Re: Cardinality of Set of Computable Numbers?

From: mitch (mitchs_at_rcnNOSPAM.com)
Date: 12/27/03 

Date: Sat, 27 Dec 2003 03:24:22 -0600

Russell Easterly wrote:

> "mitch" <mitchs@rcnNOSPAM.com> wrote in message
> news:3FEBCCD0.D42A0709@rcnNOSPAM.com...
> >
> > I submit that Surygin's research on e-sets might have some answers for
> > you.
> >
> > :-)
> >
> > mitch
> >
>
> Thanks.
> Now, I'm really scared.
> :-)
>

There are some good reasons for restricting yourself to constructive
mathematics.

I have spent the last year being ridiculed on sci.logic for things I should
not have been (hence the change in group lists on this thread). The theory
I offered over there captured an intuitionist concept referred to as
"apartness." However, since I was unaware of that connection at the
time,... well there is no need to repeat myself.

Looking at Blackburn's "Guide to Philosophical Logic" I have discovered that
this notion of apartness is fundamental to the "super-semantics" (sheaf
semantics) of intuitionist logic. In particular, it seems to have some
relation to measure theory in that apartness is needed to distinguish
elements from zero for the intuitionistic theory of reals. There is work on
constructive measures on constructive reals and work on solvable Boolean
algebras. Moreover, when I turn to Blackburn's chapter on negation, it
seems that the strongest definite concept of negation is associated with
constructive concepts. So, when the discussion of sheaf semantics ends with
an observation that most of the semantics is adequately captured by Hyland's
description of an effective topos, I get the impression that there has been
a great deal of good mathematics being ignored by curriculum committees.

But, of course, we have to work with that of which we have knowledge. I
wish I could direct you to specific papers, but as I hope you have
discerned, I am only beginning to realize how "scared" I am myself. :-)

I see you are getting many suggestions from others that should be helpful
for avoiding confusion with your specific question. That is a good thing.

:-)

mitch

Relevant Pages

  • Re: Formal Semantics of the Java "new" Operator
    ... > Whenever I encounter arbitrariness in mathematics, ... In a sense this means that the semantics is non-deterministic, ... _All_ formalizations are arbitrary in the sense that the semantics of ...
    (comp.theory)
  • Yes You Can Tell The Difference Revisited
    ... they must be relativized and the apparent predication eliminated. ... complicate our semantics in this way (for in semantics as in ... What makes mathematics objective, everyone learns in ... I'm not contending that aesthetic subjectivism is wrong, ...
    (rec.music.makers.guitar.acoustic)
  • Re: Yes You Can Tell The Difference Revisited
    ... they must be relativized and the apparent predication eliminated. ... complicate our semantics in this way (for in semantics as in ... What makes mathematics objective, everyone learns in ... I'm not contending that aesthetic subjectivism is wrong, ...
    (rec.music.makers.guitar.acoustic)
  • Re: Countable models of ZFC
    ... But if you don't like my views about the semantics of mathematics, ... language of set theory which isn't model-theoretic. ... in which the first-order language of set theory is interpreted ...
    (sci.logic)
  • Re: SUBTHREAD : fixed point
    ... rudimentary knowledge of mathematics and has shown no ability or ... the only utterances from Mitch ... On the contrary, these opinions are formed ... Bigotry has been eliminated and honest debate can begin! ...
    (sci.math)