computational philosophy of mathematics
Feb. 5th, 2005 10:53 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
guslI only believe in mathematical objects with finite information (i.e. with a finite expression). Things can expressed in any way you wish, intensionally, extensionally, whatever. So, while the natural numbers exist, not all subsets of it do.
While the set of real numbers exists (R can be expressed as the power set of N), not all of the traditional real numbers exist (only the computable ones do). So, in my definition, the cardinality of R is aleph_0 (i.e. the same as N). In fact, no set can have greater cardinality, for that would imply it had non-existing elements (since only computable things exist).
Can we design a set theory this way? How much of traditional set theory can be translated? Does we lose any good mathematics this way?
Whereas people seem to view uncomputability as a fundamental property of a problem, I tend to view it as another form of self-reference paradoxes. All "well-defined" problems are computable. Again, this is not a theorem or mathematical insight, but a re-definition, just like my "R only has countably many existing numbers" is a definition. But the point isn't just to change names and keep everything the same... it's to change the intuition that goes with the names.
While the set of real numbers exists (R can be expressed as the power set of N), not all of the traditional real numbers exist (only the computable ones do). So, in my definition, the cardinality of R is aleph_0 (i.e. the same as N). In fact, no set can have greater cardinality, for that would imply it had non-existing elements (since only computable things exist).
Can we design a set theory this way? How much of traditional set theory can be translated? Does we lose any good mathematics this way?
Whereas people seem to view uncomputability as a fundamental property of a problem, I tend to view it as another form of self-reference paradoxes. All "well-defined" problems are computable. Again, this is not a theorem or mathematical insight, but a re-definition, just like my "R only has countably many existing numbers" is a definition. But the point isn't just to change names and keep everything the same... it's to change the intuition that goes with the names.
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Re: Cardinality
Date: 2005-02-05 09:57 pm (UTC)There is no bijection. No function (algorithm) exists that can map ℕ onto ℝ. I can give you a list of Turning machines, but you cannot pick out all the ones that are real numbers.
I’m in Nijmegen now. I’m giving a talk in Eindoven Tuesday afternoon.
Re: Cardinality
Date: 2005-02-05 10:14 pm (UTC)I'd like to hear your ideas on applications of formal theories to education. I have a bunch of ideas, but they need to be beaten up.
Re: Cardinality
Date: 2005-02-05 10:36 pm (UTC)I agree the set is smaller. I’m just saying that no bijection exists. In fact no onto function exists, and a set A is countable if there is a function f:ℕ→A that is onto.
Just because the computable reals are a ‘subset’ of Turning machines, doesn’t mean a bijection exists.
Re: Cardinality
Date: 2005-02-05 10:58 pm (UTC)Interesting. How do you know? Is this a consequence of the halting problem?
Re: Cardinality
Date: 2005-02-06 10:18 am (UTC)Assume a bijection exists. Apply Cantor’s diagonalization argument and you get a contradiction.
Re: Cardinality
Date: 2005-02-06 10:25 am (UTC)