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Re: [ontolog-forum] Axiomatic ontology

To: Pat Hayes <phayes@xxxxxxx>
Cc: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Avril Styrman <Avril.Styrman@xxxxxxxxxxx>
Date: Thu, 21 Feb 2008 20:54:13 +0200
Message-id: <1203620053.47bdc8d5c68a0@xxxxxxxxxxxxxxxx>
Pat and Chris, here's some answers and qestions.    (01)

Quoting Pat Hayes <phayes@xxxxxxx>:    (02)

> >I'm claiming that finitism is better than transfinitsim
> >because it is simpler, uncontroversial, and nothing more is
> >needed.
> 
> 1. Its not simpler. It has to explain a host of 
> new difficulties, not the least of which is how 
> it can account for the hundreds of apparently 
> obvious refutations of it (such as the elementary 
> proof that cannot be a largest integer.)
> 
> 2. It is most certainly not uncontroversial. As 
> this is an elementary empirical fact, I will not 
> provide arguments for it.    (03)

If you are not satisfied with strict finitism, then take 
Aristotle's potential infinity: you can take always more 
and more, as much as is needed. However, because we are 
only weak humans, we can only ever take a finite number
of anything, even though the finite border is vague. 
This is why potential infinity can be reduced into  
finitism: no matter how much we take, we always take 
finitely many.    (04)

> 3. Continuous mathematics is certainly useful, I would claim needed.    (05)

No need for transfinitism to be continuous. Transfinitism is needed
only for being point-continuous. Trying to make the idea of point-
continuum intelligible does require transfinitism. I do not see why,
because mapping the point-continuum into nature only makes movement
impossible. This was Zeno's reasoning, and Aristotle frustrated it
by maintaining that there is no sense in thinking that continuous 
magnitudes consist of points. Again, if there is no point-continuum
in nature, then what is the use to study it in mathematics? In fact, 
at best, the point-continuum only buries the question of physical 
continuity.     (06)

As an example, think of a 1) digital thermostat and 2) an analogical 
thermostat. In digital theromostat, everything is finite. In 
analogical thermostat everything is finite, although we may 
problemize about physical continuity. In both cases, the notion
of point-continuum is totally useless. Then why have it?    (07)

> (By the way, your claim, repeated below, that 
> mathematicians believe in infinity only because 
> we were taught to as babies and are too 
> uncritical to think for ourselves, is rather 
> insulting. You might want to think about other 
> ways to try to convince people.)    (08)

I apologize if I have been insulting, that was not 
the intention. Still, they probably believe in 
transfinitism mainly because it was taught to 
them along with classical analysis, of course,
without analyzing the problems.    (09)

> >Yes, and the conception that line consists of points can be seen
> >as the very source of all transfinitism: transfinitism in needed
> >in order to make the point-continuum intelligible.
> 
> That is not the only reason, although it was 
> historically important. But note, Cantor's 
> demonstration of the uncountability of the reals 
> is all about *numbers*, not about points. It 
> nowhere refers to geometry: it uses only the idea 
> of a decimal expansion.     (010)

Perhaps the Cantor-Dedekind theorem can be seen only 
as making the case clearer, because it is easy to think 
that line consists of points, and to visualize it. In any 
case, the idea that geometrical and topological objects 
consist of points is very strong, but also totally 
unnecessary and useless: they do not need to consists 
of anything.    (011)

> One could reject the 
> set-theoretic account of the continuum, and still 
> Cantor's diagonalization is crying out for a 
> refutation from any finitist. I'd be interested 
> to see a careful refutation of it (or, for that 
> matter, of the ancient and elementary proof that 
> there is no largest number, which I gave in an 
> earlier posting.)    (012)

As you might know, many people think that the diagonal 
argument proves nothing. It is very much about the 
choice between accepting the whole transfinitism or 
refuting it. I'll be happy to post it to you.     (013)

> There is no 
> "finitism-transfinitism debate". Nobody in the 
> philosophy of mathematics takes finitism 
> seriously as a coherent foundational philosophy.    (014)

