Re: Why?

abo wrote:
david petry wrote:
Daryl McCullough wrote:
david petry says...

I'd like to see a quote from a constructivist claiming that the
cardinality of a subset of the natural numbers can be greater than the
cardinality of the natural numbers. If you can't produce such a quote,
you're talking nonsense.

In constructive mathematics, a distinction is made between being
countable (a set is countable if there is a surjection from the
naturals to that set) and being subcountable:

The essence of Cantor's claim is that the cardinality of the set of
reals is greater that the cardinality of the set of natural numbers. I
don't think you have provided a quote that shows that any
constructivist thinks that claim is constructively meaningful.

Bishop and Bridges, "Constructive Analysis", p. 27:

"(2.19) Theorem. Let (a_n) be a sequence of real numberrs, and let x_0
and y_0 be real numbers with x_0 < y_0. There there exists a real
number x such that x_0 <= x <= y_0 and ! x = a_n for all n in Z+.

Proof: <omitted>

Theorem (2.19) is the famous theorem of Cantor, that the real numbers
are uncountable. The proof is essentially Cantor's "diagonal" proof.
Both Cantor's theorem and his method of proof are of great importance."

To whomever might be interested:

I don't know of a rigorous definition of 'constructively meaningful'
(there might be one, for all I know) but I do know (even if I'm not
versed in the details) of at least one semantics for intuitionisitic
logic and soundness and completeness theorems too. There is a
definition of 'intuitionistically valid'. And as far as I can tell,
intuitionisitic validity does not depend on some kind of special
constructivist endorsement of any given axiom set. It seems to me that
what one thinks of the axioms of set theory is irrelevent to the pure
question of whether Cantor's theorem follows by intuitionisitic logic
from whatever axioms are active in the proof of the theorem. And the
impression I get (please correct me if I'm wrong) is that the entire
trail of arguments, from whatever axioms we're talking about here,
leading to the theorem that there is no bijection between a set and its
power set is intuitionistically valid, as well as the proof that the
set of natural numbers has no bijection with the set of real numbers
(though I don't know whether the definition of 'the set of real
numbers' is different in Bishop and Bridges from classical definitions
that pick out isomorphic number systems no matter how the real number
system is defined).

MoeBlee

.

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