
Re: How many real numbers are there?
Posted:
Jul 9, 2006 3:00 PM


> Of course my T does not contain every set, that would be really silly. > However, the elements of my set T satisfy the ZF axioms. They do this > by very construction, and you cannot argue about that! For example, if > x and y are in my set T, then so is x union y, {x,y}, etc.
That's true, but the converse need not be true. That is to say, there exists sets which are not the union of some S_i.
> The construction of the real numbers does not depend on the specific > details of the set theory. *** All it requires is that ZF axioms hold, > and they DO hold for elements of my set T ***
Correct.
> Therefore, we can construct the "reals" (ie, a complete ordered field) > using nothing but the elements of my set T.
Incorrect. How did you get the "therefore"? Just because sets hold for all the ZF axioms does *NOT* imply that they can be constructed using your method. This is a major flaw in your thinking, and I am not sure how you justify it even in your own mind.
> My set T has countably many elements.
Correct.
> THEREFORE the cardinality of the reals is countable!!
Incorrect, for the reasons shown above.
> If you want to dispute this, please consider the following string of > statements and tell me which one is false. Just one will suffice. > > 1. The construction of the reals (ie, a complete ordered field) does > not depend on particular details of one's set theory, only on the fact > that the ZF axioms hold
Agreed.
> 2. ZF axioms hold for elements of my set T
Agreed.
> 3. We can construct, in the usual way, the "reals" (ie, a complete > ordered field), in such a way that each "real" is an element of my set > T
This is where your step fails. Steps #4 & #5 are now moot.
> 4. My set T has countable cardinality > > 5. THEREFORE, the "reals" (ie, one particular complete ordered field, > namely the one in statement 3) have countable cardinality. > > > Note that in making the deduction in step 5 we use the lemma "A subset > of a countable set is countable".
This is a very small issue. You are worrying about a minor crack here, where a gaping hole exists at Step #3.
Jonathan Hoyle

