# Re: Rationals vs Reals in Comp

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On 24 Apr 2013, at 23:54, smi...@zonnet.nl wrote:```
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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)

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This implements (in PROLOG) the Peano axiom that every number has a successor
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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)))), ....
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Similarly, you can implement a universal machine in a finite code. But then the machine will ask sometimes for more memory space, like us.
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But some other properties of integers are valid if they are valid in every finite state machine that implement arithmetic modulo prime numbers.
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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.
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I'm not into the foundations of math, I'll leave that to Bruno :) . But since we are machines with a finite brain capacity,
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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.
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and even the entire visible universe has only a finite information content,
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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.
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we should be able to replace real analysis with discrete analysis as explained by Doron.
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That can makes sense for some application, but would contradict comp for the theoretical consequences.
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Bruno

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Saibal

Citeren Brian Tenneson <tenn...@gmail.com>:

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```Interesting read.

The problem I have with this is that in set theory, there are several
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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,
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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?

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How is the second axiom an improvement while containing words like huge,
```unknowable, and tiny??

quote
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So I deny even the existence of the Peano axiom that every integer has a
```successor. Eventually
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we would get an overflow error in the big computer in the sky, and the sum
```and product of any
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two integers is well-defined only if the result is less than p, or if one
```wishes, one can compute them
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modulo p. Since p is so large, this is not a practical problem, since the
```overflow in our earthly
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computers comes so much sooner than the overflow errors in the big computer
```in the sky.
end quote

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What if the big computer in the sky is infinite? Or if all computers are
```finite in capacity yet there is no largest computer?

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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:
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See here:

http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf

Saibal

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