Re: Cantor Confusion

mueckenh@xxxxxxxxxxxxxxxxx wrote:
MoeBlee schrieb:

mueckenh@xxxxxxxxxxxxxxxxx wrote:
MoeBlee schrieb:

David Marcus wrote:
Obviously, I could be wrong, but I think WM means map it on a line of
the list. He seems to think that because we construct the diagonal from
the list, the diagonal must be one of the lines in the list. Why he
thinks this, I have no clue.

You mean map the diagonal (or the range of the diagonal, or whatever)
onto one of the finite sequences that is in the range of the infinite
sequence of those finite sequences? I.e., map the diagonal onto a
member of the range of S? A 1-1 map? If so, yes, I would share your
bafflement as to why we should think there is such a mapping or what
contradiction there is in there not being such a mapping.

There is a mapping of the diagonal on a (each) column.

Okay, if you want to put it that way.

There is no mapping of the diagonal on any line.

Okay.

So there is a difference between the natural numbers (initial segements
of the first column) and the natural numbers (lines). There is not a
bijecton N <--> N?

I already asked you please to state exactly which of these you are
claiming:

(1) That my proof is not indeed a proof in Z set theory. That is, that
there is something in my proof that is not first order logic applied to
the axioms and previousl proven theorems of set theory.

(2) That set theory is inconsistent. That is, that there is a sentence
P and ~ P such that both are theorems of set theory.

(3) That my proof does not correspond with your own mathematical
notions.

As I said, I don't care about (3).

As to (2), then I'd like to see what you claim to be a proof in set
theory of a sentence P and a sentence ~P.

As to (1), then I'd like you to say what in my proof is not justified
by first order logic applied to the axioms of set theory.

The diagonal cannot have more elements than the width of the matrix is.

I answered that already. If you define 'the width', then it turns out
to be equal to omega, which is just what the length of the diagonal is.

The diagonal is assumed to exist such that each of its digits exists,
actually. This is established by the mapping on a column.

No it is not.

But it cannot
be established by the mapping on any line. You should recognize that
the following bijection between columns and lines shows a
contradiction, because one element is missing:

1 <--> 1
1,2 <--> 2
1,2,3 <--> 3
...
1,2,3,...n <--> n
...
1,2,3,... <--> omega

1, 2, 3 ... is NOT a line.

1,2,3,...omega <--> omega+1

1, 2, 3 ... is the sequence defined by f(n) = n+1.

So 1, 2, 3, .. omega is the union of f with {<w w>} ('w' stands for
omega).

And it is NOT a contradiction that 1, 2, 3, .. omega is equinumerous
with omega+1.

And nothing about this shows anything in my proof that is not first
order logic applied to the axioms of set theory nor any contradiction
in set theory.

Enough with your silly games.

Please refer to (1) - (3) in this post and tell me already which one(s)
you claim!

I answered that already. And the cardinality of the diagonal is not
assumed, but is proven, to be omega.

It is *assumed* by stating the axiom of infinity. Without this
assumption the length was not omega.

It is PROVEN from the axiom of infinity.

Of course I we can't even assert the EXISTENCE of a denumerable
sequence such as in this problem without adding an axiom to
Z-axiom_infinity. And the axiom of infinity is an axiom of Z set
theory.

I said my proof is in Z set theory. That includes the axiom of
infinity.

Your complaining that I use the axiom of infinity is a STUPID and RUDE
complaint, since you are now complaining that I am using something that
I said FROM THE BEGINNING that I am using. I never said I could prove
things about denumerable sets without my using the axiom of infinity.

I said clearly, FROM THE BEGINNING of my proof that it is a proof in Z
set theory. For you NOW to complain that I'm using an axiom of Z set
theory is STUPID and RUDE since I would not have bothered to even post
a proof about denumerable sequences and talk with you about it for so
long if I accepted any condition that I can't use the axiom of
infinity.

Sheesh!!!

MoeBlee

.

Relevant Pages

  • Re: Cantor Confusion
    ... > definition leads the direct way to the result of actuallity we ... primitive language of set theory. ... As I have written again and again, you do not accept the axiom of infinity. ...
    (sci.math)
  • Re: Implementable Set Theory and Consistency of ZFC
    ... This is still not correct as the Axiom of Infinity is an axiom of ZFC ... Set Theory, even less when the latter is supposed to allow the Naturals ... consistent within common mathematics. ...
    (sci.math)
  • Re: Cantor Confusion
    ... It should be comprehensible that potential infinity is possible without ... But that is not what set theory requires. ... infinity present or detectable without the axiom of infinity. ... you won't be able to show that merely dropping the ...
    (sci.math)
  • Re: Ultimate debunking of Cantors Theory
    ... Perhaps, for example, you think that specifying ... What axiom or theorem were you using to deduce ... exists without the axiom of infinity. ... thing about ZF set theory or how axioms are used to prove theorems. ...
    (sci.math)
  • Re: Some basic set theory questions
    ... remarks so that they can be generalized without reliance on some ... we don't need the axiom of infinity to prove there is ... virtually unanimously used in modern set theory textbooks) to prove ... that I'm talking about ordinary Z set theories. ...
    (sci.math)