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Re: [ontolog-forum] Finitism vs. Transfinitism

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>, Christopher Menzel <cmenzel@xxxxxxxx>
From: Avril Styrman <Avril.Styrman@xxxxxxxxxxx>
Date: Thu, 6 Mar 2008 11:34:45 +0200
Message-id: <1204796085.47cfbab5edcc8@xxxxxxxxxxxxxxxxxxx>
Lainaus Christopher Menzel <cmenzel@xxxxxxxx>:    (01)

> > "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.
> 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.    (02)

I do not need to define things such as 'number' and 'successor', 
because they are the most self-evident things in the world. 
Instead, I can use them to define other things. For 'set' and 
'member', I can use the same axiom of extensionality that is in 
ZFC -there is no transfinitism in extensionality. For cardinality 
and rank, I can use constructive definitions, such as those in 
Finitist set theory: www.cs.helsinki.fi/u/astyrman/FST.pdf    (03)

> > 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.
> 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.      (04)

The term "the natural numbers" can be used very well without 
having to commit to anything transfinite, by maintaining that 
the series is potentially infinite. It is in the heart of 
mathematics of course. I'm sure that also you understand the 
twist in having a never ending as a totality.    (05)

 * * * * * * * * * * * * * * * * * * * *
If you can show there are numbers bigger than the infinite, your head
whirls. [124] p.16.    (06)

I have always said you can’t speak of all numbers, because there’s no such
thing as ’all numbers’. But that’s only the expression of a feeling.
Strictly, one should say, . . . ”In arithmetic we never are talking about
all numbers, and if someone nevertheless does speak in that way, then he so
to speak invents something - nonsensical - to supplement the arithmetical
facts.” (Anything invented as a supplement to logic must of course be
nonsense). [122] XII.129, [121] XII.448.    (07)

. . . A searchlight sends out light into infinite space and so illuminates
everything in its direction, but you can’t say it illuminates infinity.
[122] XII.142, [121] XII.490.    (08)

The infinite number series is only the infinite possibility of finite series
of numbers. It is senseless to speak of the whole infinite number series, as
if it, too, were an extension. [122] XII.144, [121] XII.504.    (09)

If I were to say ”If we were acquainted with an infinite extension, then it
would be all right to talk of an actual infinite”, that would really be like
saying, ”If there were a sense of abracadabra then it would be all right to
talk about abracadabraic sense perception”. [122] XII.144, [121] XII.511.    (010)

But why is it easier to imagine life without end than an endless series in
space? Somehow, it’s because we simply take the endless life as never
complete, whereas the infinite series in space ought, we feel, already to
exist as a whole. [122] XII.145, [121] XII.515.    (011)

Let’s imagine a man whose life goes back for an infinite time and who says
to us: ‘I’m just writing down the last digit of p, and it’s a 2. Every day
of his life he has written down a digit, without ever having begun; he has
just finished. This seems utter nonsense, and a reductio ad absurdum of the
concept of an infinite totality. [122] XII.145, [121] XII.516.    (012)

. . . what is infinite about endlessness is only the endlessness itself.
[122] XII.145, [121] XII.519.
 * * * * * * * * * * * * * * * * * * * *    (013)

> 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.    (014)

The problems are evident, but you just deny them, similarly as a 
priest in the year 1600 would deny alternative gods. This is very 
natural for a human being: "My language is good, my country 
is good, my theory is good".     (015)

Why do you need an alternative for something that is useless? I'm not 
aiming to give an alternative, but I'm only writing a thesis about 
the problems of transfinity. The problems are not in the coherence
of the axioms, but the problems are in what the axioms say. Similarly,
Alice in the Wonderland is totally coherent, but it is only a story.    (016)

Again, tell me one thing where transfinitism is really used, other than 
in turning contradiction into contraction, proving that Cantor's set 
exists, and so forth. The proofs that require transfinitism are not 
really required in practice. They are not required in space flights, 
cosmology, physics, chemistry, computer science, you name it.     (017)

> > I have clearly argued that potential infinity is totally enough for  
> > the needs of the man kind.
> 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.    (018)

For the theory of computation, Turing machine is an implementation
of potential infinity. The memory tape of a TM is potentially 
infinite. It is not actually infinite/transfinite. The potential 
infinity is totally enough and very useful. Show me one actual 
problem or application where it is not enough, and I'll turn into
a transfinitist.     (019)

Avril    (020)

[124] Cora Diamond (editor): Wittgenstein’s Lectures on the Foundations of
Mathematics, Cambridge, 1939.    (021)

[122] Ludwig Wittgenstein: Philosophical Remarks. Edited by Rhus Rhees and
translated into English by Raymond Hargreaves and Roger White. Basil
Blackwell, 1975. First German edition 1964.    (022)

[121] LudwigWittgenstein: Filosofisia Huomautuksia (Philosophical Remarks).
Translated by Heikki Nyman. Werner Söderström, 1983.    (023)

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