# Russell Easterly

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## Registered User

 Name Russell Easterly Member for 8 months Seen 20 hours ago Website Location Seattle WA USA Age
 Mar27 awarded ● Popular Question Mar17 awarded ● Popular Question Feb20 comment Definitions for OddnessYou are right. I am slow. And you are right these are not terribly interesting. We don't really get any new classes. We just can't assume all the statements in a given class are equivalent. Feb20 comment Why Does Induction Prove Multiplication is Commutative?$\forall x(Sx=x+S0)$ is a theorem of PA and weaker theories. I don't know if it is a theorem of GA2. I am interested in theories of arithmetic where induction fails. It looks like finite non-commutative rings are models of ring theory + Not(Ind) + $\forall x(Sx=x+1)$. Feb19 comment Why Does Induction Prove Multiplication is Commutative?Matrix addition and multiplication satisfy all of the axioms of Ring Theory (RT). Non-commutative rings are not models of RT+Ind where Ind is first order induction. Abo gives an example of a phi(x) we can prove using induction that is false in matrix arithmetic. Feb18 revised Why Does Induction Prove Multiplication is Commutative?added 342 characters in body Feb18 comment Why Does Induction Prove Multiplication is Commutative?This still satisfies the successor axiom. An element doesn't even have to have a successor in GA2. Feb18 comment Why Does Induction Prove Multiplication is Commutative?I think GA2 is strong enough to prove if x has a successor then $Sx = x+S0$. I know ring theory proves this. Once we call some element 0 and some element 1 we have no choice on how successor is defined. I know nothing about non-commutative rings, but I assume they satisfy GA2's very weak successor axiom using matrix addition. Feb17 asked Why Does Induction Prove Multiplication is Commutative? Feb15 awarded ● Commentator Feb15 comment Definitions for Oddness$O2 \land ( \forall x \exists y(y^2=x) \to 0=2)$ is false in PA yet is not equivalent to O2, O3, and O4. Doesn't this prove there must be more than two classes of definitions for oddness? I think a conjunction with any statement true in all finite models works. Feb14 comment Definitions for OddnessThanks. So, all of the statements false in PA are equivalent and all the statements true in PA are equivalent. I am still curious if there are definitions of odd that aren't equivalent to one of these two classes. If not, then the complex numbers satisfy every definition of odd. Feb13 revised Definitions for Oddnessdeleted 9 characters in body; added 9 characters in body Feb13 awarded ● Editor Feb13 revised Definitions for Oddnessadded 9 characters in body Feb13 asked Definitions for Oddness Jan26 comment How are mathematical objects defined from an ultrafinitist perspective?I am not sure one can talk about lines and circles in an ultrafinite theory. The set of points equidistant from some point (using some measure) will always be a finite set. I have studied graphs where the number of points equidistant from the origin oscillates between 3*r and 4*r depending on the radius. It is easy to come up with graphs where the average ratio of equidistant points to radius approaches Pi for large r. Jan25 awarded ● Supporter Jan24 awarded ● Scholar Jan24 comment Even XOR Odd Infinities?Thanks again Emil. This has been very educational. Jan24 comment Even XOR Odd Infinities?Thanks Emil! If induction holds in the 2-adic integers does this mean the four square theorem is true in this model? If so, how would I represent -1 (...111) as four squares? Would this be a model for MA2? Jan22 answered Even XOR Odd Infinities? Jan21 comment Even XOR Odd Infinities?I became interested in MA as a tool to prove PA is inconsistent. If my conjecture is provable I think I can show MA proves PA is inconsistent. I would show every possible subset must be even or odd and can't be a model of PA. Then I would have a consistent theory that proves PA is inconsistent. Just showing MA has recursive infinite models is problematic because of Tennenbaum's theorem. Jan21 comment Even XOR Odd Infinities?I use to think so, too. I think we need successor to justify induction. For example, I don't think the complex numbers are a model of MA because it is impossible to define a consistent successor function. Just adding 1 to 0 will never reach a pure complex number like i. Jan21 comment Even XOR Odd Infinities?I ran into this problem, too. Most of my attempts start by assuming -1 is not 0 (I exclude the trivial ring). Jan20 comment Even XOR Odd Infinities?The main differences between MA and commutative ring theory are successor and induction. If my statement can be proven without induction I think it applies to all commutative rings. That would mean the integers are not a commutative ring. Jan20 comment Even XOR Odd Infinities?The smallest model of MA is the trivial ring which satisfies Ashutosh's formula. Let x,y=0. 0=1 is true in this model. We need to include the requirement that x is not 0. I would be happy if someone comes up with an even infinite set. Jan20 awarded ● Nice Question Jan19 awarded ● Student Jan19 asked Even XOR Odd Infinities?