There is a very strong anti-foundationalist flow
going on, and finitism sides with that. Also
constantly evolving and growing things like 
Martin-L÷f type theory side with finitism. 
In fact, the philosophy of mathematics is full of 
criticism of transfinitism. It is not just
Aristotle and Wittgenstein who are critical
towards transfinitism, but it is generally 
everyone who thinks over it for a while.    (015)


> >Of course, there is no
> >sense of typing Ack(Ack(5,5),5) because you cannot
> >understand its magnity, and you have no you use for it.
> 
> What can I not understand about its magnitude? I 
> know it is larger than Ack(5,5), for example. But 
> in any case, what I can understand about it and 
> whether it is any use (and how do you know what 
> uses I might have for very large numbers?) are 
> irrelevant to the proposition that it EXISTS. It 
> seems obvious to me that it exists, as does its 
> square; indeed, as does the result of raising it 
> to its own power (a much bigger number than any 
> that have been discussed on this thread so far, 
> but still one that has a successor.)    (016)

I can in no way deny that it exists, once you have 
typed it down. It is a good criterion for the 
existence of numbers. But, there are other criteria
too, such as the use of numbers. Numbers such as 
Ack(Ack(5,5),5) might as well be considered solely 
as numerals, or as character strings. Numbers like
1,2,3,4, are useful. For what do you use numbers
such as Ack(Ack(5,5),5)?    (017)

All the numbers that have a potential applicability,
are within the borders of potential infinity, that can
in one sense be reduced into finitism. Why is potential
infinity not enough for you?    (018)

> If I am allowed to write omega-n, then I can 
> refer to infinite ordinals using a finite 
> vocabulary. Are you saying that infinity cannot 
> be referred to? That is obviously false.    (019)

I could argue that you are using the string "omega-n",
and you think that you refer to something infinite
with it, but in reality, because the conception of 
omega-0 as a completed totality is controversial,
you actually only use the word omega-n to denote 
a finite totality. We can think a lot.    (020)

Does not a never ending totality bother you?
How can it be a totality, if it never ends?
How can it be completed, if it never ends?    (021)

> >PH:
> >>  Finitism seems much more intuitive when dealing
> >>  with very large numbers than when dealing with
> >>  very small ones, ie their reciprocals. If all our
> >>  series have to be finite and we cannot talk of
> >>  limits, it becomes impossible to give an adequate
> >>  foundation for calculus, for example. On the
> >>  whole, I think that the mathematicians have done
> >>  a fairly good job and we would all be better off
> >>  leaving it to them, and focusing on matters of
> >>  more direct importance to our engineering.
> >
> >We can very well talk about limits without having to
> >use anything infinite. Take the series 1/2, 1/4, 1/8, ...
> >the limit of the series is obviously 0.
> 
> You just did talk about something infinite, by 
> your casual use of the three-dots notation, and 
> saying 'the series'. That series is an infinite 
> object.    (022)

I interpret the three dots not as "ad infimum", but as 
"ad enough". The interpretation of the three dots is 
also one theme of Wittgenstein's critique of 
transfinitism, and I recommend that you'd reconsider
your attitude towards it. Did you know that e.g.
Alan Turing participated into Wittgensten's lectures
on the foundations of mathematics? His attitute in 
the lectures has some resemblance with your attitude.    (023)


> >You must
> >give a certain level of accuracy that is needed
> >for the problem at hand, like building a house or
> >assigning the accuracy of a float variable on a
> >computer program. If your accuracy is 100 digits,
> >then you only need 100 digits, and the limit of
> >the series is 0 because within the accuracy of 100
> >digits there are only zeros 0.000....0
> 
> That does not establish that the limit is zero. 
> To establish that, you have to also show that if 
> your accuracy is 10|3 digits, then.. and if your 
> accuracy is 10|4 digits, then, ... and so on. And 
> that 'and so on' has no finite stopping place.    (024)

