> Date: Tue, 9 Jun 2009 17:20:39 -0700
> From: meekerdb.domain.name.hidden
> To: everything-list.domain.name.hidden
> Subject: Re: The seven step-Mathematical preliminaries
>
>
> Jesse Mazer wrote:
>>
>>
>>> Date: Tue, 9 Jun 2009 15:22:10 -0700
>>> From: meekerdb.domain.name.hidden
>>> To: everything-list.domain.name.hidden
>>> Subject: Re: The seven step-Mathematical preliminaries
>>>
>>>
>>> Jesse Mazer wrote:
>>>>
>>>>
>>>>> Date: Tue, 9 Jun 2009 12:54:16 -0700
>>>>> From: meekerdb.domain.name.hidden
>>>>> To: everything-list.domain.name.hidden
>>>>> Subject: Re: The seven step-Mathematical preliminaries
>>>>>
>>>>
>>>>> You don't justify definitions. How would you justify Peano's axioms
>>>> as being
>>>>> the "right" ones? You are just confirming my point that you are
>>>> begging the
>>>>> question by assuming there is a set called "the natural numbers"
>>>> that exists
>>>>> independently of it's definition and it satisfies Peano's axioms.
>>>>
>>>> What do you mean by "exists" in this context? What would it mean to
>>>> have a well-defined, non-contradictory definition of some mathematical
>>>> objects, and yet for those mathematical objects not to "exist"?
>>>
>>> A good question. But if one talks about some mathematical object, like
>>> the natural numbers, having properties that are unprovable from their
>>> defining set of axioms then it seems that one has assumed some kind of
>>> existence apart from the particular definition.
>>
>> 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), then
>> I think we'd have a well-defined notion of how to program it to decide
>> the truth of every "for all" or "there exists" proposition in a way
>> that's compatible with the propositions already proved by the Peano
>> axioms. If I'm right about that, it would lead naturally to the idea
>> of something like a "unique consistent extension" of the Peano axioms
>> (not a real technical term, I just made up this phrase, but unless
>> there's an error in my reasoning I imagine mathematicians have some
>> analogous notion...maybe Bruno knows?) which assigns truth values to
>> all the well-formed propositions that are undecidable by the Peano
>> axioms themselves. So this would be a natural way of understanding the
>> idea of truths "about the natural numbers" that are not decidable by
>> the Peano axioms.
>
> I think Godel's imcompleteness theorem already implies that there must
> be non-unique extensions, (e.g. maybe you can add an axiom either that
> there are infinitely many pairs of primes differing by two or the
> negative of that). That would seem to be a reductio against the
> existence of a hypercomputer that could decide these propositions by
> inspection.
I think I remember reading in one of Roger Penrose's books that there is a difference between an ordinary consistency condition (which just means that no two propositions explicitly contradict each other) and "omega-consistency"--see
http://en.wikipedia.org/wiki/Omega-consistent_theory . I can't quite follow the details, but I'm guessing the condition means (or at least includes) something like the idea that if you have a statement of the form "there exists a number (or set of numbers) with property X" then there must actually be some other proposition describing a particular number (or set of numbers) does in fact have this property. The fact that you can add either a Godel statement or its negation to the Peano axioms without creating a contradiction (as long as the Peano axioms are not inconsistent) may not mean you can add either one and still have an omega-consistent theory; if that's true, would there be a unique omega-consistent way to set the truth value of all well-formed propositions about arithmetic which are undecidable by the Peano axioms? Again, Bruno might know...
>
> So we believe in the consistency of Peano's arithmetic because we have a
> physical model.
Well, I would say we generalize our understanding from a physical model, but once we have that understanding it's sort of generalized and doesn't depend on checking that things work for each specific number of discrete objects. For example, would you agree with the intuition that if we have a square array of idealized marbles, then simply mentally rotating it so we count them in a different order shouldn't change the total number of marbles in the array, so we can be confident that A*B = B*A for arbitrary numbers of marbles on the vertical and horizontal sides of the square?
Jesse
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Received on Tue Jun 09 2009 - 22:14:33 PDT