Hi Avril (01)
You used the phrase "potentially infinite" several times in your
discussion below. I'm not sure I understand. Could you say more about
what you mean by "potentially" so I can make sense of what a finitist
could mean by "potentially infinite"? (02)
Cheers! (03)
Bill Andersen
Ontology Works, Inc.
3600 O'Donnell Street, Suite 600
Baltimore, MD 21224
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On Mar 6, 2008, at 4:34 AM, Avril Styrman <Avril.Styrman@xxxxxxxxxxx>
wrote: (05)
> Lainaus Christopher Menzel <cmenzel@xxxxxxxx>:
>
>>> "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.
>
> I do not need to define things such as 'number' and 'successor',
> because they are the most selfevident 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
>
>
>>> All the rest of the transf. hierarchy is built on omega0. Having
>>> omega0 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.
>
> 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.
>
> * * * * * * * * * * * * * * * * * * * *
> If you can show there are numbers bigger than the infinite, your head
> whirls. [124] p.16.
>
> 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.
>
> . . . A searchlight sends out light into infinite space and so
> illuminates
> everything in its direction, but you can’t say it illuminates infini
> ty.
> [122] XII.142, [121] XII.490.
>
> 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.
>
> 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 r
> ight to
> talk about abracadabraic sense perception”. [122] XII.144, [121] XII
> .511.
>
> 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 nev
> er
> complete, whereas the infinite series in space ought, we feel,
> already to
> exist as a whole. [122] XII.145, [121] XII.515.
>
> Let’s imagine a man whose life goes back for an infinite time and wh
> o 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.
>
> . . . what is infinite about endlessness is only the endlessness
> itself.
> [122] XII.145, [121] XII.519.
> * * * * * * * * * * * * * * * * * * * *
>
>> 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.
>
> 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".
>
> 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.
>
> 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.
>
>
>>> 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.
>
> 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.
>
>
> Avril
>
>
>
>
>
> [124] Cora Diamond (editor): Wittgenstein’s Lectures on the Foundati
> ons of
> Mathematics, Cambridge, 1939.
>
> [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.
>
> [121] LudwigWittgenstein: Filosofisia Huomautuksia (Philosophical
> Remarks).
> Translated by Heikki Nyman. Werner Söderström, 1983.
>
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