Re: Solving a * 0 = 0 Plankenstein Monster "IT'S ALLLLiiiiiivvvvvvvve !!!"


hagman wrote:
huangxienchen@xxxxxxxxx schrieb:


Would you say that the solution set to

0 = 0 * a ( where a is a "given" constant)

is topologically equivalent to

0 = 0 * x (where x is a variable)

???

No, why should I?
The first equation has no unknowns, hence there is nothing to solve at
all.
The second equation has one unknown.
In general, the solution set of an equation (or a set of equations)
with n unknowns is a set of n-tuples. It's a bit tricky to say what a
set of 0-tuples should be, but apparently there is only one 0-tuple:
(). And there are only two sets of 0-tuples: {} and {()}.
If you insist that the first equation should be viewed as an equation
in 0 unknowns,
then the set of solutions is (for any given [real?] a) the set {()}.
And, as has been stated often enough, when working over the reals, the
set of solutions to the second equation is R. R, for example with
standard topology, is not homeomorphic to {()} (which allows only one
topology).

Certainly, whatever these solutions are, they are not equivalent to the
null set of solutions which you'd get from 0 = 0.

In the same sense, viewing this as an equation in 0 unknowns, the set
of solutions would be {()}.
However, the set of solutions to 0 = 1 would be the null set {}.


Yes, exactly. And of course,

0 * a = 0 * x


Well, I think that this whole sitaution does have substantive meaning.

It does not show that math is wrong, or that math melts down or
anything like that.

Much more fascinating, I think that what is means is that you have
genuine indeterminacy. It even undermines topology. That is wierd, and
beautiful I'd say.

The strange thing is that you cant really use it as a random number
generator because regardless of what number you produce or what
topological structure you generate, the original equation is reducible
to 0 = 0, and this implies that the disorder is existentially
indeterminate.

Somehow, perhaps this "could" be a possible source of disorder in
things like the logistic map, attractors, cellular automata, fractals,
etc.

Lots of speculation, but all seemingly reasonable.

.

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