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From: olcott <polcott333@gmail.com>
Newsgroups: sci.logic
Subject: Re: Deriving X from the finite set of FooBar preserving operations
 --- membership algorithm for X in L
Date: Mon, 21 Oct 2024 18:12:49 -0500
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On 10/21/2024 5:46 PM, Richard Damon wrote:
> On 10/21/24 9:31 AM, olcott wrote:
>> 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.
>>
> 
> So, you admit that the statment might be true and unprovable?
> 

There are some expressions of language that seem to
have a truth value of UNKNOWABLE.

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