On 15 Sep 2011, at 04:24, meekerdb wrote:

On 9/14/2011 5:51 PM, Mindey wrote:
"Nothingness" (as absence of things) is more than a concept. It is a
mathematical concept - an empty set. It is easy to give an example of
an empty set

I'm not so sure about that. Usually you would give some definition like "The present king of France" = { }. But if you think the universe is infinite and 'everything' exists, then there is a present king of France...somewhere. Bruno of course would say something like "The even divisors of three" = { }, but that would be circular since the definition of three is the successor of the successor of the successor of { }. And in any case he assumes arithmetic exists.

Not really. I have define three as the successor of successor of successor of 0. Which is a primitive symbol. I don't rely on set theory. Of course I could have use an axiomatic of hereditarily finite sets, instead of PA, like I could use combinators or java programs, but we have to stick on the definition, once chosen. Simple sets of numbers can be handled in PA by arithmetical definition, indeed we can define in arithmetic all recursively enumerable set, and we can talk about non recursively enumerable set too, although not always define them. Also, it is slightly misleading to say that I assume arithmetic exists. Arithmetic exists like physics exists, like music exists, etc. What I assume is that arithmetical statements are true independently of me, and of the physical laws (which is a very common assumption implicitly used in theoretical physics, analysis, etc.). (and then I assume the "yes doctor" and Church thesis). To just understand CT we need at least the believe that the excluded middle applies on the sigma_1 sentences: a machine stops or does not stop. I don't need much more.



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