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. But some other properties of integers are valid if they are
valid in every finite state machine that implement arithmetic modulo
prime numbers.
I'm not into the foundations of math, I'll leave that to Bruno :) . But
since we are machines with a finite brain capacity, and even the entire
visible universe has only a finite information content, we should be
able to replace real analysis with discrete analysis as explained by
Doron.
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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