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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: Sun, 20 Oct 2024 22:58:05 -0500
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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.

-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer