Re: Ultimate debunking of Cantor's Theory

On Jul 17, 5:25 am, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:
On 16 Jul., 23:28, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

On Jul 16, 10:22 am, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:

On 16 Jul., 18:16, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:

On Jul 13, 10:14 am, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:

Consider the list
0.0
0.1
0.11
0.111
...
and switch 0 to 1 on the diagonal. Then you have at the diagonal the
number 0.111..., but only if this number (with one digit less) is also
in the list.

And that does not refute that the set of real number is uncountable.

Let's first clear the following question: Do you agree that the
diagonal in this special case is in the list?

Let's first DEFINE 'the list' and 'diagonal'.

I take your list to be the infinite denumerable sequence of finite
sequences such that each of the finite sequences starts with 0, then a
decimal point (whatever object you'd like that to be), then a 0 or
then a finite subsequence of 1's.

I take the diagonal to be the infinite denumerable sequence that
starts with 0, then a decimal point, then all 1's.

So two questions:

(1) Is the diagonal of the list an entry in the list? No, of course
it's not, since the diagonal is a denumerable sequence and each entry
in the list is a finite sequence.

That is one statement. The other claim is that the infinite diagonal
cannot exist without an infinite number of list entries and an
infinite number of 0's adjacent to 1's. This second claim proves the
existence of a sequence of infinitely many 1's. Why do you think this
second claim is wrong?

Whose claim? Is your claim that there exists a denumerable sequence of
1's (precisely, there exist a denumerable sequence into {1})? I agree
with that claim. And do you claim that the diagonal (except for the
initial 0 and the decimal point) of your list is a denumerable
sequence of 1's? I agree with that claim.

Wow, this is getting very frustrating and tiresome with you. As in the
other thread we're talking in recenlty, I just keep coming up against
these senseless questions from you and you give out yet more undefined
and incoherent terminology when I ask for definition and explication
of the terminology you start with.

(2) Is the diagonal of the list a base 2 representation of any real
number represented in base 2 by an entry in the list? No, since the
diagonal of the list is a base 2 representation of 1, while each entry
in the list is a base 2 representation of a number less than 1.

You have not shown that your list refutes uncountability.

I did not intend to show this by the list, because, if it turns out
that Cantor's arguing fails (because the diagonal cannot exist without
infinite sequences of 1's in the list), then there is no reason to
contradict uncountability --- it simply vanishes.

Then what is the point of your list if not to refute undecidability?

Are you just playing games with terminology?

First, the diagonal argument in Z set theory proves that there are
uncountable sets, and that the set of reals is an uncountable set. The
proof is a FINITE syntactical object. There is more than enough ink,
paper, time, and energy in the universe, even enough of those owned by
a single person, to write this finite syntactical object, which is a
finite sequence of formulas, each formula being either an axiom of
first order logic with identity, an axiom of Z set theory, or derived,
by modus ponens, from two previous entries in the sequence, and with
it mechanically checkable that each entry in the sequence is as just
described.

You can disagree with the axioms or even with first order logic with
identity, but the existence of the sequence of formulas I just
described exists and is finite and checkable and writable by just a
single person with a moderate supply of ink and paper.

Do you or do you not understand that that is a fact? If we at least
know that you do or do not understand that point, then we can at least
know where we stand in this discussion.

Second, as far as modern set theory goes, to dispute the diagonal
argument, you must show some step in the argument that uses something
other than first order logic with identity applied to the axioms of Z
set theory.

Third, to prove that there are no uncountable sets is not just to
dispute the diagonal argument, but also to go on to make a PROOF that
there DOES exist a bijection from any given set onto a natural number
or onto the set of natural numbers.

I've answered your questions. Now, would you answer whether or not you
understand the three items I just mentioned?

Also, I've been answering your questionaires. Would you return the
consideration by answering these:

(1) How do you figure (without choice):

If it is not the case that S strictly dominates N, then N dominates S
(where N is the set of natural numbers? As this is needed to show that
your earlier definition of 'countable' is equivalent with the ordinary
definition you gave later.

(2) Where do you find a quote of John von Neumann as you represented
him to say certain things about set theory a couple of days ago?

MoeBlee


.

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