Re: No Unique Initial Segment And No Characteristic Expansion

From: george (greeneg_at_cs.unc.edu)
Date: 12/07/04 

Date: 7 Dec 2004 00:16:30 -0800

H7> What jumps straight out at most everyone is :
H7> Aren't all possible combinations of heads and tails for infinite
flips
H7> already been done?

No, obviously; the anti-diagonal BY DEFINITION hasn't
been done.

H7> This doesn't stir the Cantor supporter one bit. They think if the
H7> combination is on the list, it must be at some natural number
position

We DON'T just THINK this Herc:
WE *KNOW* this for an absolute DEFINITION.
That's what "list" MEANS if you are insisting that it is a
list-as-we-know-it. Now, IF you want to start talking about
"lists" of order-types LONGER THAN omega, lists where maybe
an infinite number of things can come BEFORE something on
the list as well as after, so that SOME things on the list are at
positions
that are NOT any particular natural number, WELL the, have at it.
But at a bare minimum, if you are going to KEEP calling it a "list",
it is going to HAVE to be well-ordered, that is, it is going to have to
have a first element and every element in it is going to have to have
a unique next element.

There are a great many infinities you can embed or permute the list
into
and still keep it countable; there are uncountably many countable
ordinals.
But the point remains that we are NOT believing something that MIGHT
be mistaken when we say that everything on the list is on it at some
natural number position: THAT is true BY DEFINITION, JUST as surely
as it is true that every decimal place in the decimal expansion of a
real
is some NATURAL number of places to the right of the decimal point.
THAT'S WHAT "list" MEANS, FOOL.
Obviously we're sorry for you that you're so ignoran that you didn't
know what a list was. If you want to talk about a list of things
with MORE than w elements, well, have at it, but STATE YOUR AXIOMS
and DEFINE YOUR TERMS.

Relevant Pages

  • Re: Different size infinities?
    ... of a binary finite state machine augmented with an infinite memory ... for the purposes of diagonalisation, and these digits may update to ... Any supposed number not on the lists could ... I'll borrow the word permutation to mean unique sequence. ...
    (comp.theory)
  • Re: Different size infinities?
    ... of a binary finite state machine augmented with an infinite memory ... for the purposes of diagonalisation, and these digits may update to ... Any supposed number not on the lists could ... I'll borrow the word permutation to mean unique sequence. ...
    (sci.math)
  • Re: Different size infinities?
    ... of a binary finite state machine augmented with an infinite memory ... for the purposes of diagonalisation, and these digits may update to ... Any supposed number not on the lists could ... I'll borrow the word permutation to mean unique sequence. ...
    (sci.logic)
  • Re: Humble pie.
    ... >>infinite number of integers on it. ... > which, starting from the single unit, can not in principle be discovered by ... > repeat applications of a successor function ... list of natural numbers because lists are not in its domain. ...
    (sci.logic)
  • Cantor K.O.d -- again!
    ... demonstrating why "infinite" lists are logically impossible. ... the m_th symbol of the n_th list member is a circle with m dots ... the first list member has one symbol: a circle with one dot ...
    (sci.math)