I can very well maintain that the limit is 0 in the 
domain where the needed accuracy is 100 digits. Think
of normal transactions in a shop. If 5 cents would be
the smallest coin, then an item with price 5,551$ would
be rounded up into 5,55$, if you pay in cash. No need 
for transfinity here, and no need in anywhere else 
either. The more accuracy you need, the more you get.
That is in the area of potential infinity.    (025)


> >The good part of finitism here is, that even if one
> >is a Cantorist, his programs use finite approximations
> >in any case. This example also shows that finitism
> >actually drives the thinker into thinking, and not
> >just burying the thinking into 'limits  of infinite
> >serieses'.
> 
> To turn your ad hominism on its head for a 
> moment, I wonder if you really understand the 
> notion of a limit. Since it is easy to show that 
> the notion cannot be defined in a strict finitist 
> framework (or, to be more exact, that any 
> definition of it reduces to triviality there - 
> just use induction to get to the finite end of 
> the finite series, and you will attain any 
> approximable limit), this may not be surprising.    (026)

Transfinitism and Cantor are very close, therefore 
ad hominism. It is only good to give a clear name 
to identify the problem.    (027)


> >CM:
> >>  The "ideology" in question would makes natural science impossible.
> >>  Notably, strict finitism, even coherently argued, does not permit
> >>  enough real analysis (which involves not only the infinite, but the
> >>  uncountably infinite) to do physics.
> >
> >What does it make impossible?
> 
> Real analysis; differential geometry, for example.    (028)

The analysis of real numbers is similarly the story that tells
what they are. Taking a finitist view, we only need finitely 
long rationals. Irrationals may be handled as symbols that 
denote a process that is executed with a needed accuracy:
there is no need to hold that pi is transfinitely long.    (029)

This is by the way another fynnity in transfinitism: the 
series of natural numbers is as long as pi, but pi ends, and 
the series of naturals does not end.    (030)

If all the time that has been consumed in investigating things such as 
infinitesimals and higher transfinity, would have been spent on 
investigating something constructive, then mathematics would probably be 
more sophisticated now, and more in accordance with computing.    (031)



> >CM:
> >>  It will never be, largely because it is viewed almost universally
> >>  among actual mathematicians as, at worst, a delusion, and at best a
> >>  novelty, a curiousity, like developing an internally consistent
> >>  defense of a flat earth.
> >
> >I can't see any sense in that analogy. And I believe that in the end,
> >every scientist is a finitist, simply because simpler is better than
> >complex and useless.
> 
> Funny, those are the very reasons that most 
> scientists and virtually all mathematicians are 
> NOT finitists.    (032)


To repeat, the reason why finitism is simpler is this: transfinitists
say "for all natural numbers", meaning also arbitrary naturals which
are in fact nothing but natural. Finitists say "For all natural numbers",
meaning those naturals that have a possibility of application. By doing
this, the finitists do not have to carry the problems of thinking about
something that never ends as a totality, and they also get rid of 
everything that is over and above the first-order infinity: the 
hierarchy of transfinities.    (033)


> >Fair enough. What about the Burali-Forti paradox. All Cantorist sets
> > exist as 'being', not as generating. So,
> 
> For the record, I utterly fail to follow this 
> piece of your argument. BUt never mind...    (034)

To exists as "being a completed totality" is very different to
existing as "Generating". Generation is a process, while being is 
not a process. For example, the Cantorist definition of rationals 
is the set that contains all  a/b, where a and b are naturals. 
There is no generation there. Everything in transfinitism exists 
solely as 'being'.    (035)


