (...) assumption that QM is an ATEP makes it possible to replace the assumption that elementary particles are individuals with the assumption that there are various kinds of elementary stuff that have smallest quantities — the smallest quantity of light, for example, is a photon. The problems about identity disappear, and the explanatory virtues of an ATEP are maintained. (shrink)

A system of finite mathematics is proposed that has all of the power of classical mathematics. I believe that finite mathematics is not committed to any form of infinity, actual or potential, either within its theories or in the metalanguage employed to specify them. I show in detail that its commitments to the infinite are no stronger than those of primitive recursive arithmetic. The finite mathematics of sets is comprehensible and usable on its own terms, without appeal to any form (...) of the infinite. That makes it possible to, without circularity, obtain the axioms of full Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC) by extrapolating (in a precisely defined technical sense) from natural principles concerning finite sets, including indefinitely large ones. The existence of such a method of extrapolation makes it possible to give a comparatively direct account of how we obtain knowledge of the mathematical infinite. The starting point for finite mathematics is Mycielski's work on locally finite theories. (shrink)

Let U be an admissible structure. A cPCd(U) class is the class of all models of a sentence of the form $\neg\exists\bar{K} \bigwedge \Phi$ , where K̄ is an U-r.e. set of relation symbols and φ is an U-r.e. set of formulas of L∞ω that are in U. The main theorem is a generalization of the following: Let U be a pure countable resolvable admissible structure such that U is not Σ-elementarily embedded in HYP(U). Then a class K of countable (...) structures whose universes are sets of urelements is a cPCd(U) class if and only if for some Σ formula σ (with parameters from U), M is in K if and only if M is a countable structure with universe a set of urelements and $(\mathrm{HYP}_\mathfrak{U}(\mathfrak{M}), \mathfrak{U}, \mathfrak{M}) \models \sigma$ , where HYPU(M), the smallest admissible set above M relative to U, is a generalization of HYP to structures with similarity type Σ over U that is defined in this article. Here we just note that when Lα is admissible, HYPLα(M) is Lβ(M) for the least β ≥ α such that Lβ(M) is admissible, and so, in particular, that HYPHF(M) is just HYP(M) in the usual sense when M has a finite similarity type. The definition of HYPU(M) is most naturally formulated using Adamson's notion of a +-admissible structure (1978). We prove a generalization from admissible to +-admissible structures of the well-known truncation lemma. That generalization is a key theorem applied in the proof of the generalized Spector-Gandy theorem. (shrink)

Let $\mathfrak{U}$ be an admissible structure. A $\mathrm{cPC}_d(\mathfrak{U})$ class is the class of all models of a sentence of the form $\neg\exists\bar{K} \bigwedge \Phi$, where $\bar{K}$ is an $\mathfrak{U}$-r.e. set of relation symbols and $\phi$ is an $\mathfrak{U}$-r.e. set of formulas of $\mathscr{L}_{\infty\omega}$ that are in $\mathfrak{U}$. The main theorem is a generalization of the following: Let $\mathfrak{U}$ be a pure countable resolvable admissible structure such that $\mathfrak{U}$ is not $\Sigma$-elementarily embedded in $\mathrm{HYP}(\mathfrak{U})$. Then a class $\mathbf{K}$ of countable structures (...) whose universes are sets of urelements is a $\mathrm{cPC}_d(\mathfrak{U})$ class if and only if for some $\Sigma$ formula $\sigma$ (with parameters from $\mathfrak{U}$), $\mathfrak{M}$ is in $\mathbf{K}$ if and only if $\mathfrak{M}$ is a countable structure with universe a set of urelements and $(\mathrm{HYP}_\mathfrak{U}(\mathfrak{M}), \mathfrak{U}, \mathfrak{M}) \models \sigma$, where $\mathrm{HYP}_\mathfrak{U}(\mathfrak{M})$, the smallest admissible set above $\mathfrak{M}$ relative to $\mathfrak{U}$, is a generalization of $\mathrm{HYP}$ to structures with similarity type $\Sigma$ over $\mathfrak{U}$ that is defined in this article. Here we just note that when $\mathrm{L}_\alpha$ is admissible, $\mathrm{HYP}_{\mathrm{L}\alpha}(\mathfrak{M})$ is $\mathrm{L}_\beta(\mathfrak{M})$ for the least $\beta \geq \alpha$ such that $\mathrm{L}_\beta(\mathfrak{M})$ is admissible, and so, in particular, that $\mathrm{HYP}_{\mathbb{HF}}(\mathfrak{M})$ is just $\mathrm{HYP}(\mathfrak{M})$ in the usual sense when $\mathfrak{M}$ has a finite similarity type. The definition of $\mathrm{HYP}_\mathfrak{U}(\mathfrak{M})$ is most naturally formulated using Adamson's notion of a +-admissible structure (1978). We prove a generalization from admissible to +-admissible structures of the well-known truncation lemma. That generalization is a key theorem applied in the proof of the generalized Spector-Gandy theorem. (shrink)

