December 6th, 2017, 07:15 AM  #41 
Senior Member Joined: May 2016 From: USA Posts: 923 Thanks: 369  But that standard arithmetic is a ring. And your construction is not a ring. So there is an inconsistency. That's enough of my time.

December 6th, 2017, 07:17 AM  #42 
Senior Member Joined: Aug 2012 Posts: 109 Thanks: 0  
December 6th, 2017, 08:25 AM  #43  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,236 Thanks: 2412 Math Focus: Mainly analysis and algebra  Quote:
\begin{align*} I_1 &= I_1 + I_2 &\text{because $I_2$ is an identity element} \\ I_2 &= I_1 + I_2 &\text{because $I_1$ is an identity element} \\ Thus \\ I_1 &= I_2 &\text{by transitivity}\end{align*} So there is only one identity element for each operation. We can't even escape by doing away with commutativity. The only way to get out of this is for $I_1 + I_2$ to take more than one value, but then the questions, for any given scenario, are which and why? Last edited by v8archie; December 6th, 2017 at 08:30 AM.  
December 6th, 2017, 08:54 AM  #44 
Senior Member Joined: Jun 2015 From: England Posts: 766 Thanks: 223  Last edited by skipjack; December 6th, 2017 at 10:45 AM. 
December 6th, 2017, 09:12 AM  #45  
Senior Member Joined: Aug 2012 Posts: 109 Thanks: 0  Quote:
Nothing says that two additive identity elements Must be equal Therefore as you asked earlier... TRANSITIVITY does not work for the additive identity elements/but remains the same elsewhere I_1 can NOT replace I_2 This does NOT mean addition doesn't work for the additive identity elements Last edited by Conway51; December 6th, 2017 at 09:14 AM.  
December 6th, 2017, 09:13 AM  #46 
Senior Member Joined: Aug 2012 Posts: 109 Thanks: 0  Your reply is clearly flaming.
Last edited by skipjack; December 6th, 2017 at 10:49 AM. 
December 6th, 2017, 09:23 AM  #47  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,236 Thanks: 2412 Math Focus: Mainly analysis and algebra  Quote:
Your system appears to be degenerating into incoherence. Given any number, nobody can tell which version of the number it is. Last edited by greg1313; December 6th, 2017 at 09:45 AM.  
December 6th, 2017, 09:43 AM  #48 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,767 Thanks: 1016 Math Focus: Elementary mathematics and beyond  

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