[FOM] Micro Set Theory
Paul Tarau
paul.tarau at gmail.com
Fri May 17 11:17:36 EDT 2013
While encoding with powers of primes is not size-proportionate, more efficient bijective encodings exists - see
http://code.google.com/p/bijective-goedel-numberings/
With such encodings, formal notations that are about as large as the printed size of the naturals in V_7 correspond
to Goedel numbers of about the same size as the notations themselves.
On the other hand, I agree with Andrej's point that structure is more important than size. See
http://code.google.com/p/giant-numbers/
for a representation that can accommodate numbers in V_n for significantly larger values of n, that have a "recursively regular" structure.
Paul Tarau
On May 16, 2013, at 3:20 AM, Andrej Bauer <andrej.bauer at andrej.com> wrote:
> I would say that size is not the only thing that matters, the structure is important too.
>
> It is important that a formalization respects, or at least does not mutilate, the inherent structure of the thing you're formalizing. For example, it may be cool to represent lists of numbers as a product of powers of prime numbers, but that sort of thing is going to be thoroughly useless in practice. In practice, the formal language must allow direct and abstract expression of the concepts involved.
>
> Which is why the 2^65536 elements of V_7 will be of little help when you try to encode a list of 9 elements.
>
> Suppose that we lived in a world of fantasy where someone gave us an oracle for V_7. We would devise (unnatural) encoding schemes for asking the oracle all sorts of questions, and it would give us answers. Soon there would be just one question we cared about: how does it work? We would take it apart and study its inner mechanism to learn --- about the structure of V_7! Because, to understand how such a magnificent piece of machinery could answer everything there is to know about V_7, it to understand the structure of V_7. Not the size.
>
> With kind regards,
>
> Andrej
>
> On Tue, May 14, 2013 at 3:53 PM, Joe Shipman <JoeShipman at aol.com> wrote:
> If V_0 is the empty set and V_(i+1) is the power set of V_i, then the elements of V_7 have infeasible size and can code objects smaller than 2^65536, so that an oracle for the theory of V_7 would allow answers to any mathematical question we care about (for example "Does the Riemann Hypothesis have a proof from ZF + your favorite large cardinal axiom of length < 10^1000 ?"), thereby rendering mathematicians obsolete.
>
> V_4 is trivially uninteresting; it has only 16 elements so brute force programming will let us answer any question about it involving up to 10 or so quantifiers.
>
> An oracle for V_6 would be very helpful since it would (I think but need to verify) let us solve instances of PSPACE-complete problems of length in the thousands of bits, for example calculating Ramsey numbers, but would it make mathematicians almost as obsolete as an oracle for V_7 would?
>
> Would an oracle for V5 tell us anything interesting at all?
>
> -- JS
>
> Sent from my iPhone
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