On 24 Apr 2013, at 23:54, smi...@zonnet.nl wrote:
Perhaps one should define things such that it can be impolemented by
any arbitrary finite state machine, no mater how large. Then, while
there may not be a limit to the capacity of finite state machines,
each such machine has a finite capacity, and therefore in none of
these machines can one implement the Peano axiom that every integer
has a successor.
Number(0)
Number(s(x)) := Number(x)
This implements (in PROLOG) the Peano axiom that every number has a
successor
What you say is that the existential query "Number(x)?" will lead the
PROLOG machine into a non terminating computation. It will generates
0, s(0), s(s(0)), s(s(s(0))), s(s(s(s(0)))), ....
Similarly, you can implement a universal machine in a finite code. But
then the machine will ask sometimes for more memory space, like us.
But some other properties of integers are valid if they are valid in
every finite state machine that implement arithmetic modulo prime
numbers.
Not the fundamental recursion properties. If you fix the prime number,
you will stay in an ultrafinistic setting, without recursion, without
universal machine, without any fertile theorems of computer science
which makes sense even if it means that the machines, when implemented
in a limited environment will complain, write on the walls, or will
build a rocket to explore space and expand their memory by themselves.
I'm not into the foundations of math, I'll leave that to Bruno :) .
But since we are machines with a finite brain capacity,
In the long run, it is a growing one. And we have infinite capacities
relatively to our neighborhood. We don't stop to expand ourselves.
and even the entire visible universe has only a finite information
content,
If the physical universe is finite, but very big, we are still
universal machine. But doomed for some long run. No worry if comp is
true, as comp precludes a finite physical universe.
we should be able to replace real analysis with discrete analysis as
explained by Doron.
That can makes sense for some application, but would contradict comp
for the theoretical consequences.
Bruno
Saibal
Citeren Brian Tenneson <tenn...@gmail.com>:
Interesting read.
The problem I have with this is that in set theory, there are several
examples of sets who owe their existence to axioms alone. In other
words,
there is an axiom that states there is a set X such that (blah, blah,
blah). How are we to know which sets/notions are meaningless
concepts?
Because to me, it sounds like Doron's personal opinion that some
concepts
are meaningless while other concepts like huge, unknowable, and
tiny are
not meaningless. Is there anything that would remove the opinion
portion
of this?
How is the second axiom an improvement while containing words like
huge,
unknowable, and tiny??
quote
So I deny even the existence of the Peano axiom that every integer
has a
successor. Eventually
we would get an overflow error in the big computer in the sky, and
the sum
and product of any
two integers is well-defined only if the result is less than p, or
if one
wishes, one can compute them
modulo p. Since p is so large, this is not a practical problem,
since the
overflow in our earthly
computers comes so much sooner than the overflow errors in the big
computer
in the sky.
end quote
What if the big computer in the sky is infinite? Or if all
computers are
finite in capacity yet there is no largest computer?
What if NO computer activity is relevant to the set of numbers that
exist
"mathematically"?
On Monday, April 22, 2013 11:28:46 AM UTC-7, smi...@zonnet.nl wrote:
See here:
http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf
Saibal
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