0
$\begingroup$

Philosophers sometimes ask which axioms are "true". Of course, no one believes that there is a serious question as to whether the axiom of commutativity for groups, or the Parallel Postulate, is true. In the former case, we're just talking about the class of models of the axioms. In the latter, we talk about a different geometrical space by giving a different answer to the Parallel Postulate question. But philosophers worry about the axioms of set theory and arithmetic, as if those were different. In general, I don't see the motivation for this. I don't see why we should regard a "universe" satisfying Determinacy differently from geometrical space satisfying the negation of the Parallel Postulate. But it does seem hard to hold this view across the board. After all, the view would be something like "every (first-order) consistent set of axioms is equally legitimate", and the claim that a theory is consistent is a Pi_1 arithmetic sentence which one can consistently deny! So, assuming that there is an objective fact (not sure how to put it) as to whether at theory like PA is consistent, is the most radical but coherent view in the neighborhood that "all Pi_1 sound theories are equally legitimate"? How do you think about this issue?

Following Carl's suggestion (below), my concrete question is the following. Is there anything subtly inconsistent in the idea that PA + Con(PA) stands to Euclidean geometry as PA + ~Con(PA) stands to Hyperbolic Geometry -- equally true, simply true of different structures?

$\endgroup$
  • 1
    $\begingroup$ There is much that can be said, but I think that the current phrasing of this question tends towards "let's have a discussion about..." rather than a concrete question. That kind of open-ended, subjective question is discouraged, even if there is no firm definition of which questions fit into the profile. Is there a way to ask a concrete question to get towards what you are interested in here? $\endgroup$ – Carl Mummert Feb 26 '18 at 1:11
  • $\begingroup$ Thanks, Carl. How about: is there anything subtly inconsistent in the idea that PA + Con(PA) stands to Euclidean geometry as PA + ~Con(PA) stands to Hyperbolic Geometry -- equally true, simply true of different structures? $\endgroup$ – davidp Feb 26 '18 at 1:19
  • 2
    $\begingroup$ There is a difference here: arithmetic and set theory are useful as foundations, whereas geometry and group theory are not. You generally want to know what holds in your foundations. Moreover, arithmetic has a very canonical model that the term "true" is defined according to it. $\endgroup$ – Asaf Karagila Feb 26 '18 at 8:09
  • $\begingroup$ Thanks, Asaf. Two questions. What do you mean: "you generally want to know what holds in your foundations?" You mean you want there to be an objective fact in the way there is not in geometry, say? If so, why? Second, what do you mean by saying that "true" is defined in terms of the canonical model of arithmetic? Do you mean that the Tarski definition presupposes that we mean finite by "finite" because it speaks of sequences and so forth, and these are finite? $\endgroup$ – davidp Feb 26 '18 at 19:25

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.