Umm ... Can someone disprove my proof that there are aleph-1 number of real numbers? Even comments to make my proof more rigorous are welcome.
How large is the infinity of real numbers
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$\begingroup$
set-theory
infinity
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2The statement $n(B)=\aleph_1$ is the continuum hypothesis, and is not provable from the axioms of ZFC. Also, the function you define in terms of decimal expansion is not well-defined, since $.1000\cdots=.0111\cdots$. – 2012-10-04
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1It seems like you are trying to show that there are as much real numbers as sets of natural numbers, Since nonen of these sets is provably equal to $\aleph_1$, this is of no help in settling the question. – 2012-10-04
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1One usually defines $\aleph_1$ as the Hartogs number (http://en.wikipedia.org/wiki/Hartogs_number) of $\aleph_0$, but your argument does not seem to be using this definition, so it might help to resolve your confusion if you tell us what definition of $\aleph_1$ you are using. – 2012-10-04
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2Are you perhaps confusing the definition of $\aleph_n$ with the definition of $\beth_n$? – 2012-10-04
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0@Trevor: While it is a reasonable (and common) mistake, not many people outside of set theory know what are $\beth$ numbers actually. Some people in set theory don't know what the $\gimel$ function is, and if we make a jump to the end of the Hebrew alphabet then most people don't know what Tav (no LaTeX symbol) is, either. – 2012-10-05
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0To the question in the title, the answer is "big. Really big. You just won't believe how vastly, hugely, mind-bogglingly big it is," courtesy Douglas Adams. – 2012-10-05
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0@Asaf, so far as I can tell, TeX symbols for Hebrew letters stop at gimel. A shame --- I so much want to do $\lamed$. – 2012-10-05
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0@Gerry: There is also $\daleth$. – 2012-10-05
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0@Asaf, ah, I tried \dalet, didn't think of \daleth. – 2012-10-05
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0@Gerry: That was my first try as well, but I figured that if it's \beth then it will be \daleth as well. As for \lamed, if you want a script Lamed, then $\delta$ is close enough... (This is why freshmen often confuse delta with lambda.) – 2012-10-05