On 10/21/2024 4:40 AM, Mikko wrote:
> On 2024-10-21 03:58:05 +0000, olcott said:
>
>> On 10/20/2024 10:26 PM, Richard Damon wrote:
>>> On 10/20/24 5:59 PM, olcott wrote:
>>>> On 10/20/2024 2:13 PM, Richard Damon wrote:
>>>>> On 10/20/24 11:32 AM, olcott wrote:
>>>>>> On 10/20/2024 6:46 AM, Richard Damon wrote:
>>>>>>
>>>>>>> A "First Principles" approach that you refer to STARTS with an
>>>>>>> study and understanding of the actual basic principles of the
>>>>>>> system. That would be things like the basic definitions of things
>>>>>>> like "Program", "Halting" "Deciding", "Turing Machine", and then
>>>>>>> from those concepts, sees what can be done, without trying to
>>>>>>> rely on the ideas that others have used, but see if they went
>>>>>>> down a wrong track, and the was a different path in the same system.
>>>>>>>
>>>>>>
>>>>>> The actual barest essence for formal systems and computations
>>>>>> is finite string transformation rules applied to finite strings.
>>>>>
>>>>> So, show what you can do with that.
>>>>>
>>>>> Note, WHAT the rules can be is very important, and seems to be
>>>>> beyond you ability to reason about.
>>>>>
>>>>> After all, all a Turing Machine is is a way of defining a finite
>>>>> stting transformation computation.
>>>>>
>>>>>>
>>>>>> The next minimal increment of further elaboration is that some
>>>>>> finite strings has an assigned or derived property of Boolean
>>>>>> true. At this point of elaboration Boolean true has no more
>>>>>> semantic meaning than FooBar.
>>>>>
>>>>> And since you can't do the first step, you don't understand what
>>>>> that actually means.
>>>>>
>>>>
>>>> As soon as any algorithm is defined to transform any finite
>>>> string into any other finite string we have conclusively
>>>> proven that algorithms can transform finite strings.
>>>
>>> So?
>>>
>>>>
>>>> The simplest formal system that I can think of transforms
>>>> pairs of strings of ASCII digits into their sum. This algorithm
>>>> can be easily specified in C.
>>>
>>> So?
>>>
>>>>
>>>>>>
>>>>>> Some finite strings are assigned the FooBar property and other
>>>>>> finite string derive the FooBar property by applying FooBar
>>>>>> preserving operations to the first set.
>>>>>
>>>>> But, since we have an infinite number of finite strings to be
>>>>> assigned values, we can't just enumerate that set.
>>>>>
>>>>
>>>> The infinite set of pairs of finite strings of ASCII digits
>>>> can be easily transformed into their corresponding sum for
>>>> arbitrary elements of this infinite set.
>>>
>>> So?
>>>
>>>>
>>>>>>
>>>>>> Once finite strings have the FooBar property we can define
>>>>>> computations that apply Foobar preserving operations to
>>>>>> determine if other finite strings also have this FooBar property.
>>>>>>
>>>>>>> It seems you never even learned the First Principles of Logic
>>>>>>> Systems, bcause you don't understand that Formal Systems are
>>>>>>> built from their definitions, and those definitions can not be
>>>>>>> changed and let you stay in the same system.
>>>>>>>
>>>>>>
>>>>>> The actual First Principles are as I say they are: Finite string
>>>>>> transformation rules applied to finite strings. What you are
>>>>>> referring to are subsequent principles that have added more on
>>>>>> top of the actual first principles.
>>>>>>
>>>>>
>>>>> But it seems you never actually came up with actual "first
>>>>> Principles' about what could be done at your first step, and thus
>>>>> you have no idea what can be done at each of the later steps.
>>>>>
>>>>> Also, you then want to talk about fields that HAVE defined what
>>>>> those mean, but you don't understand that, so your claims about
>>>>> what they can do are just baseless.
>>>>>
>>>>> All you have done is proved that you don't really understand what
>>>>> you are talking about, but try to throw around jargon that you
>>>>> don't actually understand either, which makes so many of your
>>>>> statements just false or meaningless.
>>>>
>>>> When we establish the ultimate foundation of computation and
>>>> formal systems as transformations of finite strings having the
>>>> FooBar (or any other property) by FooBar preserving operations
>>>> into other finite strings then the membership algorithm would
>>>> seem to always be computable.
>>>>
>>>> There would either be some finite sequence of FooBar preserving
>>>> operations that derives X from the set of finite strings defined
>>>> to have the FooBar property or not.
>>>>
>>>
>>> But you don't understand that if you need to answer a question that
>>> isn;t based on a computable function, you get a question that you can
>>> not compute.
>>>
>>> Remember, a problem statement is effectively asking for a machine to
>>> compute a mapping from EVERY POSSIBLE finite string input to the
>>> corresponding answer.
>>>
>>> By simple counting, there are Aleph_0 possible deciders (since we can
>>> express the algorithm of the system as a finite string, so we must
>>> have only a countable infinite number of possible computations.
>>>
>>> When we count the possible problems to ask, even for a binary
>>> question, we have Aleph_0 possible inputs too, and thus 2 ^ Aleph_0
>>> possible mappings (as each mapping can have a unique combinations of
>>> output for every possible input).
>>>
>>> It turns out that 2 ^ Aleph_0 is Aleph_1, and that is greater than
>>> Aleph_0.
>>>
>>> This means we have more problems than deciders, and thus there MUST
>>> be problems that can not be solved.
>>>
>>
>> The problem is always:
>> Can this finite string be derived in L by applying FooBar
>> preserving operations to a set of strings in L having the
>> FooBar property?
>>
>> With finite strings that express all human knowledge that
>> can be expressed in language we can always reduce what could
>> otherwise be infinities into a finite set of categories.
>>
>>> When we look at the problem of proof finding, the problem is that
>>> from the finite number of statements, we can build an arbitrary
>>> length finite string that establishes the theorem. Trying to find an
>>> arbitrary length finite s
>>
>> Human knowledge expressed in language just doesn't seem
>> to work that way. When you ask someone a question as long
>> as they are not brain damaged they give you a succinct answer.
>
> Answers like "I don't know" and "What are you talking about" are
> fairly common.
>
For the Golbach conjecture IDK is the only correct answer.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer
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