> >there should also exist the greatest ordinal
> >OMEGA, the universal closure of ZFC. But the axioms say there are
> always
> >greater and greater ordinals. This is a case of A and not A.
> 
> No, its a case where one has to admit that naive 
> intuitions break down. That is why this is no 
> longer classified as a paradox. There is no 
> greatest ordinal (in fact, I don't find this 
> surprising at all: why would one think there 
> was??) The set of all sets is not itself a set 
> (or perhaps, does not exist); and many other 
> examples like this are all handled in the same 
> way. So this is not a paradox in the sense that 
> Chris was asking for. Again, you might not like 
> this way of thinking, but it is the way that has 
> been used, and it makes sense and appears to be 
> internally consistent.
> 
> >In order to justify his cause, Cantor turned the paradox into a
> >definition: it is in the essence of the 'truly' infinite to be such
> that
> >it is the greatest and simultaneously not the greatest. A Cantorist
> >can of course invent more and more supplements to logic, such
> >as the 'class' of all ordinals, but this is only an escape route
> >from an evident paradox.
> 
> But one an say that about almost any new 
> mathematical insight. Such rhetoric isn't 
> constructive or useful (and it certainly isnt 
> doing mathematics.)
> 
> >I just wait to see in what exact way you disagree, because it seems as
> >though you have already decided that you will disagree before even
> >reading this. The reason for this is that you believe that 
> >transfinitism is true, and you want to defend it.
> 
> We think it is true because there are so many 
> convincing arguments for it, and it seems 
> obviously, intuitively, true.    (036)

You just cannot overlook the Burali-Forti paradox and say it is 
nonsense. It is about the most fundamental feature of classical
analysis, ZFC incorporated, that the law of the excluded middle 
holds. Every theorem, that has any sense in it, is either true
or false. Every set in ZFC either exists or it does not exists.
The set of the natural numbers exists as being a completed 
totality. So does its power set, and so on. And this is the very
paradox. Everything in ZFC should exists as a completed totality,
but the axioms say that there are always a greater and greater
sets. This is the paradox, and it does not help to call it a
theorem that there is no greatest set/ordinal. It is as if you
had a computer program with a clear misfunctionality, and 
then you would maintain "Hey, it is not a bug, it is a feature".    (037)

Another way to explain it: first, the Cantorists consider 
the set of the natural numbers as a completed totality,
and they build the strange ZFC system to justify that. 
The finitist and Aristotelian argues that anything that 
is neverending, unendlichen, cannot be considered as a 
totality. The Cantorists do not care. But, in the end, the 
Cantorists notice that their own hierarchy cannot be 
completed, and because they hold on to their religion, they
just say "Hey, it is not a bug, it is a feature". In reality,
the natural numbers are not any less infinite than the 
Cantorist hierarchy: the Cantorist hierarchy is just an
unnecessary structure.     (038)

        I would say, ''I wouldn't dream of trying to 
        drive anyone from this paradise.'' I would do 
        something quite different: I would try to show 
        you that it is not a paradise - so that you'll 
        leave of your own accord. I would say, ''You're 
        welcome to this; just look about you.'' 
        (For if one person can see it as a paradise, 
        why should not another see it as a joke?)
                -Ludwig Wittgenstein, Lectures on the 
                Foundations of Mathematics p. 103.    (039)

Chris argued about the same thing here:    (040)

> > What about the Burali-Forti paradox.  All Cantorist  
> > sets exist as 'being', not as generating.
> 
> This is not mathematics.  There is no such thing in mathematics as  
> "Cantorist sets".  Here's what we've got:  For any purported  
> collection of things, ZFC either (i) proves that it exists (as a set),  
> (ii) proves that it doesn't exist, or (iii) does not decide the matter  
> either way.  Those are the only facts pertinent to set existence.   
> Your talk of "existence as being" and "existence as generating" is  
> muddle-headed philosophical codswallop.    (041)

The idea of generation is from Brouwer's intuitionism, which
must be seen as a middle option between finitism and transfinitism.
There, e.g. the real numbers are freely generating. This of course
faces the problem of the speed of the generation, that dives the 
generation into finitism or into transfinitism.    (042)

That the set of natural numbers/the inductive set exist, is 
stated by the axiom of infinity in ZCF. It surely exists. But the 
whole hieararchy does not exist as a totality. And this is the
BF-paradox, which I already explained to Pat above.     (043)

