On 09 Dec 2011, at 19:57, Stephen P. King wrote:
Dear Bruno,
On 12/9/2011 11:55 AM, Stephen P. King wrote:
On 12/9/2011 9:43 AM, Bruno Marchal wrote:
Assuming different instances of boolean algebra is assuming more
than the natural numbers (like assuming finite and infinite sets).
Are two Boolean algebras that have different propositional
content one and the same? If this is true then there is no
variation is algorithms, it is to say that all algorithms are
identical in every way.
Let me answer this differently. Does not the postulation of the
primitive existence of numbers not equivalent to postulating an
infinite set.
Not at all. As I said we need to postulate 0 and the successor rules
(and the + and * laws). Every "existing" object (that is the object
that you can prove to exist) are finite. The set N is not part of
arithmetic.
Are not the Integers an (countable) infinite set?
Yes, but that is not part of the theory. But you can prove in the
theory that there is no biggest numbers, or that for all numbers n you
can find a bigger one. You can also prove the existence of numbers who
believes in infinite sets, but you cannot prove the existence of an
infinite set in arithmetic.
Arithmetic is the simplest (universal with respect to computations)
theory. The one that Hillbert was hoping we could reduce all math to
it, but since Gödel we know that we cannot even reduce arithmetical
truth, or computer theoretical truth, to it.
Bruno
http://iridia.ulb.ac.be/~marchal/
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