Date: Mar 31, 2013 12:53 PM
Author: fom
Subject: Re: Matheology § 224
On 3/31/2013 11:49 AM, Ross A. Finlayson wrote:

> On Mar 31, 9:17 am, fom <fomJ...@nyms.net> wrote:

>> On 3/31/2013 10:52 AM, Ross A. Finlayson wrote:

>>

>>

>>

>>> Well, you see Virgil has introduced a term in context the "binary

>>> rational path": in cooperative communication that is so defined

>>> there, because that every initial segment is the initial segment of a

>>> rational, and that the language of "rational paths" is unbounded,

>>> doesn't offer for him the conclusion of his arguments. So, he expects

>>> that to be understood as his definition in passing, or he can point to

>>> it later, as to differentiating his personal definition from the

>>> general definition, as so qualified.

>>

>> Yeah.

>>

>> He should not have done that.

>>

>> The Baire space has the required property in

>> relation to rational numbers -- correspondence

>> with eventually constant sequences.

>>

>> It gets confusing when you are trying to deal

>> with WM's misrepresentations.

>

> Where "the" Baire space for Rene-Louis Baire is N^N as opposed to the

> general property of a space being Baire, consider whether there are

> ordinals between n, for any n in N, and N. N^n <-> N, N^N <-> P(N).

> if there are no ordinals between n and N, are there no cardinals

> betwen those of N^n and N^N? Because, cardinals have initial

> ordinals. Are there limit ordinals between those of w^n and w^N?

> Obviously enough it's consistent with ZF that there are, though, there

> are none between n, for all n e N, and N.

>

> Then, compared to the language of the expansions of 2^w from the

> alphabet {0,1} as (0|1)\infty, items from N^N are in a language (n e

> N)\infty.

>

> What's the 1-1 and onto function from 2^w to N^N?

The elements of the Baire space coincide with

real numbers according to the system of continued

fractions.