Russell Easterly
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Registered User
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Mar 27 |
awarded | ● Popular Question |
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Mar 17 |
awarded | ● Popular Question |
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Feb 20 |
comment |
Definitions for Oddness You 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. |
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Feb 20 |
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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) + $\forall x(Sx=x+1)$. |
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Feb 19 |
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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. |
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Feb 18 |
revised |
Why Does Induction Prove Multiplication is Commutative? added 342 characters in body |
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Feb 18 |
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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. |
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Feb 18 |
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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. |
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Feb 17 |
asked | Why Does Induction Prove Multiplication is Commutative? |
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Feb 15 |
awarded | ● Commentator |
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Feb 15 |
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. |
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Feb 14 |
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Definitions for Oddness Thanks. 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. |
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Feb 13 |
revised |
Definitions for Oddness deleted 9 characters in body; added 9 characters in body |
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Feb 13 |
awarded | ● Editor |
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Feb 13 |
revised |
Definitions for Oddness added 9 characters in body |
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Feb 13 |
asked | Definitions for Oddness |
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Jan 26 |
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. |
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Jan 25 |
awarded | ● Supporter |
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Jan 24 |
awarded | ● Scholar |
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Jan 24 |
comment |
Even XOR Odd Infinities? Thanks again Emil. This has been very educational. |
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Jan 24 |
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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? |
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Jan 22 |
answered | Even XOR Odd Infinities? |
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Jan 21 |
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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. |
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Jan 21 |
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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. |
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Jan 21 |
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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). |
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Jan 20 |
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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. |
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Jan 20 |
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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. |
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Jan 20 |
awarded | ● Nice Question |
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Jan 19 |
awarded | ● Student |
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Jan 19 |
asked | Even XOR Odd Infinities? |
