The reals are uncountable

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The reals are uncountable

Postby mathsman1963 » Sat Sep 12, 2015 9:44 am

People forget ( such is the fame of the diagonal argument) that Cantor produced another uncountability argument a few years prior. This is probably the nicest version of it:

Suppose that there is a sequence {rn} which exhausts the real numbers. Then we can choose a closed interval [a1,b1] which avoids r1. Next, choose a sub-interval [a2,b2]⊂[a1,b1] which avoids r2. Continue in like manner to obtain a decreasing sequence of intervals {[an,bn]} such that for all n, [an,bn] avoids rn.
Now, the an form an increasing sequence, bn a decreasing sequence, and for all n we have an≤bn. It follows that supan≤infbn and in particular, there exists a point x∈[supan,infbn]. Then for all n we have x≠rn, since x∈[an,bn] and we said that [an,bn] avoids rn. But this contradicts the exhaustive nature of {rn}.
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Re: The reals are uncountable

Postby davar55 » Mon Sep 14, 2015 10:19 am

mathsman1963 wrote:People forget ( such is the fame of the diagonal argument) that Cantor produced another uncountability argument a few years prior. This is probably the nicest version of it:

Suppose that there is a sequence {rn} which exhausts the real numbers. Then we can choose a closed interval [a1,b1] which avoids r1. Next, choose a sub-interval [a2,b2]⊂[a1,b1] which avoids r2. Continue in like manner to obtain a decreasing sequence of intervals {[an,bn]} such that for all n, [an,bn] avoids rn.
Now, the an form an increasing sequence, bn a decreasing sequence, and for all n we have an≤bn. It follows that supan≤infbn and in particular, there exists a point x∈[supan,infbn]. Then for all n we have x≠rn, since x∈[an,bn] and we said that [an,bn] avoids rn. But this contradicts the exhaustive nature of {rn}.

Looks right to me. The assumption that is disproven is that the reals can be listed in a sequence
(not necessarily in order of size or value, which is of course impossible since between any two
reals are an infinite number of more reals), which is the same as saying they can't be put into
a one-to-one correspondence with the counting numbers.
The next mathematical leap is to say there is a cardinality associated with the integers, which is
shared by the rational fractions, but which is not shared by the reals and which must therefore
be greater for the reals. It can then be shown that the cardinality of the imaginaries (the
product of reals times i ) is the same as that of the reals, and with a little effort so is the cardinality of the complex numbers a+bi for a and b real.
Hence the alephs.
The next issue is, is that all? Is there an even greater cardinality? It takes more math, and it's
only important in math, but therefore in anything supported by higher math. Where the issue
becomes relevant outside of pure math I leave alone.
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Re: The reals are uncountable

Postby davar55 » Sat Nov 14, 2015 12:28 am

This proves that there are at least two different levels of infinity.
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Re: The reals are uncountable

Postby davar55 » Fri May 13, 2016 9:07 pm

The infinite cardinality of the integers is called Aleph-null or A0.

The infinite cardinality of the reals is called Aleph-one or C (for Continuum).

The infinite cardinality of the functions from R to R is called Aleph-two or F (for Functions).

Cantor proves that the first two are not equal, that A0 < C.

We must also show C < F.
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