Starting fresh with a peace offering

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Starting fresh with a peace offering

Postby JeffJo » Thu Oct 08, 2015 2:02 pm

What a great idea. Start fresh. I will, if you will.

Definition: An infinite set X is said to be countable, or countably-infinite, if there is a function x(n) that defines a 1:1 correspondence between the infinite set of natural counting numbers N and the members of X.

Assertion: There is at least one infinite set which is not countable.

Proof (note that one minor typo, using d(n,n) where I meant d(n), has been corrected since I first published this. I'm a horrible typist, so I reserve the right to do it again):
  1. Let T be the set of all infinite binary strings. (Note that an infinite string can be considered to be an infinite set of characters, with duplicates allowed.)
  2. If there is a string t in T that does not have countably-infinite bits, the assertion is already demonstrated to be true. So we only need to prove the assertion under the assumption that every string t in T has countably-infinite bits.
  3. For any infinite subset S of T that is countable:
    1. By the definition of countability, there is a function s(n1) that defines a 1:1 correspondence between the set of natural counting numbers N and the members of S.
    2. Also by the definition of countability, for any string s(n1) there is a function f_n1(n2) that defines a 1:1 correspondence between the set of natural counting numbers N and the bits of s(n1).
    3. So there is a function b(n1,n2) where b(n1,n2)=f_n1(n2).
    4. There is an infinite string d whose nth bit is d(n)=~b(n,n).
    5. The function d(n) defines a 1:1 correspondence between the set of natural counting numbers N and the bits of d, so d is countably infinite and must be in T.
    6. For every n in N, d(n) is different than b(n,n), so s(n) is different than d.
    7. d is not in S.
  4. Since there a string missing from any countably-infinite subset S, and that string is in T, T is not countable.

QED.

+++++

Now, there and be three ways to discount such a claimed proof.
  1. You can try to prove a contradictory assertion by a different method.
  2. You can try to prove a different assertion, known to be incorrect, by the same method.
  3. You can find a logical flaw in the proof.
I'm not ruling out a fourth method, but the purpose should be stated in a similar manner first.

But note that method 1 only proves there is a flaw in one of the proofs. In particular, a proof that deals with only finite strings is irrelevant.

And method 2 needs to be meticulous in showing that the method is the same. There are no lists, no real numbers, and no fractional numbers in this proof. If you use them, your proof is different. The function b(*,*) works on an infinite domain, so it is not an "array," finite portions are irrelevant, and it and cannot be considered to have an aspect ratio.

If you think a point has been misunderstood in the past, try to make it clearer. Just don't add twelve other tangential points that circle around the main point. In fact, number your points, and I will tell you what I think of each. Just be advised that it may be"irrelevant"because of another point I made.
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Re: Starting fresh with a peace offering

Postby davar55 » Fri Oct 09, 2015 12:43 pm

I agree with this proof, it's really just Cantor's with
a different veneer. I'm still trying to see the crux
of his objection to that or this. I should note that
Cantor uses digits (decimal or binary) to the right
of a "decimal" point so as to bring in the reals; an
"infinite string of characters from a finite alphabet"
is a more modern formulation. The fact that digits
are small numbers lets it lead to more interesting
math than just the bare fact of different infinite
cardinalities.
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Re: Starting fresh with a peace offering

Postby JeffJo » Fri Oct 09, 2015 3:09 pm

davar55 wrote:I agree with this proof, it's really just Cantor's with a different veneer.

Ssshhh! Robert isn't supposed to notice that it quite literally is Cantor's proof. It was not about real numbers at all, it was about "a set M of elements of the form E=(x1,x2,...,xn,...), where each xn is either m or w." The theorem itself is "If E1,E2,..., En,... is any simply infinite sequence of elements of the set M, then there is always an element E0 of M that corresponds to no En." With a simple change in terminology, this is exactly the assertion I made, and the proof must be similar (although I created mine, and haven't read Cantor's which is in German; it has to be similar).

I'm still trying to see the crux of his objection to that or this.
He doesn't want to believe you can work with the concept of "infinity," so he must find a flaw in the reasoning that created the basis for infinite arithmetic.

But he thinks that the way teachers explain what diagonalization accomplishes to adolescents is the definition of the algorithm itself. Those are the lists he keeps talking about.

I should note that Cantor uses digits (decimal or binary) to the right of a "decimal" point so as to bring in the reals
I've pointed it out many times - he refuses to notice.
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Re: Starting fresh with a peace offering

Postby davar55 » Mon Nov 02, 2015 4:15 pm

The real numbers R have other properties beyond those
of the rationals Q other than just higher cardinality.

A sequence of rationals might approach a limit not in Q;
a sequence of reals approaching a limit has that limit in R.

The reals are continuous; the rationals leave gaps.

Q is a subset of R, and the closure of Q is R.

The set of subsets of Q has the same cardinality as R.

The set of subsets of R has a greater cardinality than that of R (or of Q).

IS that a peace offering or what? :)
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Re: Starting fresh with a peace offering

Postby davar55 » Fri May 13, 2016 9:11 pm

Love the idea of offering peace.
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