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Re: [ontolog-forum] Finitism vs. Transfinitism

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Tue, 4 Mar 2008 11:12:26 -0600
Message-id: <D0A6B2B7-0A84-4A81-AC32-A460722E1A18@xxxxxxxx>
Not debating here, only correcting demonstrably false and hence  
potentially misleading claims.    (01)

On Mar 4, 2008, at 7:21 AM, Avril Styrman wrote:
>>> There is no objective way to decide that the convention that you
>>> have learned is somehow better than the below convention:
>>> 1. Having a set {1,2,3, ..., n}, the cardinality of the set is n.
>>> 2. The cardinality of the set grows as n grows.
>>> 3. If the cardinality is infinite, there must be an infinite n
>> Unbelievable.  Ok, tell you what.  In ZFC, proposition 3 is  
>> provably false.  So you obviously don't accept ZFC. If you hold on  
>> to ZFC, then you perhaps can prove that 3. is wrong, but I do not  
>> hold on to ZFC, and therefore 3. makes sense to me.    (02)

> I gave the axiom that makes 3. intelligible:
> "Having any set such as {1,2,3,...} that starts with number 1, and  
> has only successors of 1 as members, one after another, then, if the  
> set has a cardinality x, then x is also a member of the set"
> This axiom totally as objective as the interpretation of complete  
> induction in ZFC.    (03)

Your "axiom" is incoherent twaddle and will remain so until you  
provide rigorous axioms or definitions for "number", "successor",  
"set", "member", and, especially, "cardinality" that are as rigorous  
as those found in ZFC.  The only reason you are able to keep talking  
is that you refuse actually to cash your claims as mathematics.  Your  
stock and trade is vagueness and ambiguity.  The minute you try to  
turn it into real mathematics your "axiom" will vanish like a puff of  
smoke -- not that I expect you to try.    (04)

> This is another thing that I'll gladly post you. But, as an example,  
> the axiom of infinity. It simply states that the inductive set  
> exists as a completed totality. This gives the first transfinite  
> ordinal omega-0.    (05)

Not by itself; you need several other axioms to prove the existence  
and uniqueness of omega.    (06)

> All the rest of the transf. hierarchy is built on omega-0. Having  
> omega-0 includes the very controversy of having a neverending as a  
> totality.    (07)

There is no controversy among real mathematicians.  The actual  
infinite is at the heart of nearly all contemporary mathematics,  
including in particular the real analysis that underlies physics.  The  
existence of the transfinite is a simple consequence of the axioms of  
ZF set theory.  Although it cannot be proved mathematically, due to  
Gödel's theorem, over a century of rigorous testing and use suggests  
there is every reason to believe these axioms are consistent and only  
cranks like you argue there are "problems" without the least  
mathematical evidence or competence.  If you think there are problems  
with ZF -- despite the fact that you cannot prove its inconsistency --  
then the only rational mathematical response is to provide a  
theoretical alternative so there is actually something to discuss.   
But you obviously lack the ability to do this.    (08)

> I have clearly argued that potential infinity is totally enough for  
> the needs of the man kind.    (09)

You have done no such thing.  Your talk of "potential infinity" is  
useless to science and a distraction to this forum until you actually  
provide a mathematical theory that realizes the idea and demonstrate  
its adequacy for natural science and the theory of computation.    (010)

Chris Menzel    (011)

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