> > So, there should also exist the greatest ordinal OMEGA, the  
> > universal closure of ZFC.
> 
> Ignorant rubbish.  It makes no sense to talk about what *should* exist  
> in ZFC, only what does or doesn't.  It is in fact a *simple* theorem  
> of ZFC that there is no largest ordinal.  Period.  That is, we have  
> case (ii) above.  The Burali-Forti paradox simply never gets off the  
> ground in ZFC.  That was the *point* of *axiomatizing* set theory in  
> the first place.  What possesses you even to talk about ZFC when you  
> clearly have never studied it and don't understand even its most  
> elementary theorems?    (044)

As I already answered Pat, it does not make it any better to say
"Hey, it is not a bug, it is a feature". The law of the excluded 
middle together with the ZFC axioms give birth to the BF paradox.
There is nothing more special in it, than what Cantor said: it 
is in the nature of the truly infinite that it is not exhausted,
and thereby Cantor turned a paradox into a definition. Also this
is not logic, because all that your counter argument proves, is 
that it is totally a subjective option whether the BF is or is 
not a paradox. And this proves also that the whole (subjective) 
framework should be abandoned.     (045)

> > But the axioms say there are always greater and greater ordinals.
> 
> Well, it's a theorem, but yes.
> 
> > This is a case of A and not A.
> 
> It is exactly not such a case.  It is a case of not-A.  Period.    (046)

You give a too friendly interpretaion of ZFC. It is in the nature 
of ZFC that all that exists, exists as a completed totality. Assume
that the hierarchy does not exists as a completed totality:
you must assume that because there is no greatest ordinal. There
is no greatest ordinal, but there simultaneously must always be 
greater and greater ordinals. This immediately brings up two more 
questions.    (047)

1) Can you see the analogy with hierarchy of transfinity and 
the natural numbers?     (048)

        There is no greatest natural number, but the set of 
        natural numbers is still considered as a completed totality.    (049)

        There is no greatest ordinal number, and the collection of 
        ordinals is not considered as a completed totality.    (050)

For a finitist and for an Aristotelian, anything that is never
ending is as never ending as anything else. In contrast, a 
Cantorist aims to 'govern' the never ending with the ZFC system.
They end up wondering about the nature of the hierarchy that 
is created with the system itself (plus with additional axioms).
It appears that the transfinite hierarchy has similar features
to a Cantorist, than the natural numbers have for a Finitist/
Aristotelian. For example, it is uncompleted, and cannot be 
considered as a totality.    (051)

2) if the hierarchy is not complete, and similarly there is no
greatest set, then there must be some sort of a generation
going on. How fast does the hierarchy generate? This is a question you
must answer, if you maintain that the series is not complete.    (052)


> > In order to justify his cause, Cantor turned the paradox into a  
> > definition: it is in the essence of the 'truly' infinite to be such  
> > that it is the greatest and simultaneously not the greatest. A  
> > Cantorist can of course invent more and more supplements to logic,  
> > such as the 'class' of all ordinals, but this is only an escape  
> > route from an evident paradox.
> 
> Utter nonsense.  You know neither the history nor the mathematics.   
> Get a set theory text and study it.  Go get Hallett's _Cantorian Set  
> Theory and Limitation of Size_ and read it instead of inventing  
> fictional histories that suit your dogmas.    (053)

I did not invent this myself. If I recall right, I read it from 
A.W Moore's "The Infinite". Your most powerful weapon is to say
''that is nonsense''.     (054)



> >  I believe that finitism is a better view, but
> >I have also a better justification: there are no abracadabraic things
> in
> >finitism, and it is all that a scientist needs. Your turn to show
> something
> >where finitism is not enough.
>     (055)

PH:
> We already have.    (056)

Show one actual application of physics or of any other branch
of enquery that necessitates transfinitism.    (057)

