Re: Rational numbers, irrational numbers: each dense in real numbers

On Oct 1, 11:28 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Sep 28, 5:51 pm, "Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:



On Sep 27, 11:01 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Sep 26, 9:10 pm, "Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:

On Sep 26, 5:44 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
Re post on Sep 25, 6:31 pm, by "Ross A. Finlayson" <r...@tiki-
lounge.com>:

(1) Now that I've shown you the basic conceptual error that makes your
previous argument incorrect, it's good that you finally recognize that
error (I hope you do).

(2) You have a new argument now. I'll work with this one. Upon
identifying the error(s), I'll consider one more of your attempts, and
upon identifying the error(s) in it, I'll call you out on three
strikes swinging. You'll not have proven in ZFC that there exists an
uncountable set that bijects into a countable set.

(3) For your latest argument, let's be clear at each step. Let's
speficy exactly each formulation along the way, starting fresh. Please
tell me if you disagree with any of the following:

* You claim to prove, in ZFC, that there exists an uncountable ordinal
and a bijection from that ordinal into a subset of the rational
numbers in the real interval (0 1).

* Your method is to use some form of transfinite recursion on an
uncoutable ordinal such that you claim thereby to define a bijection
from that uncountable ordinal into a subset of the rational numbers in
the real interval (0 1).

* So, the first matter to settle is whether you are arguing that for
ANY arbitrary uncountable ordinal there is such a bijection, or
whether you have a particular uncountable ordinal in mind. It seems to
me, that, since you have not mentioned a particular uncountable
ordinal, then you're argument is about an arbitrary uncountable
ordinal.

So, let S be an uncountable ordinal. You are to show that there exists
a bijection from S into a subset of the rationals in the real interval
(0 1).

....
All of that is meaningless without specification of an actual
recursive definition using an actual recursion schema.

....


I'm under the impression that a transfinite recursion schema is the
provision of class functions for the ordinal zero, (other) limit
ordinals, and successors, as ordinals, as is done above.

Let there be a family of choice functions F_(x,y) for each x,y E (0,1)
such that each C_(x,y) E F_(x,y) returns an irrational number p such
that y < p < x and p is irrational.

Then, let p_0 = C_(1, 0), and t_0 = C_(p_0,0). (Hopefully that
notation is clear: y < C_(x,y) < x, C_(x,y) is irrational.) Then,
the class function for zero is defined. For each successor ordinal:
t_alpha+ = t_alpha, where alpha+ is shorthand for alpha+1 in ordinal
arithmetic, and p_alpha+ = C_(p_alpha, t_alpha). Then, for limit
ordinals lambda, where the alpha's are from those successors of the
previous limit ordinal, p_lambda is a function of the previous limit
ordinal's successors, p_lambda = C_(t_alpha, 0), and t_lambda is a
C_(p_lambda, 0).

The domain of those class functions has definition for ordinals less
than the initial ordinal of the cardinality greater than the
cardinality of the irrationals.

Then, if the irrationals are uncountable, there is a well-ordering of
the uncountable set {(p_i, i)} by i's natural well-ordering as an
ordinal.

So, there is a transfinite recursion schema.

Furthermore, for each p_i there exists a distinct q_i from the
rationals Q that are in (0,1). The value of q_0 is a rational between
p_0 and 1, then q_alpha+ is a rational between p_alpha+ and p_alpha,
and q_lambda is a rational between t_alpha and p_lambda. Given
trichotomy of the reals the q_i's are distinct.

(Here, there is a slight difference from the intermediate development,
with the limit ordinals' p_lambda's being strictly less than t_alpha,
although that was indicated in allusion, and q_i > p_i to simplify
definition.)

Then, construct a function f:P->Q as the set of ordered pairs {(p_i,
q_i), i E X}, defined for an arbitrarily large uncountable ordinal X.
That's an injection, a 1-1 mapping, from an uncountable set of
irrationals to a subset of the rationals. In your terminology, for i
E S, {(i,q_i)} is a bijection from S to a subset of the rationals.

If you think that's funny I also described a method to select natural
integers at uniform random by selecting real numbers from the unit
interval at uniform random by fair coin tosses, and EF is the CDF of a
uniform probability distribution over the naturals. (The real numbers
are more than standard.)

Ross

--
Finlayson Consulting

.

Relevant Pages

  • Re: Rational numbers, irrational numbers: each dense in real numbers
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  • Re: Question
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  • Re: Question
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  • Re: RAF: Rational numbers, irrational numbers: each dense in real numbers
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  • Re: Non-denumerable ordinals
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