Re: Cantor Confusion

In article <N_YSg.1208$3E2.403@xxxxxxxxxxxxxxxxxxxxx>,
Poker Joker <Poker@xxxxxxxxx> wrote:

"Arturo Magidin" <magidin@xxxxxxxxxxxxxxxxx> wrote in message
news:efgfhd$261u$1@xxxxxxxxxxxxxxxxxxxxx
In article <1159410937.013643.192240@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<the_wign@xxxxxxxxx> wrote:
Cantor's proof is one of the most popular topics on this NG. It
seems that people are confused or uncomfortable with it, so
I've tried to summarize it to the simplest terms:

1. Assume there is a list containing all the reals.
2. Show that a real can be defined/constructed from that list.
3. Show why the real from step 2 is not on the list.
4. Conclude that the premise is wrong because of the contradiction.

This is hardly the simplest terms. Much simpler is to do a ->direct<-
proof instead of a proof by contradiction.

1. Take ANY list of real numbers.
2. Show that a real can be defined/constructed from that list.
3. Show that the real from step 2 is not on the list.
4. Conclude that no list can contain all reals.


How can it be simpler if the list can be ANY list instead of a
particular one.

Because a direct proof is simpler than a proof by contradiction.

ANY list opens up more possiblities than
a single list.

Any list does not require you to assume that there is a "single list"
which some some particular property.

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

.

Relevant Pages

  • Re: Review of Mueckenheims book.
    ... >> proof by contradiction is wrong ... It is said that Euclid assumed ... Cantor assumed a complete list of reals and showed the existence ... I do not know of such an axiom. ...
    (sci.math)
  • Cantors diagonal proof wrong?
    ... When I was shown Cantor's diagonal proof that the number of reals was not ... infinity, you are making some basic assumptions on what an idea is. ... of the size of an infinite set is a contradiction in itself). ... So you just reverse the digits in the integer to create the real. ...
    (sci.math)
  • Re: infinitely many nns = infinite nns?
    ... proofs, one which shows that there is a FINITE distance (or difference, ... if you must) between any two reals on the line segment (which is why I ... What "infinite sum of finite distances"? ... We are still waiting for you to present the contradiction. ...
    (sci.logic)
  • Re: A question about Caontors proof of the uncountability of the reals
    ... understanding Cantor's diagonalization proof: assume the reals are ... countable, show that this assumption leads to a contradiction, conclude ... Cantor's proof of the uncountability of the reals per se, ... Does this answer your objection? ...
    (sci.math)
  • Re: A question about Caontors proof of the uncountability of the reals
    ... countable, show that this assumption leads to a contradiction, conclude ... the uncountability of the reals per se, but about the validity of proof by ... N -> R be an arbitrary mapping. ... that is not in the range of f, and hence f is not a surjection. ...
    (sci.math)