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April 25th, 2018, 02:09 PM  #11 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 397 Thanks: 27 Math Focus: Number theory 
f(x) maps uncountable x to uncountable f(x); x is equal to or greater than f(x). g(y) maps countable y to countable g(y); y is equal to or greater than g(y). h(z) maps finite z to finite h(z); z is equal to or greater than h(z). ? 
April 25th, 2018, 02:15 PM  #12  
Senior Member Joined: Aug 2012 Posts: 2,157 Thanks: 631  Quote:
I really don't see what you're doing. Can you give an example? What is y for example? What does it mean that y is greater than g(y)? When you say y is greater than g(y) do you mean arithmetically greater? Or the cardinality is greater? Your exposition is very murky. It's certainly true that the cardinality of a set must be greater than or equal to the cardinality of any function's image of the set, since each element of the domain gets mapped to exactly one element of the range, and more than one element of the domain may be mapped to the same element of the range. Or do you mean these are decreasing maps? If f(x) = x  1, then f(5) = 4 and in general y > f(y). Which meaning are you intending? Last edited by Maschke; April 25th, 2018 at 02:21 PM.  
April 25th, 2018, 02:55 PM  #13  
Senior Member Joined: Sep 2015 From: USA Posts: 2,312 Thanks: 1224  Quote:
Take two numbers as close as you like. They can be so close that expressing them takes up every bit of matter in the universe and they differ only in the last bit. There are still infinite numbers between the two. Do this with a countable number of universes. Still infinite numbers between the two. Countable sets can never have elements so close together that there isn't a further continuum between the closest members.  
April 25th, 2018, 03:06 PM  #14  
Senior Member Joined: Aug 2012 Posts: 2,157 Thanks: 631  Quote:
I got lost trying to parse that. There are infinitely many rationals between any two rationals. But there aren't any integers strictly between 3 and 4. So countable sets may be discrete or they may be densely ordered.  
April 25th, 2018, 03:52 PM  #15  
Senior Member Joined: Sep 2015 From: USA Posts: 2,312 Thanks: 1224  An interval, or the countable union thereof. A set with nonzero measure. Quote:
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My point was that given any countable set of real numbers, generally the rationals, there are infinitely many numbers between any two of them no matter how small the distance between them is. This shouldn't be any sort of point of contention, it's simply an expression of the density of the reals on the rationals. (and vice versa) You're arguing just to argue.  
April 25th, 2018, 04:35 PM  #16  
Senior Member Joined: Aug 2012 Posts: 2,157 Thanks: 631  Quote:
The OP specifically asked about countable and uncountable sets in general, and said nothing about Lebesgue measure or the real numbers. Quote:
Now by distance, you are assuming the sets in question have a metric. And by using the word between, you are assuming there's an order. There are many countable ordinals whose order structure as ordinals is quite wild. IMO you are assuming way more than the original question asked. What if I biject the rationals to the natural numbers and use that order? Then there is no rational between any two others. So again, you are assuming the usual order on the real numbers, things that the OP said nothing about. I do agree that the OP may be thinking about the real numbers. I'm not much of a mind reader. I feel that responses to the question should help the OP clarify their own ideas, not just slather on more unspoken assumptions. My opinion. Quote:
I can see how you might think that. On the contrary, I'm arguing because people are assuming things the OP never mentioned, such as distance, order, and density. The OP asked if you can somehow map a countablye set on an uncountable set using a countable sequence of maps, and the answer to that is no, as I explained earlier. I don't think it helps the OP's thought process to assume all their unspoken assumptions, without pointing out that these are special cases. If they use the word map, that has a very specific meaning in math and it doesn't hep the OP to pretend you can somehow map a countable set on an uncountable one by taking limits. That is very far off the point of what the OP asked, even if that's (perhaps, perhaps not) what they meant. Perhaps when you say I'm arguing just to argue, what you really mean is that I can't read people's minds, I can only read what they write. Last edited by Maschke; April 25th, 2018 at 04:38 PM.  
April 25th, 2018, 08:22 PM  #17 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 397 Thanks: 27 Math Focus: Number theory 
Maschke, I respect your authority that there is no way of expressing the uncountable reals (no topological "wormholes") by operating on the countable reals, but that the reverse exists  both relations perhaps by definition. Is continuity between the two what needs to be (dis)proved ? May one say that sets of any two cardinalities exhibit continuity? I dare say that real numbers between the two sets might be fundamentally different, unlike not only in sets but also in numerical character  not just singularities. Can one "approach" the continuum (lower limit given by 2^Alephnull?) more closely by approaching 1+^Alephnull from above? 
April 25th, 2018, 09:15 PM  #18  
Senior Member Joined: Aug 2012 Posts: 2,157 Thanks: 631  I plead guilty to excessive pickiness, which might have been the case here. I found your question ambiguous and confusing, but others apparently understood you better than I did. I should probably quit while I'm behind here. Quote:
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In fact the only simple operation that jumps cardinalities is exponentiation (or taking powersets, same thing). In fact Paul Cohen, who proved the independence of the Continuum Hypothesis, said that he believed that $2^{\aleph_0}$ must be far greater than $\aleph_0$ because exponentiation is a very powerful operation. For example between $2^4 = 16$ and $2^5 = 32$ we have many cardinalities. Why shouldn't it be the same for the transfinite cardinals? Of course this is only a heuristic argument and in no way a proof; but it does show how a great set theorist thinks about the size of the continuum. Cohen pointed out that taking successors is a weak operation and exponentiation is strong; so we can not naively expect the cardinal successor of $\aleph_0$ to be $\aleph_1$. Last edited by Maschke; April 25th, 2018 at 09:18 PM.  
April 25th, 2018, 11:09 PM  #19 
Senior Member Joined: May 2015 From: Arlington, VA Posts: 397 Thanks: 27 Math Focus: Number theory 
Thank you for your tolerance; I am learning gradually. Sorry for the Katzenjammer. My continuum extends the typical lower limit given by 2^Alephnull. Generalize it more closely by approaching 1^Alephnull (not a sum, but a limit of Alephnull's infinitude as an exponent of its base 1+ [from above] where the former diverges more rapidly that the latter converges). 
April 26th, 2018, 10:28 AM  #20  
Senior Member Joined: Aug 2012 Posts: 2,157 Thanks: 631  Quote:
On the other hand, the settheoretic definition of $1^{\aleph_0}$ is that it's the set of functions from $\aleph_0$ to $1$. The set $1$ is defined as $\{0\}$. There's only one such function, namely the constant function that maps each element of $\aleph_0$ to $0$. So that's two proofs that $1^{\aleph_0} = 1$. Last edited by Maschke; April 26th, 2018 at 10:53 AM.  

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