We suggest a new approach for addressing the problem of establishing an axiomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the basis of our knowledge of ω itself, or of generally agreed upon intuitions about the true nature of the mathematical universe, what the right strengthening of the Axiom of Infinity is—which large (...) cardinals ought to be derivable? It was shown in the 1960s by Lawvere that the existence of an infinite set is equivalent to the existence of a certain kind of structure-preserving transformation from V to itself, not isomorphic to the identity. We use Lawvere's transformation, rather than ω, as a starting point for a reasonably natural sequence of strengthenings and refinements, leading to a proposed strong Axiom of Infinity. A first refinement was discussed in later work by Trnková—Blass, showing that if the preservation properties of Lawvere's tranformation are strengthened to the point of requiring it to be an exact functor , such a transformation is provably equivalent to the existence of a measurable cardinal. We propose to push the preservation properties as far as possible, short of inconsistency. The resulting transformation V→V is strong enough to account for virtually all large cardinals, but is at the same time a natural generalization of an assertion about transformations V→V known to be equivalent to the Axiom of Infinity. (shrink)

One of the most distinctive and intriguing developments of modern set theory has been the realization that, despite widely divergent incentives for strengthening the standard axioms, there is essentially only one way of ascending the higher reaches of infinity. To the mathematical realist the unexpected convergence suggests that all these axiomatic extensions describe different aspects of the same underlying reality.

This paper discusses a sequence of extensions ofNFU, Jensen's improvement of Quine's set theory “New Foundations” (NF) of [16].The original theoryNFof Quine continues to present difficulties. After 60 years of intermittent investigation, it is still not known to be consistent relative to any set theory in which we have confidence. Specker showed in [20] thatNFdisproves Choice (and so proves Infinity). Even if one assumes the consistency ofNF, one is hampered by the lack of powerful methods for proofs of consistency (...) and independence such as are available for use withZFC; very clever work has been done with permutation methods, starting with [18] and [5], and exemplified more recently by [14], but permutation methods can only be applied to show the consistency or independence of unstratified sentences (see the definition ofNFUbelow for a definition of stratification). For example, there is no method available to determine whether the assertion “the continuum can be well-ordered” is consistent with or independent ofNF. There is one substantial independence result for an assertion with nontrivial stratified consequences, using metamathematical methods: this is Orey's proof of the independence of the Axiom of Counting fromNF(see below for a statement of this axiom).We mention these difficulties only to reassure the reader of their irrelevance to the present work. Jensen's modification of “New Foundations” (in [13]), which was to restrict extensionality to sets, allowing many non-sets (urelements) with no elements, has almost magical effects. (shrink)

This paper discusses a sequence of extensions ofNFU, Jensen's improvement of Quine's set theory “New Foundations” (NF) of [16].The original theoryNFof Quine continues to present difficulties. After 60 years of intermittent investigation, it is still not known to be consistent relative to any set theory in which we have confidence. Specker showed in [20] thatNFdisproves Choice (and so proves Infinity). Even if one assumes the consistency ofNF, one is hampered by the lack of powerful methods for proofs of consistency (...) and independence such as are available for use withZFC; very clever work has been done with permutation methods, starting with [18] and [5], and exemplified more recently by [14], but permutation methods can only be applied to show the consistency or independence of unstratified sentences (see the definition ofNFUbelow for a definition of stratification). For example, there is no method available to determine whether the assertion “the continuum can be well-ordered” is consistent with or independent ofNF. There is one substantial independence result for an assertion with nontrivial stratified consequences, using metamathematical methods: this is Orey's proof of the independence of the Axiom of Counting fromNF(see below for a statement of this axiom).We mention these difficulties only to reassure the reader of their irrelevance to the present work. Jensen's modification of “New Foundations” (in [13]), which was to restrict extensionality to sets, allowing many non-sets (urelements) with no elements, has almost magical effects. (shrink)

We use NF to designate the system known as Quine's New Foundations, and NF + AF to designate the same system with a suitable axiom of infinity adjoined. We use ML to designate the revised system appearing in the third printing of Quine's “Mathematical Logic”. This system ML is just the systemPproposed by Wang in [4], and essentially includes NF as a part.The pripcipal results of the present paper are:A. In NF the axiom of infinity is (...) equivalent to the definability of an ordered pair having the same type as its constituents.B. Considering the formulas of NF as having translations in the formalism of ML in the sense defined in [4], we find that a formula of NF is provable in NF if and only if its translation is provable in ML. From the fact that the axiom of infinity is provable in ML, one might be tempted to conclude that the axiom of infinity is provable in NF. It is explained why this is not a valid inference. Consequently, it would appear that there is a sense in which NF + AF is stronger than ML. (shrink)

Related Works: Original Paper: M. Randall Holmes. Strong Axioms of Infinity in NFU. J. Symbolic Logic, Volume 66, Issue 1 , 87--116. Project Euclid: euclid.jsl/1183746361.

