On Tue, May 22, 2012 at 7:36 AM, Stephen P. King <stephe...@charter.net>wrote:

>  On 5/21/2012 6:26 PM, Russell Standish wrote:
>
> On Mon, May 21, 2012 at 07:42:01AM -0400, Stephen P. King wrote:
>
>  On 5/21/2012 12:33 AM, Russell Standish wrote:
>
>  On Sun, May 20, 2012 at 12:06:05PM -0700, meekerdb wrote:
>
>  On 5/20/2012 9:27 AM, Stephen P. King wrote:
>
>  4) What is the cardinality of "all computations"?
>
>  Aleph1.
>
>
>  Actually, it is aleph_0. The set of all computations is
> countable. OTOH, the set of all experiences (under COMP) is uncountable
> (2^\aleph_0 in fact), which only equals \aleph_1 if the continuity
> hypothesis holds.
>
>  Hi Russell,
>
>     Interesting. Do you have any thoughts on what would follow from
> not holding the continuity (Cantor's continuum?) hypothesis?
>
>
>  No - its not my field. My understanding is that the CH has bugger all
> impact on quotidian mathematics - the stuff physicists use,
> basically. But it has a profound effect on the properties of
> transfinite sets. And nobody can decide whether CH should be true or
> false (both possibilities produce consistent results).
>
>
> Hi Russell,
>
>     I once thought that consistency, in mathematics, was the indication of
> existence but situations like this make that idea a point of contention...
> CH and AoC <http://en.wikipedia.org/wiki/Axiom_of_choice> are two axioms
> associated with ZF set theory that have lead some people (including me) to
> consider a wider interpretation of mathematics. What if all possible
> consistent mathematical theories must somehow exist?
>

Joel David Hamkins introduced the "set-theoretic multiverse" idea
(link<http://arxiv.org/abs/1108.4223>).
The abstract reads:

"The multiverse view in set theory, introduced and argued for in this
article, is the view that there are many distinct concepts of set, each
instantiated in a corresponding set-theoretic universe. The universe view,
in contrast, asserts that there is an absolute background set concept, with
a corresponding absolute set-theoretic universe in which every
set-theoretic question has a definite answer. The multiverse position, I
argue, explains our experience with the enormous diversity of set-theoretic
possibilities, a phenomenon that challenges the universe view. In
particular, I argue that the continuum hypothesis is settled on the
multiverse view by our extensive knowledge about how it behaves in the
multiverse, and as a result it can no longer be settled in the manner
formerly hoped for."


>
>
>  Its one reason why Bruno would like to restrict ontology to machines,
> or at most integers - echoing Kronecker's quotable "God made the
> integers, all else is the work of man".
>
>
>
>
>     I understand that, but this choice to restrict makes Bruno's Idealism
> even more perplexing to me; how is it that the Integers are given such
> special status, especially when we cast aside all possibility (within our
> ontology) of the "reality" of the physical world? Without the physical
> world to act as a "selection" mechanism for what is "Real", why the bias
> for integers? This has been a question that I have tried to get answered to
> no avail.
>

I think Bruno gives such high status to the natural numbers because they
are perhaps the least-doubt-able mathematical entities there are. The very
fact that talks of a "set-theoretic multiverse" exist makes one ask, how
real are sets? Do set theories tell us more about our minds than they do
about the mathematical world? (Obviously, as David Lewis pointed out, you
need something like a set theory in order to do mathematics at all, and as
Russell says, for the average mathematician it really doesn't matter.)

Also: *No one here has questioned the reality of the physical world. *Should
I append this statement to every email until you stop countering it?


>
>
>
>   This is the origin of Bruno's claim that COMP entails that physics is
> not computable, a corrolory of which is that Digital Physics is
> refuted (since DP=>COMP).
>
>
>      Does the symbol "=>" mean "implies"? I get confused ...
>
>
>  Yes, that is the usual meaning. It can also be written (DP or not COMP).
>
>
>     "=>" = "or not"]
>

Actually "a implies b" is defined as "not a or b".


>
>     I am still trying to comprehent that equivalence! BTW, I was reading a 
> related
> Wiki article <http://en.wikipedia.org/wiki/Transposition_%28logic%29> and
> found the sentence "the truth of "A implies B" the truth of "Not-B implies
> not-A"". That looks familiar... Didn't I write something like that to
> Quentin and was rebuffed... I wrote it incorrectly it appears...
>
>
>  Of course in Fortran, it means something entirely different: it
> renames a type, much like the typedef statement of C. Sorry, that was
> a digression.
>
>
>     That's OK. ;-) I suppose that it is a blessing to be able to "think in
> code". ;-)
>
>
>
>
> --
> Onward!
>
> Stephen
>
> "Nature, to be commanded, must be obeyed."
> ~ Francis Bacon
>
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-- 
Joseph Knight

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