Re: Why does everyone do it?

On Tue, 12 Aug 2008 11:29:49 +0200, Han de Bruijn
<Han.deBruijn@xxxxxxxxxxxxxx> wrote:

David C. Ullrich wrote:

On Fri, 08 Aug 2008 14:15:53 +0200, Han de Bruijn
<Han.deBruijn@xxxxxxxxxxxxxx> wrote:

David C. Ullrich wrote:

In article <7096$489ada71$82a1e228$29626@xxxxxxxxxxxxxxxx>,
Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> wrote:

David C. Ullrich wrote:

Of course the question of whether Cantor's proof of the
uncountability of the reals is correct is a very important
question.

But it's also a trivial question. Yes, the proof is correct.

Preaching like the pope, saying "Of course the question of whether G*d
exists is a very important question. But it's also a trivial question.
Yes, G*d exists."

There's a big difference. The pope cannot prove that God exists.

The pope _can_ prove that God exists, _given_ his _own_ assumptions and
within his _own_ albeit informal overly correct logic.

I _can_ prove that |P(A)| > |A|. And the proof is very simple.

Sure. But what has _this_ to do with the uncountability of the reals ?

For heaven's sake. Have you ever considered learning some math
before pontificating on it?

Mike Kelly has already "teached" me that the reals have same cardinality
as the power set of the naturals. If it's _that_ what you mean .. But in
a digital computer, I see integers, and floats, nothing power set alike.
And yet it works, good enough for me at least.

That computer has no problem proving all the theorems of set
theory, including theorems about infinite sets.

There's nothing infinite inside your computer, fine. So what?
If "computers are infinite" were a theorem of set theory that
would be a problem for set theory. But there's no such theorem.

Why in the world should mathematics be restricted to what
you can see in your computer?

A possible answer is that you would _define_ mathematics
that way. Among many possible replies to _that_:

(i) You can define mathematics any way you like - mathematicians
take a broader view.

(ii) Even if we decided that the theory of computation was the
only mathematics worth discussing, we should note that aspects
of that theory become simpler if we assume that our computer
is infinitely large (for example the output tape on a standard
Turing machine is infinitely long). That simplifies things because
we don't need to worry about out-of-memory errors in our
theoretical discussions - then of course when we attempt to
apply these results to real computers we bear in mind that
the real computer has finite size.

For example, the theory of actual IEEE floats is much more
complicated than the theory of the real real numbers. When
people are trying to understand how floats behave they
start with how reals work and then note places where floats
are different. Understanding floats _without_ the theory
of real reals would be _much_ harder.

(iii) Even if we decided that the theory of computation was
the only mathematics worth discussing, and even if we decided
for no good reason that we must never consider a hypothetical
machine with infinite memory, instead introducing irrelevant
complications at _every_ step in our reasoning, it's still true
that given a theorem of set theory a computer with enough
space is going to have no problem proving that theorem.

Han de Bruijn

David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.

Relevant Pages

  • Re: A wise decision
    ... Cantor showed how infinite sets could have different cardinality, ... i.e. irrationals like e.g. pi as well as the "other" reals. ... i.e. mistakes already at the level of mathematics. ...
    (sci.math)
  • Re: Is continuum completely filled up?
    ... of infinite nutural numbers, we cannot show all of them. ... I tried to show that different idea from Cantor's can explane reals ... As the basis of his proof, he explane as a example the bijection between ... standard of infinity and foundation of mathematics. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, ... number of reals on the line. ... to the subsequent logic that claims such a set cannot have infinite values. ... standard orderings, since sets in general don't come with little tags ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... If a quantitative set is mapped in ascending order from the naturals, with each increment in the domain, the range increases by some amount. ... Like it's the number of unit intervals, and the number of reals in the unit interval. ... You are using a form of infinite induction, making a claim for an infinite set based on all finite initial segments of it. ... don't have a definition for an arbitrary set of its "standard ordering" ...
    (sci.math)
  • Re: Cantors diagonal proof wrong?
    ... > with language, and what exists in the physical world. ... infinite sets are each infinite, if only as an example to give to ... from the reals would have to have characteristics thus that Cantor's, ... mathematics is that it is so simple. ...
    (sci.math)