Related Works: Original Paper: M. Randall Holmes. Strong Axioms of Infinity in NFU. J. Symbolic Logic, Volume 66, Issue 1, 87--116. Project Euclid: euclid.jsl/1183746361.

In this paper we introduce a theory of finite sets FST with a strong negation of the axiom of infinity asserting that every set is provably bijective with a natural number. We study in detail the role of the axioms of Power Set, Choice, Regularity in FST, pointing out the relative dependences or independences among them. FST is shown to be provably equivalent to a fragment of Alternative Set Theory. Furthermore, the introduction of FST is motivated in view (...) of a non-standard development. MSC: 03E30, 03E35. (shrink)

We will give a simple philosophical "proof" of the negation of Cantor's continuum hypothesis (CH). (A formal proof for or against CH from the axioms of ZFC is impossible; see Cohen [1].) We will assume the axioms of ZFC together with intuitively clear axioms which are based on some intuition of Stuart Davidson and an old theorem of Sierpinski and are justified by the symmetry in a thought experiment throwing darts at the real number line. We will in fact show (...) why there must be an infinity of cardinalities between the integers and the reals. We will also show why Martin's Axiom must be false, and we will prove the extension of Fubini's Theorem for Lebesgue measure where joint measurability is not assumed. Following the philosophy--if you reject CH you are only two steps away from rejecting the axiom of choice (AC)--we will point out along the way some extensions of our intuition which contradict AC. (shrink)

Abstract:As a contribution to the critical appreciation of a central thesis in Russell’s philosophical logic, I consider the Third Man objection to Platonic realism in the philosophy of mathematics, and argue that the Third Man infinite regress, for those who accept its assumptions, provides a worthy substitute for Whitehead and Russell’s Axiom of Infinity in positing a denumerably infinite set or series onto which other sets, series, and formal operations in the foundations of mathematics can be mapped.

I wish to thank Klaus Kühnle who streamlined in [8] several of my definitions and proofs concerning the subject matter of this paper. Some ideas and results arose from discussions with Klaus Leeb. Jan Johannsen discovered some mistakes in an earlier version.

In the early 1900s, Russell began to recognize that he, and many other mathematicians, had been using assertions like the Axiom of Choice implicitly, and without explicitly proving them. In working with the Axioms of Choice, Infinity, and Reducibility, and his and Whitehead’s Multiplicative Axiom, Russell came to take the position that some axioms are necessary to recovering certain results of mathematics, but may not be proven to be true absolutely. The essay traces historical roots of, and (...) motivations for, Russell’s method of analysis, which are intended to shed light on his view about the status of mathematical axioms. I describe the position Russell develops in consequence as “immanent logicism,” in contrast to what Irving (1989) describes as “epistemic logicism.” Immanent logicism allows Russell to avoid the logocentric predicament, and to propose a method for discovering structural relationships of dependence within mathematical theories. (shrink)

In this paper is used to denote Jensen's modification of Quine's ‘new foundations’ set theory () fortified with a type‐level pairing function but without the axiom of choice. The axiom is the variant of the axiom of counting which asserts that no finite set is smaller than its own set of singletons. This paper shows that proves the consistency of the simple theory of types with infinity (). This result implies that proves that consistency of, and (...) that proves the consistency of. (shrink)

Those incompleteness theorems mean the relation of (Peano) arithmetic and (ZFC) set theory, or philosophically, the relation of arithmetical finiteness and actual infinity. The same is managed in the framework of set theory by the axiom of choice (respectively, by the equivalent well-ordering "theorem'). One may discuss that incompleteness form the viewpoint of set theory by the axiom of choice rather than the usual viewpoint meant in the proof of theorems. The logical corollaries from that "nonstandard" viewpoint (...) the relation of set theory and arithmetic are demonstrated. (shrink)

It is proved that in a suitable intuitionistic, locally classical, version of the theory ZFC deprived of the axiom of infinity, the requirement that every set be finite is equivalent to the assertion that every ordinal is a natural number. Moreover, the theory obtained with the addition of these finiteness assumptions is equivalent to a theory of hereditarily finite sets, developed by Previale in "Induction and foundation in the theory of hereditarily finite sets." This solves some problems stated (...) there. The analysis is undertaken using for each of these results a limited fragment of the relevant theory. (shrink)

We measure, in the presence of the axiom of infinity, the proof-theoretic strength of the axioms of set theory which make the theory look really like a “theory of sets”, namely, the axiom of extensionality Ext, separation axioms and the axiom of regularity Reg . We first introduce a weak weak set theory as a base over which to clarify the strength of these axioms. We then prove the following results about proof-theoretic ordinals:1. and ,2. and (...) . We also show that neither Reg nor affects the proof-theoretic strength, i.e., where T is Basic plus any combination of Ext and. (shrink)