It is well known that, in the terminology of Moschovakis, Descriptive set theory (1980), every adequate normed pointclass closed under ∀ω has an effective version of the generalized reduction property (GRP) called the easy uniformization property (EUP). We prove a dual result: every adequate normed pointclass closed under ∃ω has the EUP. Moschovakis was concerned with the descriptive set theory of subsets of Polish topological spaces. We set up a general framework for parts of descriptive set theory and prove results (...) that have as special cases not only the just-mentioned topological results, but also corresponding results concerning the descriptive set theory of classes of structures. Vaught (1973) asked whether the class of cPCδ classes of countable structures has the GRP. It does. A cPC(A) class is the class of all models of a sentence of the form ¬∃K̄φ, where φ is a sentence of L∞ω that is in A and K̄ is a set of relation symbols that is in A. Vaught also asked whether there is any primitive recursively closed set A such that some effective version of the GRP holds for the class of cPC(A) classes of countable structures. There is: The class of cPC(A) classes of countable structures has the EUP if ω ∈ A and A is countable and primitive recursively closed. Those results and some extensions are obtained by first showing that the relevant classes of classes of structures, which Vaught showed normed, are in a suitable sense adequate and closed under ∃ω, and then applying the dual easy uniformization theorem. (shrink)

Let A be an admissible set. A sentence of the form ∀R̄φ is a ∀1(A) (∀s 1(A),∀1(Lω1ω)) sentence if φ ∈ A (φ is $\bigvee\Phi$ , where Φ is an A-r.e. set of sentences from A; φ ∈ Lω1ω). A sentence of the form ∃R̄φ is an ∃2(A) (∃s 2(A),∃2(Lω1ω)) sentence if φ is a ∀1(A) (∀s 1(A),∀1(Lω1ω)) sentence. A class of structures is, for example, a ∀1(A) class if it is the class of models of a ∀1(A) sentence. Thus (...) ∀1(A) is a class of classes of structures, and so forth. Let Mi be the structure $\langle i, 0$. Let Γ be a class of classes of structures. We say that a sequence $J_1,\ldots,J_i,\ldots, i < \omega$, of classes of structures is a Γ sequence if $J_i \in \Gamma, i < \omega$, and there is I ∈ Γ such that M ∈ Ji if and only if [M, Mi] ∈ I, where [,] is the disjoint sum. A class Γ of classes of structures has the easy uniformization property if for every Γ sequence $J_1,\ldots,J_i, \ldots, i < \omega$, there is a Γ sequence $J'_1,\ldots,J'_i,\ldots, i < \omega$, such that $J'_i \subseteq J_i, i < \omega, \bigcup J'_i = \bigcup J_i$, and the J'i are pairwise disjoint. The easy uniformization property is an effective version of Kuratowski's generalized reduction property that is closely related to Moschovakis's (topological) easy uniformization property. We show over countable structures that ∀1(A) and ∃2(A) have the easy uniformization property if A is a countable admissible set with an infinite member, that ∀s 1(Lα) and ∃s 2(Lα) have the easy uniformization property if α is countable, admissible, and not weakly stable, and that ∀1(Lω1ω) and ∃2(Lω1ω) have the easy uniformization property. The results proved are more general. The result for ∀s 1(Lα) answers a question of Vaught (1980). (shrink)

Based on her earlier ground-breaking axiomatization of quantified modal logic, the papers collected here by the distinguished philosopher Ruth Barcan Marcus cover much ground in the development of her thought, including influential essays on moral conflict, on belief and rationality, and on some historical figures.