Potential infinity, abstraction principles and arithmetic (Leniewski Style)

This paper starts with an explanation of how the logicist research program can be approached within the framework of Leśniewski’s systems. One nice feature of the system is that Hume’s Principle is derivable in it from an explicit definition of natural numbers. I generalize this result to show that all predicative abstraction principles corresponding to second-level relations, which are provably equivalence relations, are provable. However, the system fails, despite being much neater than the construction of Principia Mathematica (PM). One of the key reasons is that, just as in the case of the system of PM, without the assumption that infinitely many objects exist, (renderings of) most of the standard axioms of Peano Arithmetic are not derivable in the system. I prove that introducing modal quantifiers meant to capture the intuitions behind potential infinity results in the (renderings of) axioms of Peano Arithmetic (PA) being valid in all relational models (i.e. Kripke-style models, to be defined later on) of the extended language. The second, historical part of the paper contains a user-friendly description of Leśniewski’s own arithmetic and a brief investigation into its properties

Some new results on decidability for elementary algebra and geometry

We carry out a systematic study of decidability for theories of (a) real vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces, Banach spaces and metric spaces, all formalised using a 2-sorted first-order language. The theories for list (a) turn out to be decidable while the theories for list (b) are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic. We find that the purely universal and purely existential fragments of the theory of normed spaces are decidable, as is the AE fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbert's 10th problem show that the EA fragments for metric and normed spaces and the AE fragment for normed spaces are all undecidable.Comment: 79 pages, 9 figures. v2: Numerous minor improvements; neater proofs of Theorems 8 and 29; v3: fixed subscripts in proof of Lemma 3

Envelopes, indicators and conservativeness

A well known theorem proved (independently) by J. Paris and H. Friedman states that BΣn +1 (the fragment of Arithmetic given by the collection scheme restricted to Σn +1‐formulas) is a Πn +2‐conservative extension of IΣn (the fragment given by the induction scheme restricted to Σn ‐formulas). In this paper, as a continuation of our previous work on collection schemes for Δn +1(T )‐formulas (see [4]), we study a general version of this theorem and characterize theories T such that T + BΣn +1 is a Πn +2‐conservative extension of T . We prove that this conservativeness property is equivalent to a model‐theoretic property relating Πn ‐envelopes and Πn ‐indicators for T . The analysis of Σn +1‐collection we develop here is also applied to Σn +1‐induction using Parsons' conservativeness theorem instead of Friedman‐Paris' theorem. As a corollary, our work provides new model‐theoretic proofs of two theorems of R. Kaye, J. Paris and C. Dimitracopoulos (see [8]): BΣn +1 and IΣn +1 are Σn +3‐conservative extensions of their parameter free versions, BΣ–n +1 and IΣ–n +1.Junta de Andalucía TIC-13

A note on parameter free Π1-induction and restricted exponentiation

We characterize the sets of all Π2 and all $\mathcal {B}(\Sigma _{1})$equation image (= Boolean combinations of Σ1) theorems of IΠ−1 in terms of restricted exponentiation, and use these characterizations to prove that both sets are not deductively equivalent. We also discuss how these results generalize to n > 0. As an application, we prove that a conservation theorem of Beklemishev stating that IΠ−n + 1 is conservative over IΣ−n with respect to $\mathcal {B}(\Sigma _{n+1})$equation image sentences cannot be extended to Πn + 2 sentences.Ministerio de Educación y Ciencia MTM2005-08658Ministerio de Educación y Ciencia MTM2008-0643

Algebraic combinatorics in bounded induction

In this paper, new methods for analyzing models of weak subsystems of Peano Arithmetic are proposed. The focus will be on the study of algebro-combinatoric properties of certain definable cuts. Their relationship with segments that satisfy more induction, with those limited by the standard powers/roots of an element, and also with definable sets in Bounded Induction is studied. As a consequence, some considerations on the Π1-interpretability of IΔ0 in weak theories, as well as some alternative axiomatizations, are reviewed. Some of the results of the paper are obtained by immersing Bounded Induction models in its Stone-Cech Compactification, once it is endowed with a topology.Ministerio de Ciencia, Innovación y Universidades PID2019-109152GB-I0