Wikipedia:Reference desk/Archives/Mathematics/2016 August 11

= August 11 =

Peano Arithmetic: Are there propositions provable in PA, yet not in the following similar system?
I'm asking, because I'd like to replace the (first order) axiom of induction by a new axiom claiming that every positive number can - be subtracted from every bigger (or not smaller) number - and divide some number lying in the "immediate local neighborhood" of the bigger number, i.e. $$\forall x \forall y \Biggl((y=0)\lor(y>x)\lor\exists z(x=y+z)\land\exists m\exists n\Bigl((m<y)\land(x+m=y\cdot n)\Bigr)\Biggr)$$, and I hope this new axiom is not weaker than the (first order) axiom of induction. HOTmag (talk) 09:43, 11 August 2016 (UTC)
 * Yes. First order PA is not finitely axiomatizable, which is why the axiom of induction is actually an axiom schema, necessarily.
 * I don't have a reference handy other than Simpson's book, but for every n there is a model of (Robinson Arithmetic) + ($$\Sigma_n$$-induction) + (not $$\Sigma_{n+1}$$-induction). Recall that PA is (Robinson Arithmetic) + ($$\Sigma_n$$-induction for all n).  So consider A a putative finite axiomatization of PA.  Since proofs are finite, A is provable from a finite fragment of the standard axiomatization of PA, which means that there's some n such that A is provable from (Robinson Arithmetic) + ($$\Sigma_n$$-induction).  So there is a structure that models A but not PA, and thus A is not an axiomatization of PA.
 * So there is some n such that your axiom cannot prove $$\Sigma_n$$-induction. I suspect n is 1 or 2.--2406:E006:178C:1:781F:A651:32E2:B81E (talk) 10:20, 11 August 2016 (UTC)
 * I still wonder how such a proposition (unprovable in my axiom system) looks like. HOTmag (talk) 10:39, 11 August 2016 (UTC)
 * Okay. Your axiom can be proved with bounded induction, which is weaker than $$\Sigma_1$$-induction.  It's known that $$\Sigma_2$$-induction can prove that Ackermann's function is total, but $$\Sigma_1$$-induction cannot.  So this is a statement unprovable in your axiom system.--2406:E006:178C:1:781F:A651:32E2:B81E (talk) 10:51, 11 August 2016 (UTC)
 * All right. Now I wonder if there are propositions, provable in $$\Sigma_1$$-induction, yet not in the system I've suggested. HOTmag (talk) 11:16, 11 August 2016 (UTC)