> >  > >Cantorists only try to escape the evident implications of
> >>  >complete induction. Sure, they don't want to have infitely
> >>  >big naturals, so they just deny them. This is the doctrine
> >>  >of the Cantorists: just deny the
> >>  >implications of your own theory.
> >>
> >>  ? What makes you feel that this is an
> >>  implication? It seems simply obvious to me that
> >>  the set of natural numbers contains only finite
> >>  numbers but is itself infinite. I think this has
> >>  been obvious to me since I was a fairly young
> >>  child. There is nothing paradoxical about it. It
> >>  has nothing to do with naming conventions.
> >
> >This is exactly the case: the Cantorist interpretation
> >of complete induction has been taught to you as a child,
> >and that's why it is hard to get rid of it. It is the
> >same as every child, and most of the adults, holds that
> >"my country is good and my language is good".
> 
> This is starting to get beyond merely insulting, 
> and verging on the insane. I knew that there were 
> infinitely many numbers (though I might not have 
> expressed it that way) a long time before I had 
> heard of Cantor. Im not sure exactly, but I know 
> it was before I was seven years old, because I 
> had an argument with a friend about it at that 
> age. His view was that there was a largest number 
> but that only God knew what it was. I said God 
> could add one to it, and he said that God could, 
> but wouldn't ever actually do it, so it would 
> always be the largest.    (058)

I could also say that it is as insulting from you to
say things that you have said about Wittgenstein, as
it is insulting to a muslim to say to say similar 
things about Mohammed. But, I am not insulted in any 
way, except I think that this discussion is very useful, 
and deserves attention, and is not just some crackpottery.
It was not my intention in any place to be insulting, and
I really appreciate your comments. That feelings are hot,
is only a sign that the topic is hot!    (059)

In any case, you just described the idea of Aristotle's
potential infinity: you can always take one after 
another, as long as you want and as fast as you can.
But since you are only a human, you can only ever take 
finitely many, and therefore the potential infinity
can in this sense be reduced into finitism.     (060)


> >>  Trying
> >>  to clarify the relevant moral concepts to him would simply, in his
> >>  eyes, confirm his belief that it is you who is confused.  I'm afraid
> >>  that those who find the inference from "S is infinite" to "S has an
> >>  infinite member" compelling are just as confused about relatively
> >>  simply mathematical concepts, and that both arguing with them and
> >>  trying to unconfuse them is just as fruitless.
> >
> >You have learned the convention: omega-0 is the order type of an
> >inductive set, such as the set of the natural numbers. Along this
> >convention, you have learned that there are infinitely many
> >naturals, but these are all finite. 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
> >
> >If there is a neverending amount of members, all different, and
> >the next always greater than the first, it implies that there
> >are also infinitely big/long members.
> 
> NO IT DOESN'T !!! Here, YOU explain to US why you 
> believe this, which seems so obviously false. Why 
> can't see that the integers are an obvious 
> counter-example?    (061)

It is not very important to me just how it is: just how the complete
induction should be interpreted, because I argue the there is no
real need to have complete induction at all. My strategy is to
show that the interpretation of complete induction is _subjective_.
Because it is subjective, it cannot have anything to do with 
logic, that should not be subjective at all.      (062)

        Anything invented as a supplement to logic 
        must of course be nonsense 
        -Wittgenstein, Philosophical Remarks XII.129.    (063)

I explained another subjective interpretaion of complete
induction to Chris below.    (064)


> >You can talk of all natural numbers, meaning
> >those that make sense. Just forget the 'unnatural'
> >natural numbers.
> 
> You have to explain what you mean by 'make sense' 
> and 'unnatural'. And you have to do this in 
> mathematical terms.    (065)

By saying that arbitrary numbers are unnatural, I mean 
that they are practically very close to transifinite 
numbers. For example, an arbitrary number cannot be
typed within the known universe with probability 1,
not even by using the Ackermann notation. I said
with probability 1, because when it is thought that 
the arbitrary number is randomly selected from within
0 and omega-0, numbers such as 1 and 2 could as well
be selected, but because we select from within a 
tranfinite totality, the probability that the number 
selected cannot be typed within the known universe 
is 1. You gave another sort of a formulation of 
arbitrary below, and there are problems with it.    (066)


