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Topic: Finitely definable reals.
Replies: 52   Last Post: Jan 18, 2013 2:37 PM

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 DBatchelo1 Posts: 236 Registered: 12/12/04
Re: Finitely definable reals.
Posted: Jan 13, 2013 4:13 PM

On Friday, January 11, 2013 4:16:39 AM UTC-5, zuhair wrote:
> Lets say that a real r is finitely definable iff there is a predicate P that is describable by a Finitary formula that is uniquely satisfied by r. Formally speaking: r is finitely definable
I think this would be more helpful if "finitely definable" were defined more carefully.
The first stab would be to say that it means computable. ie there is a Turing machine that will compute the real.
This is not adequate; there are recursively enumerable sequences of computable numbers taht do not have a computable LUB. So; extend the definition. Assume that one has access to an oracle that can solve the halting for Turing Machines.
Unfortunately this only moves the problem. One finds one needs a more poerful oracle to solve the new halting problem - and so on.
This hierarchy is called the Kleene hierarchy. The set of real numbers thus defined is called the set of hyper arithmatic numbers. (For a fuller, better description , search the web. Wikepedia has a definition.)
I believe that the set of hyperarithmaqtic reals is what you mean by finitely definable.
Dick