My Math Forum Rudin Theorem 2.15 and Countability of the Real Numbers

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 July 27th, 2016, 07:58 PM #21 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,635 Thanks: 2620 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
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 Originally Posted by JeffM1 Frankly, I am not quite sure what Mashke is saying, and I certainly cannot resolve what appear to be his contradictory citations. (The one says that the other is, or perhaps used to be, wrong.)
The cites are actually not contradictory. The informal argument is that since only countably many reals are definable, there must be undefinable reals. This is true in the standard model of the reals.

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 finite-length 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 finite-length 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:
 Originally Posted by JeffM1 I took Mashke to mean that all the real numbers in [0, 1) cannot be expressed by a countably infinite number of digits
No, that's not what I said. Of course every real in the unit interval has a decimal representation. A decimal representation is by definition a function from $\displaystyle \mathbb N \rightarrow \{\text{digits}\}$, where the set of digits is 0 through 9 for decimal, 0 or 1 for binary, etc. Each decimal representation is interpreted as an infinite sum, which converges by virtue of the way infinite sums are defined. No problem there. Nor do "uncountable sums" make sense, since it's not hard to prove that if an uncountable sum is finite, all but countably many of the terms must be zero anyway. So countable sums and countable decimal expressions are the only ones that make sense.

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: 7,635 Thanks: 2620 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
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 Originally Posted by v8archie The logic in this thread has far bigger problems than whether "definable" is being used in its technical sense or not.
Agreed, and people should not read too much into my digression regarding definability.

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
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Quote:
 Originally Posted by v8archie The logic in this thread has far bigger problems than whether "definable" is being used in its technical sense or not.
At least Zylo isn't discussing perpetual motion machines. The threads would be 10 times more painful if that were the case!

Silver linings bro...

 August 8th, 2016, 09:05 AM #27 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,635 Thanks: 2620 Math Focus: Mainly analysis and algebra Isn't the principal of conservation of mass/energy sufficient flatten the idea of perpetual motion?

 Tags 215, countability, numbers, real, rudin, theorem

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