Re: Real Discontinuity in Cantor Diagonal

On Aug 27, 5:07 pm, Balthasar <nomail@invalid> wrote:
Am Wed, 27 Aug 2008 09:35:56 -0700 (PDT) schrieb slartibartfast:







Maybe you have problems to comprehend the method of indirect proof. Where
"impossible things" are formulated as "facts" in the course of the proof
[which itself is starting with a FALSE claim, the assumption] till we
arrive at a (obvious) contradiction.

It's right that THERE is no "entire infinite list of all reals". But *that*
is what we want to prove in the first place. Hence we may start with the
ASSUMPTION that such a list exists:

        Assume that there is an infinite list of all reals.

Then we proceed with the proof, i.e. we derive conclusions from this
assumption (by applying the diagonal argument, btw.). Finally we get a
contradiction. And by RAA we conclude that the negation of our assumption
holds, i.e.

        There is no infinite list of all reals.

I do not have a problem with indirect proofs, ...

Ah? At least this was my impression. Well, let's see.



but the reduction ad absurdum proof that the diagonal argument
appears to be may be only that: an appearance.

Uh?



for the contradiction drawn via it, is contingent upon the
diagonalizability of something called a list [...] of "ALL"
the reals, it is this contingent assumption that is at issue which is
different from the fact that such an entity would produce a
contradiction IF diagonalized.

Yeah. You obviously have problems with indirect (or any mathematical)
proofs. :-)



a valid reduction argument has all its steps applicable and THEN draws
a contradiction.

If you say so.



if the "entity" (the list of all reals - whatever
that could mean) could be shown to be diagonalizable THEN (an not
before) the diagonal argument could validly be used to show that such
a list would be in fact an impossibility.

We do not have to SHOW that. We just APPLY the "diagonal argument". And
right, all steps in the proof are _allowed_, i.e. sanctioned by the axioms,
definitions and rules of derivations of our system we are working in. (In
this case a certain axiomatic set theory, say ZFC).



To put it another way;
Assume that there is an infinite list of all reals
Assume that such a list canNOT be diagonalized.
now what?

Huh? The second assumption is a meta-claim, I'd say, while the first is a
object claim (i.e. a claim formulated at object level). You are mixing up
levels here.

ok Balthasar, what is a "meta-claim" if not just a "claim about a
cliam".
couldn't one equally call assumption 2 a claim and that claim that
assumption is false a "meta-claim"? if not why not?



what do you do now? the argument cannot proceed!

Sure it can. We just can "apply diagonalization". This way "showing" that
your meta-assumption does not hold.

But let's -instead- consider a _direct_ proof:

... consider an ARBITRARY (but fixed) list ofrealnumbers. Now apply the
diagonal argument. This way we get arealnumber which is NOT in the
(considered) list. Since the list was arbitrary this shows that for _every_
list ofrealnumbers there is (at least) onerealnumber that is not in the
list.

HOWEVER THAT is not the diagonal argument.

It is.



... for in none of these cases is there a contradiction.

Yeah. That's why this proof is called a DIRECT one (in contrast to an
_indirect_ one).



its not a contradiction to say somerealisn't in a list.

Right. We "construct" arealnumber which is not in the list by
"diagonalization". That's why I wrote:

        "*I* would tend to avoid the indirect proof [...]. It clouds
         the fact that the diagonal argument is eminently _constructive_
         (if formulated in a direct way)."

you could equally well say for any arbitrary (but finite) list of
NATURAL numbers there is a natural number that is not in the list.

Sure. So what? We don't use "diagonalization" in this case.

True, but whether the number is outside the arbitrary (but finite)
list by diagonalization (in the case of reals) or by simple counting
(in the case of a list of natuirals) is not relevant .....OR IS IT????

in other words... why does the fact that it is "diagonalization" that
puts an element outside of an arbitrary (but finite) list, rather than
merely by dint of the arbitrary list's finiteness matter?

how does that show that those elements are uncountable since any such
real number can just be included in a list "one place higher"?
This can happen in the Naturals: each Natural that is not in the list
can always simply be put in a 1:1 correspondence with an n in a list
one place higher.

In your direct "proof", a diagonally constructed real from any
arbitrary (but finite) list, can likewise always be included higher up
in some bigger list.

How does "always being outside the list" imply that "the reals can not
be put in 1:1 correspondence with the maturals " any more than the
very same property:"always being outside the list" implying that "the
naturals can not be put in 1:1 correspondence with the naturals "

How does the REASON for an element's being outside a arbitrary list
matter?

I do not see how the fact that a real is always outside an arbitrary
list of reals
makes the reals UNcountable, since it is also true that there is
always a natural outside any arbitrary list of naturals. and the
naturals are countable.

what's the difference between the two?

for any list of n naturals there exists a natural not in that list.
for any list of n reals there exists a real not in that list.

why / how do you draw differing conclusions as to countanility in the
two cases?

what more is diagonalization saying other than "for each n reals
there's another real"? how is that any different from saying "for each
list of n naturals there's another natural"?







the diagonalization only BECOMES the diagonal ARGUMENT of Cantor, when
the list is NOT any arbitrary list of reals ...

Nonsense. You really should read CANTOR before stating false claims.

B.- Hide quoted text -

- Show quoted text -

.

Relevant Pages

  • Re: Cardinality of Real Numbers
    ... Cantor's first assumes the existance of a bijection between the ... >> natural numbers and the reals. ... From this, a contradiction is reached ... >naturals to the reals, and shows that there is some real not in the ...
    (sci.math)
  • Re: Cardinality of Real Numbers
    ... >>> natural numbers and the reals. ... From this, a contradiction is reached ... Cantor's first starts with an arbitrary injection from the ... >>naturals to the reals, and shows that there is some real not in the ...
    (sci.math)
  • Re: Cardinality of Real Numbers
    ... >>> natural numbers and the reals. ... From this, a contradiction is reached ... >>naturals to the reals, and shows that there is some real not in the ... The proof shows that there exists c not in the sequence. ...
    (sci.math)
  • Re: Cantor and the binary tree
    ... >> all the reals in this interval, and suppose further that we can ... >> This is a contradiction; therefore our hypothesis (that I can have ... > Indeed it shows that there are more numbers than digits, ... > more reals this way than naturals, it would seem, but it doesn't ...
    (sci.math)
  • Re: abundance of irrationals!)
    ... All I know is that what we know about infinite ... > the sets I call finite have larges members. ... The set of all finite naturals is not infinite, ... Sets defined by mapping functions from the naturals to the reals which have ...
    (sci.math)