> >>  >It is not just an ideology like some religion. It is  evident
> >>  >that everything paradoxical should be pruned off from logic.
> >
> >PH:
> >>  Of course. But one can work paradox-free without adopting strict
> >>  finitism.
> >
> >And this is the point: being paradox-free. Transfinitism is the very
> >swamp of contradictions, as I have shown above
> 
> It isn't, and you havn't.    (067)

What is your atitude towards Burali-Forti now?    (068)

> >, while finitism manages
> >to do all that needs to be done, without paradoxes.
> 
> Its rife with paradoxes. There's the 
> largest-number paradox. All finite sets of 
> numbers have a largest member. If the set of all 
> numbers is finite, then there is a largest 
> number, call it N.  N+1 is larger than N: 
> paradox. Show me how to get past that one, and 
> I'll give you another.    (069)

Take the Aristotelian view, which can in one sense
be reduced into finitism.    (070)


> >  > >Do you disagree with Wittgenstein?
> >>
> >>  Yes. Almost everything he wrote, in fact.
> >
> >Do you disagree with Aristotle too:
> 
> Yes.    (071)

Come on Pat, you can't be serious. These guys 
are about the most insightful in the history.
You have a sense of humour after all.    (072)

> 
> >  > >If the set has infinitely many members, there should be infinitely
> big
> >>  >memebers too.
> >
> >PH:
> >>  No, that is simply a mistake. There are indeed
> >>  arbitrarily large members, and in fact infinitely
> >>  many of them, but there need not be infinitely
> >>  large members.
> >
> >The conception of arbitrary is severely rotten. It
> >is supposed to be a number randomly selected from within
> >the set of all natural numbers.
> 
> No. Nothing about random. I can rephrase the 
> point without using the a-word: for any given 
> size, there is a number in the set larger than 
> that size. That is all I meant by 'arbitrary'.    (073)

So, once you have specified some size n, there is always 
a larger size. If you can specify n, it is not arbitrary.
If you specify n+1, it is not arbitrary either. If you say
that for all natural numbers x(n > x), then n cannot be 
a natural number...  Do you mean that arbitrary is something 
that we weak humans cannot type down?     (074)

> >We can talk about
> >a vague border for the intelligible natural numbers.
> 
> You can talk about it, but I'd like to know what 
> you mean. I have no idea what a vague border 
> could be. Give us some of this much-vaunted 
> finitist mathematics. What is a vague boundary? 
> What properties does it have, what theorems are 
> true of it?    (075)

You first take the Aristotelian potential infinity.
But, because we humans always can only ever take 
only finitely many things, one after another, the 
potential infinity can in this sense be reduced 
into finitism. How much can we ever take, that is
the vague finite border.     (076)

This holds for it: type a number down in decimal notation.
It is either below the border, or it is the border.
Of course, the border depends on the version of finitism.
Because Ack(Ack(Ack(5,5),5),5) cannot be understood properly,
I do not see why it should be qualified it as a natural 
number. What does 'natural' mean for you?    (077)


> >PH:
> >>  No, its not nonsense. I can imagine any number. I
> >>  claim this is true. Prove me wrong. You have to
> >>  show that a number exists such that I cannot ever
> >  > imagine it. Your move.
> >
> >This only proves that the intepretation of induction is subjective:
> >one says it is nonsense, and the other says that it is nonsense to
> >say that it is nonsense.
> >
> >But ok, if you can imagine it, then explain somehow the meaning of
> >Ackerman(Ackerman(5 5) 5), and how you will use it. What is its
> >application and utility?
> 
> I said nothing about application or utility. I 
> said only that I can imagine it (you havn't 
> refuted this claim) and that it exists (which I 
> understand you to be denying?)    (078)

You can imagine a textual string Ackerman(Ackerman(5 5) 5),
but you can imagine njnba÷bnja÷jd÷bfk as well. So, what is 
the use of postulating Ackerman(Ackerman(5 5) 5)? Application
and utility are central argumants for finitism: postulate only
those numbers that have a possibility of application. You can
also take e.g. all 500 -page books that have all 500 -page
combinations of all characters that are known. Also your 
Ackerman(Ackerman(5 5) 5) is there. All that we can imagine
is there, and it is still finite.    (079)