By obtaining several new results on Cook-style two-sorted bounded arithmetic, this paper measures the strengths of the axiom of extensionality and of other weak fundamental set-theoretic axioms in the absence of the axiom of infinity, following the author’s previous work [K. Sato, The strength of extensionality I — weak weak set theories with infinity, Annals of Pure and Applied Logic 157 234–268] which measures them in the presence. These investigations provide a uniform framework in which three (...) different kinds of reverse mathematics–Friedman–Simpson’s “orthodox” reverse mathematics, Cook’s bounded reverse mathematics and large cardinal theory–can be reformulated within one language so that we can compare them more directly. (shrink)

The axiom of infinity can be expressed by stating the existence of sets satisfying a formula which involves restricted universal quantifiers only, even if the axiom of foundation is not assumed.

Alexander R. Pruss examines a large family of paradoxes to do with infinity - ranging from deterministic supertasks to infinite lotteries and decision theory. Having identified their common structure, Pruss considers at length how these paradoxes can be resolved by embracing causal finitism.

This volume is based on the talks given at the Workshop on Infinity and Truth held at the Institute for Mathematical Sciences, National University of Singapore, from 25 to 29 July 2011. The chapters are by leading experts in mathematical and philosophical logic that examine various aspects of the foundations of mathematics. The theme of the volume focuses on two basic foundational questions: (i) What is the nature of mathematical truth and how does one resolve questions that are formally (...) unsolvable within the Zermelo Fraenkel Set Theory with the Axiom of Choice, and (ii) Do the discoveries in mathematics provide evidence favoring one philosophical view over others? These issues are discussed from the vantage point of recent progresses in foundational studies. The final chapter features questions proposed by the participants of the Workshop that will drive foundational research. The wide range of topics covered here will be of benefit to students, researchers and mathematicians interested in the foundations of mathematics. (shrink)

This article is concerned with reflection principles in the context of Cantor’s conception of the set-theoretic universe. We argue that within such a conception reflection principles can be formulated that confer intrinsic plausibility to strong axioms of infinity.

Two strategies to infinity are equally relevant for it is as universal and thus complete as open and thus incomplete. Quantum mechanics is forced to introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can demonstrate that essential properties of quantum information, entanglement, and quantum computer originate directly from infinity once it is involved in quantum mechanics. Thus, thеse phenomena can be elucidated as both complete and incomplete, after which choice is the border (...) between them. A special kind of invariance to the axiom of choice shared by quantum mechanics is discussed to be involved that border between the completeness and incompleteness of infinity in a consistent way. The so-called paradox of Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted entirely in the same terms only of set theory. Quantum computer can demonstrate especially clearly the privilege of the internal position, or “observer”, or “user” to infinity implied by Henkin’s proposition as the only consistent ones as to infinity. (shrink)

This paper deals with the treatment of infinity and finiteness in mereology. After an overview of some first-order mereological theories, finiteness axioms are introduced along with a mereological definition of “x is finite” in terms of which the axioms themselves are derivable in each of those theories. The finiteness axioms also provide the background for definitions of “ T makes an assumption of infinity”. In addition, extensions of mereological theories by the axioms are investigated for their own sake. (...) In the final part, a definition of “x is finite” stated in a second-order language is also presented, followed by some concluding remarks on the motivation for the study of the extensions of mereological theories dealt with in the paper. (shrink)

It is controversial whether Wittgenstein's philosophy of mathematics is of critical importance for mathematical proofs, or is only concerned with the adequate philosophical interpretation of mathematics. Wittgenstein's remarks on the infinity of prime numbers provide a helpful example which will be used to clarify this question. His antiplatonistic view of mathematics contradicts the widespread understanding of proofs as logical derivations from a set of axioms or assumptions. Wittgenstein's critique of traditional proofs of the infinity of prime numbers, specifically (...) those of Euler and Euclid, not only offers philosophical insight but also suggests substantive improvements. A careful examination of his comments leads to a deeper understanding of what proves the infinity of primes. (shrink)

Peano arithmetic cannot serve as the ground of mathematics for it is inconsistent to infinity, and infinity is necessary for its foundation. Though Peano arithmetic cannot be complemented by any axiom of infinity, there exists at least one (logical) axiomatics consistent to infinity. That is nothing else than a new reading at issue and comparative interpretation of Gödel’s papers (1930; 1931) meant here. Peano arithmetic admits anyway generalizations consistent to infinity and thus to some (...) addable axiom(s) of infinity. The most utilized example of those generalizations is the complex Hilbert space. Any generalization of Peano arithmetic consistent to infinity, e.g. the complex Hilbert space, can serve as a foundation for mathematics to found itself and by itself. (shrink)