Dopico, Pablo (2023) A defence of Isaacson's thesis, or how to make sense of the boundaries of finite mathematics. [Preprint]

Text
A defence of Isaacson's thesis, or how to make sense of the boundaries of finite mathematics.pdf Download (467kB)  Preview 


Text
A defence of Isaacson's thesis, or how to make sense of the boundaries of finite mathematics.pdf Download (468kB)  Preview 
Abstract
Daniel Isaacson has advanced an epistemic notion of arithmetical truth according to which the latter is the set of truths that we grasp on the basis of our understanding of the structure of natural numbers alone. Isaacson's thesis is then the claim that Peano Arithmetic (PA) is the theory of finite mathematics, in the sense that it proves all and only arithmetical truths thus understood. In this paper, we raise a challenge for the thesis and show how it can be overcome. We introduce the concept of purity for arithmetical theories: an arithmetical theory is pure when it only proves arithmetical truths. Then, we argue that, under Isaacson's thesis, some PAprovable truths—including transfinite induction claims for infinite ordinals and consistency statements—are seemingly not arithmetical in Isaacson's sense, and hence that Isaacson's thesis might entail the impurity of PA. Nonetheless, we conjecture that the advocate of Isaacson's thesis can avoid this undesirable consequence: the arithmetical nature, as understood by Isaacson, of all contentious PAprovable statements can be justified. As a case study, we explore how this is done for transfinite induction claims with infinite ordinals below ε_0. To this end, we show that the PAproof of such claims exclusively employs resources from finite mathematics, and that ordinals below ε_0 are finitary objects despite being infinite.
Export/Citation:  EndNote  BibTeX  Dublin Core  ASCII/Text Citation (Chicago)  HTML Citation  OpenURL 
Social Networking: 
Item Type:  Preprint  

Creators: 


Keywords:  Isaacson's thesis, Peano Arithmetic, arithmetical truth, finite mathematics  
Subjects:  Specific Sciences > Mathematics > Foundations Specific Sciences > Mathematics > Proof Specific Sciences > Mathematics 

Depositing User:  Mr Pablo Dopico  
Date Deposited:  03 Jan 2024 22:56  
Last Modified:  03 Jan 2024 22:56  
Item ID:  22904  
Subjects:  Specific Sciences > Mathematics > Foundations Specific Sciences > Mathematics > Proof Specific Sciences > Mathematics 

Date:  30 December 2023  
URI:  https://philsciarchive.pitt.edu/id/eprint/22904 
Monthly Views for the past 3 years
Monthly Downloads for the past 3 years
Plum Analytics
Actions (login required)
View Item 