July 29th, 2016, 12:19 AM  #1 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 424 Thanks: 27 Math Focus: Number theory  Cardinality of all operations on rational numbers
What is the cardinality of all operations of addition, multiplication and exponentiation upon rational numbers?

July 29th, 2016, 09:16 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,664 Thanks: 2644 Math Focus: Mainly analysis and algebra 
Cardinality applies to sets, not operations. However, there are only a countably infinite number of finite sums, multiplications, exponentiation, etc. using only rational numbers. Since nobody knows (for sure) how to evaluate infinite sums, products, exponentiations, etc. it doesn't necessarily make sense to talk about them. But there are uncountably many infinite summation, multiplication, exponentiation, etc. expressions. All of this comes from applications of Cantor's diagonal argument or the cardinality of countably infinitely many countably infinite sets. 
July 29th, 2016, 09:37 AM  #3  
Banned Camp Joined: Dec 2013 Posts: 1,117 Thanks: 41  Quote:
Helping someone is not useful when you "enslave" him. You are not helping him. You are just making him behave like you. You are parrot and you are trying to make others parrots like you.  
July 29th, 2016, 10:36 AM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,664 Thanks: 2644 Math Focus: Mainly analysis and algebra 
If you disagree with my analysis, post some coherent counterargument rather than hurling abuse.

July 29th, 2016, 10:51 AM  #5 
Banned Camp Joined: Dec 2013 Posts: 1,117 Thanks: 41 
Loren is not dumb and you are too dumb to understand what he has in mind. His question in fact is " what if we apply the concept of cardinality to operations not sets? You are reacting like robot because you are assuming that Loren knows nothing about set theory (which is itself controversial). Contempt contempt contempt at work 
July 29th, 2016, 11:34 AM  #6 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 424 Thanks: 27 Math Focus: Number theory 
Actually, v8archie's response is generally helpful  somewhat at my level of understanding. I needed some reminding and application. I too am sensitive to criticism, but usually benefit when it is constructive. He knows from previous conversations that I am able to comprehend most of what he says. Repetition here helps. His comment about evaluating infinite sums (and products and exponents) was insightful. Overall, you identified my concern: "[Loren's] question in fact is 'what if we apply the concept of cardinality to operations not sets?'" Or sets of operations. Thanks, mobel. 
July 29th, 2016, 11:56 AM  #7 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,948 Thanks: 1139 Math Focus: Elementary mathematics and beyond  mobel, please be polite. Personal attacks are not helpful to anyone.

July 29th, 2016, 12:12 PM  #8 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,664 Thanks: 2644 Math Focus: Mainly analysis and algebra  I'm pleased to hear it. How are you thinking to apply cardinality to operations? There can be only countably many additions in a single expression because each adds another term. Similarly for multiplication. You can only raise something to a countably infinite number of powers too. In fact all operations, because they are evaluated in sequence, can be used at most a countably infinite number of times in a single expression. 
July 29th, 2016, 01:21 PM  #9 
Senior Member Joined: Jun 2015 From: England Posts: 915 Thanks: 271 
I gotta say that we are using the term 'operation' rather loosely here. I guess you mean binary operation? A binary operation mean an operation that combines two elements of the source set (rationals in this case) to form an output. As archie has observed we can cascade such operations to three, four or more elements but we can only do that for as many elements as there are in the set and the rationals are countably infinite. If we are allowed to exponentiate to powers that are not in the set (eg to the power of any real number) then there is no countable restriction like this. 
July 29th, 2016, 10:11 PM  #10 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 424 Thanks: 27 Math Focus: Number theory 
I hear you, v8archie and studiot. The number of elements for possible operations N on the rationals (Q  countably infinite elements) is primarily countably infinite. What about the possible cardinality for the set of operations with elements numbering as N=2^Q, as studiot may be suggesting? 

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cardinality, numbers, operations, rational 
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