Re: Deriving X from the finite set of FooBar preserving operations --- membership algorithm for X in L

On 10/20/24 11:58 PM, olcott wrote:

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:
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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.
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The actual barest essence for formal systems and computations
is finite string transformation rules applied to finite strings.

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So, show what you can do with that.
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Note, WHAT the rules can be is very important, and seems to be beyond you ability to reason about.
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After all, all a Turing Machine is is a way of defining a finite stting transformation computation.
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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.

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And since you can't do the first step, you don't understand what that actually means.
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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.

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So?
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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.

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So?
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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.

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But, since we have an infinite number of finite strings to be assigned values, we can't just enumerate that set.
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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.

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So?
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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.
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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.
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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.
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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.
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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.
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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.

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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.
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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.
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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.
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Remember, a problem statement is effectively asking for a machine to compute a mapping from EVERY POSSIBLE finite string input to the corresponding answer.
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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.
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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).
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It turns out that 2 ^ Aleph_0 is Aleph_1, and that is greater than Aleph_0.
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This means we have more problems than deciders, and thus there MUST be problems that can not be solved.
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 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.

But searching the infinite space of possible strings can not be always done in finite time.
Remember, we can express all of infinity with just two characters, (0 and 1) in there unlimited combinations.
Part of your problem is you just don't understand what infinity is, because you just don't understand how logic actually works, and some attributes of infinity are not self-evident.

 

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.
 

If they know it.
Note also, you premise confuses knowledge with truth. You could store everything we know in a computer database, and perhaps program a computer to work to "evolve it" to discover things we didn't understand before (but would need a good filter so you don't fill with a lot of truths like 1+3 = 4)
The problem with such a system is it doesn't tell us if  a statement its TRUE< but if it is KNOWN. True statements that haven't been discovered yet will not be in its database, and if that database is based on human knowleged which comes from observations, it WILL contain errors due to errors in observations.
After all, if done at some points in time, it would have the "fact" that it was known that the Earth was flat. (and also that it was round).
You are just proving that you don't understand the difference between facts and knowledge, and thus much of what you claim to be true is actually just a lie based on your own misunderstandings.