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

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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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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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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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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 And, as posted in comp.theory, you are just showing that you don't understand the nature of the problem, and that determining whether or not such a string exists can be uncomputable, as it requires searching an infinite set.