Subject: Re: The seven step-Mathematical preliminaries
Date: Wed, 10 Jun 2009 18:03:26 +0200

On 10 Jun 2009, at 01:50, Jesse Mazer wrote:

Isn't this based on the idea that there should be an objective truth about 
every well-formed proposition about the natural numbers even if the Peano 
axioms cannot decide the truth about all propositions? I think that the 
statements that cannot be proved are disproved would all be ones of the type 
"for all numbers with property X, Y is true" or "there exists a number (or some 
finite group of numbers) with property X" (i.e. propositions using either the 
'for all' or 'there exists' universal quantifiers in logic, with variables 
representing specific numbers or groups of numbers). So to believe these 
statements are objectively true basically means there would be a unique way to 
"extend" our judgment of the truth-values of propositions from the judgments 
already given by the Peano axioms, in such a way that if we could flip through 
all the infinite propositions judged true by the Peano axioms, we would *not* 
find an example of a proposition like "for this specific number N with property 
X, Y is false" (which would disprove the 'for all' proposition above), and 
likewise we would not find that for every possible number (or group of numbers) 
N, the Peano axioms proved a proposition like "number N does not have property 
X" (which would disprove the 'there exists' proposition above). 
We can't actual flip through an infinite number of propositions in a finite 
time of course, but if we had a "hypercomputer" that could do so (which is 
equivalent to the notion of a hypercomputer that can decide in finite time if 
any given Turing program halts or not), 
>Such an hypercomputer is just what Turing called an "oracle". And the haslting 
>oracle is very low in the hierarchy of possible oracles.And Turing results is 
>that even a transfinite ladder of more and more powerful oracles that you can 
>add on Peano Arithmetic,  will not give you a complete theory. Hypercomputing 
>by constructive extension of PA, with more and more powerful oracles does not 
>help to overcome incompleteness, unless you add non constructive ordinal 
>extension of "hypercomputation".This is the obeject of the study of the 
>degrees of unsolvability, originated by Emil Post.

Interesting, thanks. But I find it hard to imagine what kind of finite 
proposition about natural numbers could not be checked simply by plugging in 
every possible value for whatever variables appear in the 
proposition...certainly as long as the number of variables appearing in the 
proposition is finite, the number of possible ways of substituting specific 
values for those variables is countably infinite and a hypercomputer should be 
able to check every case in a finite time. Does what you're saying imply you 
can you have a proposition which somehow implicitly involves an infinite number 
of distinct variables even though it doesn't actually write them all out? Can 
all propositions about arbitrary *real* numbers (which are of course 
uncountably infinite) be translated into equivalent propositions about whole 
numbers in arithmetic? Or am I taking the wrong approach here, and the reason a 
hypercomputer can't decide every proposition about arithmetic unrelated to the 
issue of how many distinct variables can appear in a proposition? 
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