Re: Ultimate debunking of Cantor's Theory

In article <1184553192.373756.80920@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Calvin <crice5@xxxxxxxxxxxxxx> wrote:

On Jul 15, 10:24 pm, Calvin <cri...@xxxxxxxxxxxxxx> wrote:
On Jul 15, 10:13 pm, "Stephen J. Herschkorn" <sjhersc...@xxxxxxxxxxxx>
wrote:
I think you mistook the objection. The objection was to the fact that
your construction of the set of natural numbers used circular logic.

Yes, I suppose by that you mean that there can't be a
sequence 1,2,3,... without first having the natural
numbers.

But supposed I hadn't mentioned a sequence, and had just
defined the set of natural numbers as {1,1+1,1+1+1,...).
That wouldn't be circular, would it? I know that is not
accepted set notation, but I think my meaning is clear.

Or take the von Neumann definition of the naturals in which 0 is {},
1 is {{}} = {0}, and, recursively, for each natural n, the next is
successor(n) = (n union {n}).

So that the cardinality of every natural is the number of its
predecessors, and no natural is forced to be a member of itself.
.

Relevant Pages

  • The Origin of a Sequence
    ... THE ORIGIN OF A SEQUENCE ... implications of the first part for the understanding of natural number as ... In mathematics, so far as I can tell, a sequence is always ... But let us go back to our idea of the Naturals ...
    (sci.math)
  • Re: infinity
    ... You really are an idiot. ... There is a cardinality for the set of them, ... It is all of those things as a set, but as a sequence, it has no end. ... >> naturals and the set of evens have the same cardinality is at odds ...
    (sci.math)
  • Re: Cantor Confusion
    ... But everyone says they make an adequate model for the naturals, ... If the sequence consists of segments of the form or, there ... is no segment which is diagonal in direction. ... I'm suggesting a definition of the curve as a sequence of pairs ...
    (sci.math)
  • Re: Dial 999 for the real number line
    ... the number of elements in a set is either infinite or it ... are an actually infinite set of naturals. ... You want to believe that even though the sequence of naturals is ... an infinite decimal expansion is ...
    (sci.math)
  • Re: Continuum hypothesis
    ... >>> your own set theory and discuss what is true in it, ... >> may be better things to work with, after all) Can you accept a sequence ... > just the same as the naturals? ... > uniform distribution on the naturals. ...
    (sci.math)
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