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by lisper
3194 days ago
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> Wow, I thought it was pretty clear when I said that self-contained == not a function. I don't know how you've interpreted that to mean 'is finite'... Because your definition doesn't make sense. Everything in mathematics is a function. I was trying to do you the courtesy of interpreting what you said in a way that made sense rather than just pointing out that you don't seem to understand what a function is. > The program above is not a function; it has no inputs; it has no free variables; etc. hence it is a finite, self-contained program. Now it sounds like what you're trying to get at is the concept of a "constant function" (https://en.wikipedia.org/wiki/Constant_function). Constant functions are functions, and they can have inputs. So now it seems that you've just spent all this time and effort to point out that diagonalization is not a constant function. That's true, but I don't think it's a particularly interesting observation. |
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No, everything in mathematics is precisely what it is defined to be by whoever is doing the defining. If you want to introduce extraneous extra concepts into a theory, then that's fine; for example, Russell and Whitehead seemed to enjoy contorting set-based constructions into areas where they're completely unnecessary.
However, care should be taken to remain compatible with the original definitions. In this case, there is a clear distinction between "a foo" and "a function which takes some input and returns a foo which depends on which particular value was provided as input". To introduce functions as a 'universal substrate' such that everything is a function, then complain that every "foo" is a function, then introduce the concept of 'constant function' serves no purpose other than to obfuscate perfectly clear definitions under a mountain of irrelevant one-upmanship.
> So now it seems that you've just spent all this time and effort to point out that diagonalization is not a constant function.
That's what most of the time and effort have been spent on, yes. That's not at all what I was trying to "point out" though. I was simply assuming that this was common knowledge (for those familar with Cantor's proof, at least), and so I defined some terms like "self-contained" precisely so I that I could ignore algorithms like diagonalization.
> That's true, but I don't think it's a particularly interesting observation.
I agree it's not interesting. But I wouldn't even call it an "observation". Again, it's simply a trivial consequence of the definitions I gave, and I gave those definitions precisely so that I can ignore diagonalization, for this reason.
My actual observations are along the lines of "the steps of Cantor's proof can be rearranged in these ways, which give rise to these alternative results". Diagonalization doesn't matter for any of that, but you seem to keep trying to:
- Insert diagonalization into those rearrangements, despite it being a type error to do so (because diagonalization is a function, not a number; modulo whatever 'constant function' axioms would make you feel better)
- Complain that those rearrangements are "wrong" (they're not; they're my definitions, so they're "right" by definition!).
- Restate over and over various ways that my definitions do not correspond to Cantor's proof. Which is the entire point.
- "Prove wrong" those definitions, by showing that they're incompatible with various aspects of Cantor's proof (e.g. the order of steps) and diagonalization (e.g. its type). Which is completely backwards, since those definitions were chose precisely so that Cantor's proof and diagonalization will not work for them.