Re: infinity
- From: "Jonathan Hoyle" <jonhoyle@xxxxxxx>
- Date: 3 Oct 2005 23:40:37 -0700
>> In Hodges' paper there is at least one plain logical error.
>> Should he have to correct that? Hodges addresses plain
>> ol' logical errors, not the thousands and thousand of pages
>> of superspecific technicalities about transfinite cardinals or
>> infinity as, for example, appear in the vigorous debates
>> among interested parties here in sic.math.
I don't know what logic error you are talking about, but for sake of
argument, if he did make a mistake in logic somewhere in the paper,
that does not make him a crank. (Confusing mistake-making and
crankhood is one of the common attributes of a crank.) To see examples
of cranks better, I recommend picking up the very entertaining book
"Mathematical Cranks" by Underwood Dudley.
>> There are theories, for example anti-foundational theories,
>> where Cantor's results about powersets and basically the
>> missing element OR requirement for dual representation do
>> not generally hold. Is Aczel a crank, Jon?
Certainly not, and again this is the distinction that cranks are either
unwilling or unable to see. All mathematicians agree that if you begin
with ZFC you get certain theorems (such as Cantor's Power Set Axiom),
and also all mathemaaticians agree that if you begin with ZF + AFA (an
Anti-Foundation Axiom), you get different theorems. Different axioms
yield different results. No one disputes this.
>> ZF is inconsistent, where quantification over sets implies
>> a universal set.
This, for example, would be a crankish staatement.
>> Then, if you're sophisticated about set theory, then there is
>> the concern of Russell, and Ord, Burali-Forti, a paradox in ZF.
You are clearly misinformed. The Burali-Forti "paradox" is nothing
more than a proof by contradiction that the collection of all ordinals
is a proper class, not a set. As for Russell's paradox, this is
avoided with the Axiom of Foundation. (In non-Well Founded Set
Theories, Russell's "paradox" holds, but is not a paradox.)
<Mathematics and Physics crank remarks snipped>
>> Why do I say that choice is a theorem of ZF?
Presumably because you too are a crank. There is a mathematical proof
that the Axiom of Choice is independent of ZF.
<More ranting snipped>
>> Hoyle, transfinite cardinals aren't useful. I'd be interested
>> to know some way that they were useful, keeping in mind
>> that I don't think they're right for measure, probability,
>> computability and bounds, etc.
Presumably you don't think so because they are excellent examples of
where transfinite cardinals are useful. Allowing countable additivity
(while disallowing uncountable additivity) is one of the best uses for
transfinite cardinals. A number of problems in probability and measure
theory are handled using this. Demonstrably correct results are
arrived at by their use.
>> Infinite sets are infinite.
By the Reflexive Law of Equality, you are correct.
>> Infinite sets are equivalent.
If by "equivalent" you mean "have the same cardinality", then you are
incorrect, as proven by Georg Cantor. Not all finite sets are
equivalent, so assuming that all infinite sets would be is an
unjustified step, one which was proven incorrect a century ago.
Hope that helps.
Jonathan Hoyle
.
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- Re: infinity
- From: Jonathan Hoyle
- Re: infinity
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