Re: Is continuum completely filled up?

On Sun, 28 Jan 2007 02:07:20 +0900, toshiaki wrote:

"Dave Seaman" <dseaman@xxxxxxxxxxxx> wrote in message
news:eovoec$8j5$1@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx

The "extended reals" is a term that often means the two-point
compactification of the reals, as used in analysis. The extended reals
do satisfy the least upper bound property and therefore are
order-complete, although they do not form a field.

If you mean something else, such as the hyperreals or the surreals, then
it's true that they are not complete, but I don't see what that has to do
with the discussion.

Wikipedia article "Surreal number says that, surreals constitute the biggist
ordered field, in which all other ordered field are enbedded.

Correct. However, the surreals do not satisfy the least upper bound
property, and therefore are not a "complete" ordered field according to
the definition. The set of infinitesimals, for example, is a nonempty
set and is bounded above by 1, but has no least upper bound.

We can define rational line though it is hard to deal with, and
cannot
in
plane.
I don't understand any of that. Why are the rationals hard to deal
with,
and why can't we consider QxQ, the subset of the plane consisting of
points with rational coordinates?
Most of us have no problem dealing with two-dimensional objects. The
Pythagorean theorem has been around for at least 2500 years.

I think that in QxQ, we can not get diagonal line of unit square?

If we consider Q^2 as a subset of R^2, then the diagonal you mention does
not lied entirely within Q^2. So what?

I don't know clearly the connection of proofs which cover from ZFC to
diagonal argument. If the axiom of power set implys uncountable reals,
problems originate ZF.

The axiom of infinity implies the existence of N. The power set axiom
implies the existence of P(N), which is uncountable by Cantor's theorem.
Notice that we haven't even mentioned the reals yet. You have not found
any problems in set theory.

Not a few people might find problem in this stage, because P(N) is
uncountable already. And I think that from discription of axiom infinity
only, we can only derive conclusion that each naturals have the next. The
idea that N contain infinite members is another consideration.

The axiom of infinity guarantees that an infinite set exists. In fact,
the axiom guarantees that there is a set big enough to contain all of the
naturals. We can then use separation to show that N is a set.

Real infinity and counting are different things, as well as geometrical
point and its decimal expression. Though it is free to define that both are
equeverent in mathematics.

Typically (as an engineer) I view a set of bits as defining an
addressable range, so when I say a set of bits I meant the set of
possible different strings that can be represented by those bits.

I think so, too.

Andy Smith's statement was ambiguous. I was attempting to make the
distinction clear.

I thougt that above statement means that every bits of infinte strings
are
not clearly defined.

An infinite string is a mapping f: N -> {0,1}. Which infinite string do
you think is not clearly defined?

The existence of N is not clear.

It's clear if you accept the axioms of ZF. You have not demonstrated a
contradiction.



--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
.

Relevant Pages

  • Re: Cantors "diagonal argument". My Objection.
    ... Infinity is NOT relevant to the proof ... you DENY the axiom of infinity (if you replace it with an axiom ... It DOESN'T MATTER whether you restrict these reals to being between 0 ... This IS NOT specific to Cantor IN ANY way except that he got there ...
    (sci.logic)
  • Re: abundance of irrationals!) - rectangles of area 1.bmp [0/1]
    ... >>> But the axiom of infinity and the principle of induction do get you to ... >>> infinity in finitely many steps. ... > rationals all of different sizes but the ratioals and the reals of the ... that it therefore constitutes an enumeration of the reals. ...
    (sci.math)
  • Re: Non-terminating numbers
    ... > of a series of discontinuities? ... The reals have the least upper bound property, ... Trivially true for the reals. ... There is no such thing as infinity -1. ...
    (sci.math)
  • Re: Cantor Confusion
    ... rationals that comes arbitrarily close to other such sequences and so ... diagonal proof was not about the reals at all. ... *based* on the axiom of infinity, ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... set and is bounded above by 1, but has no least upper bound. ... The axiom of infinity guarantees that an infinite set exists. ... keeping diary is always after the affare. ...
    (sci.math)
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