3224Re: [learningfromeachother] Re: Fruitful non-excluded middle
- May 2, 2011On Mon, May 2, 2011 at 15:10, <ms@...> wrote:
> Edward, Thank you for explaining! I've included your examples here:I forgot to mention that the general case of figuring things out is
> And here is a link to the semiotic square:
> Now I'm working on writing up a brief "general method" for "figuring
> things out". Then I'll show how I apply that to making a living.
formally undecidable, and cannot be reduced to a fully general method.
There are innumerable examples. One of the first was the Halting
Problem for Turing machines, and thus for all computers we know of.
The problem is that the set of theorems in a first-order theory is
recursively enumerable (there is a process for generating them, one by
one, that will eventually get to each and every one) but the set of
non-theorems is not. So if our theorem generator has not produced a
particular statement or its denial as a theorem, we do not know which
set it falls into: theorem, denial of a theorem, or undecidable
Any system that can represent all statements and proofs in a
first-order theory is also undecidable. This includes tiling problems
and Diophantine equations (Hilbert's Tenth Problem).
Second- and higher-order theories with arithmetic are inherently
undecidable, because they have more than countably many theorems, and
we can only prove a countable subset.
> Andrius Kulikauskas, ms@...--
> 2011.04.25 20:30, Edward Cherlin rašė:
>> 2011/4/25 Andrius Kulikauskas <ms@...>:
>>> Edward, thank you for your letters! They are very helpful. I ask you
>>> also to think of examples where methods, or a kind of thinking, proved
>>> fruitful. You mention the excluded middle. For example, the Lithuanian
>>> semiotician Algirdas Julius Greimas developed the semiotic square
>>> (related to Aristotle's logical square), for example: White Black
>>> Not-Black Not-White. Where Not-White might be "colorlessness" and
>>> Not-Black might be "grey" if I remember correctly. But for my purposes,
>>> I want to document examples where such thinking was actually fruitful.
>> Yale Professor Fred B. Fitch's book, Symbolic Logic presents a system
>> of logic that can be proven consistent. Dropping the law of Excluded
>> Middle was essential to the construction. Gödel's theorem depends on
>> Excluded Middle, so it doesn't apply to this proof of consistency.
>> If R is the set of all sets that are not members of themselves (with
>> further precision required that does not concern us here), then R is a
>> member of R if and only if R is not a member of R. In the presence of
>> Excluded Middle, this results in contradiction. In its absence, it is
>> merely undecidable both in terms of provability and of truth.
>> This idea can be followed into a realm of multiple-valued logics.
>> Buddhist logic considers the possibilities
>> Does not exist
>> Both exists and does not exist
>> Neither exists nor does not exist
>> None of the above
>> as one of many ways of stating that meditation does not work the way you
>>> I don't want to confuse fruitful and nonfruitful approaches! And I also
>>> want to relate each way of thinking with the kinds of results it yields.
>>> I'm always wondering how I could make a living from documenting and
>>> sharing "ways of figuring things out". Perhaps I should do that for
>>> business and economics.
>> It's known as becoming a professor or a published writer.
>> Separately, however, you would be welcome to contribute to our
>> analysis of business and economics for schoolchildren in developing
>> countries, where the dogmas of conventional economics are revealed to
>> be the airiest fantasies.
>>> Thank you for thinking along with me!
>>> Andrius Kulikauskas
>>> (773) 306-3807
>>> Twitter: @selflearners
>>> Each letter sent to Learning From Each Other enters the PUBLIC DOMAIN
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Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin
Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.
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