Re: the need for relevance

Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> writes:

Jesse F. Hughes wrote:

Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> writes:

There is one and only one real world. And we _both_ live in
it. Whether you like it or deny it or not. Anything that doesn't
match with the real world is contradictory to it.
Is this the working definition, then?
A contradiction in a theory T is a statement P which does not match
with the real world.
Is this what you mean when you say that ZFC has contradictions? If
so, perhaps you can go on to show me how it is that the claim "There
is no uniform probability distribution on N" is a contradiction. In
what way does that not match the real world? What does "match the
real world" mean? Many thanks. Illuminating as ever.

I've been illuminating this over and over, on my website and in the
'sci.math' newsgroups. If the real world is consistent - and it IS,
because otherwise it would cease to exist - then mathematical stuff,
not giving outcomes that match with the real world, is inconsistent.

I have also explained there what "match with the real world" _means_:
there is a pathway from the Idealizations back to the physical world,
called Materialization, which is the inverse process of Idealization.
It is evident that materialization is not possible with pure fantasies,
giving kind of a watershed between the "good" and the "bad" axioms.

*This* is an explanation, is it? Sorry, but it doesn't help me out at
all. You have not said anything specific about what "Materialization"
actually *is*.

It is demonstrated in "Implementable Set Theory" that all axioms of ZFC
are consistent with the bit mapped implementation, where the axiom of
Infinity seems to be the only exception.

So: A materialization is a model in a computer? Is this what you
mean?

If it is claimed that the bit mapped implementation actually is the
meaning of / what SETS ARE, then Implementable Set Theory offers us
a means to _judge_ the axiom system (not the other way around). That
judgement is negative for the axiom of Infinity, not quite
unexpectedly.

Right. If we demand that our theory has a model in a finite state
machine, it will follow that our theory has a finite model. Remind me
why this is a good demand.

I've published more than enough about "uniform probability distribution
on N". In a nutshell: it's trivial for finite serts {0,1,2,3, ... ,n}
that a uniform probability distribution exists. Now replace the finite
set by an unfinished set {0,1,2,3, ... ,n,n+1, .. } and assume no upper
bound for (n) as soon as your calculation is finished.

Not sure what that last sentence means, nor why this is relevant to
the claim that there is no uniform distribution on N.

Results are that e.g. the probability of a natural being divisible
by 3 is 1/3 , because the limit for of the finite set result
approaches this value. There is nothing of a mystery or "undefined"
about all this.

The results are that *only if* you have a principle that the uniform
distribution on N is the same thing as the limit of the uniform
distributions on the proper initial segments of N. But this is not a
self-evident principle, so where did you come up with that?

Yes, I know. You think it's self-evident that N is "potentially
infinite" and that to determine if a property holds on a potentially
infinite set, you check to see how it holds on the finite subsets and
"take a limit". But that doesn't always work, I guess. Here's a
property P: "X is finite (i.e., not potentially infinite)."

P is true of {0}.
P is true of {0,1}.
P is true of {0,1,2}.
....
P is true of {0,1,2,3,...,n}.

so by your reasoning, it follows that P is true of N and hence that N
is not potentially infinite. Geez, where did I go wrong?

--
Jesse F. Hughes
"The Cantorians are conducting a campaign of psychological warfare
against humanity."
-- David Petry, on why set theory is evil.
.

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