July 27th, 2016, 07:58 PM  #21 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra 
Maschke was talking about the use of the word "defined", going on to talk about definable numbers. There are countably many of these, and I believe the definition of a definable number is one that can be defined using a finite number of symbols.

August 1st, 2016, 09:32 AM  #22  
Senior Member Joined: Aug 2012 Posts: 2,357 Thanks: 740  Quote:
Note that even though every real has a decimal representation, you can only write down countably many of them using finite strings of symbols. That's (more or less) the idea of definability. For example, $\displaystyle \pi$ can be uniquely characterized by several different finitelength strings of symbols, so $\displaystyle \pi$ is definable. On the other hand if you flipped a fair coin countably many times and interpreted the result as the binary representation of some real, chances are that real would be completely random. No finitelength description could characterize it. Such a number is not definable, even though it is a perfectly good real number. Hamkins is making a technical point that there are nonstandard, countable models of the reals in which every real is definable. Since Hamkins's models are countable, they do not contradict the fact that there are only countably many definable reals. So his point does not contradict the Wiki article. Also note that even in a countable model of the reals, the reals are uncountable. How can that be? Uncountability says that there's no bijection from the naturals to the reals. In a countable model, there is no such bijection inside the model. From "outside" the model we can see that the reals are countable; but inside the model, there's no bijection; so that from inside the model, the reals are uncountable and Cantor's proof goes through. That's why when one is being precise, cardinality is not about "how many," but rather about the existence of bijections within some model. Quote:
But you can only write down countably many of those reals, if by "write down" we limit ourselves to finite strings of symbols. And since all mathematical reasoning is done using finite strings, that's a reasonable metric. My only point here is that the word "definable" is a technical term and should not be casually used in this thread; since no matter how you slice it, there are only countably many definable reals. Last edited by Maschke; August 1st, 2016 at 09:46 AM.  
August 1st, 2016, 10:31 AM  #23 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra 
The logic in this thread has far bigger problems than whether "definable" is being used in its technical sense or not.

August 1st, 2016, 10:37 AM  #24  
Senior Member Joined: Aug 2012 Posts: 2,357 Thanks: 740  Quote:
But it's my observation that the best tactic for countering "alternative" math is to increase clarity and precision. Tossing in definability is a step away from clarity and towards imprecision, which is why I posted as I did.  
August 1st, 2016, 03:26 PM  #25 
Senior Member Joined: May 2016 From: USA Posts: 1,310 Thanks: 551 
OK. I clearly am wrong although I thought the cited article asserted pretty clearly that wikipedia was wrong (at the time, which may no longer be true). In any case, it is a side issue. And I am happy to understand what you were saying. Thanks, 
August 8th, 2016, 08:18 AM  #26  
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,157 Thanks: 732 Math Focus: Physics, mathematical modelling, numerical and computational solutions  Quote:
Silver linings bro...  
August 8th, 2016, 09:05 AM  #27 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2655 Math Focus: Mainly analysis and algebra 
Isn't the principal of conservation of mass/energy sufficient flatten the idea of perpetual motion?


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