> >That part of mathematics that the human kind actually
> >needs e.g. in space flights, hospitals, nuclear plants,
> >computers, experimental fission facilities, AI, does
> >certainly not need transfinitism.
> 
> They most certainly need real analysis and 
> differential geometry. If you can show how to 
> re-create these fields (and, say, topology, 
> catastrophe theory, etc.) within a strictly 
> finitist philosophy of mathematics, you will at 
> the very least be able to publish many papers in 
> journals devoted to the philosophy of mathematics.    (080)

As said before, finitism gives a different answer to
''what are real numbers?'' than Dedekind-cuts. Also,
if differential geometry is actually used in some 
real-life application, nothing intinite is actually 
used. Infinity might be used in thinking about 
differential geometry, but similarly as in differential
and integral calculus, noting infinite is actually 
needed: we get along with finite approximations. There
is no need for infinitesimals. In fact, the 
point-continuum leads to having infnitesimal, even though
Cantor explicitly called infinitesimals the cholera 
bacillus of mathematics.    (081)


> >  > the question is, just how big a number can we understand?
> >>
> >>  (1) What exactly do you mean by 'understand'? And
> >>  (2) why does this matter? We can certainly refer
> >>  to and reason about and prove properties of very,
> >>  very large numbers: much larger than could
> >>  possibly be physically represented as a numeral.
> >
> >What use can there be for a number that cannot even
> >be typed down? If there is no use, why postulate it?
> 
> I DID write it down. And the 'use' is that its a 
> lot easier to allow it to exist than to deny its 
> existence. It doesn't need to be 'postulated': 
> its existence follows from the ordinary 
> assumptions of arithmetic. Denying its existence 
> is what would take a great deal of explaining. 
> See below.    (082)

So, can you by now settle with the potential infinity?    (083)

> >We can define a vague border on how big a number we
> >can ever apply in practice. We can think that our proofs
> >are limited to cover only that range. By doing this, we can
> >give up the unnatural natural numbers, and simultaneously
> >we give up also the whole hierarchy of transfinities.
> 
> Consider N!, where N is an integer. Clearly N! is 
> greater than N, for any N. Moreover, as N 
> increases, N! increases much faster, so that the 
> amount by which N! is greater than N increases as 
> N increases. This means that if there is an upper 
> limit to the set of numbers, even if it is 
> 'vague', that N! crosses that upper boundary a 
> lot sooner than N does. So there must be a lot of 
> numbers between the largest N such that N! 
> vaguely exists and the largest N, all of which 
> exist but their factorials don't. How can that 
> happen? Now re-run this argument for any other 
> rapidly-increasing function of N, say 2|N or N|N, 
> or even Ack(N,N). What happens to these functions 
> past the point where their value would be too 
> large to be a 'real' integer, yet their argument 
> is still quite small? Do they somehow "stop 
> working"? But they are all defined by finitely 
> describable, even quite compact, algorithms which 
> perform ordinary arithmetic operations, so what 
> is there to go wrong? Or do they actually produce 
> results, but these very, very big results are not 
> counted as "numbers"? Why not? But more to the 
> point, whatever you call them, if they exist then 
> classical infinitary mathematics seems to apply 
> to them. Or... what?? You have to say SOMETHING 
> about arguments like this, other than that you 
> don't like their conclusions.    (084)

So, you consider above the process of generation,
like, having 3, we have also 3!=6, and having 6,
we have 6!, and so on. And this is nothing more
than 1,2,3,... The question is, that do you consider
the series as being 1) a completed totality, or do you
consider it as 2) generating. 1) has the problems of
the subjective complete induction, and 2) begs the 
question of how fast does it generate? If it generates
in a finite speed, we are in the area of finitism.
If the speed is transfinite, we are again in the 
area of 1).    (085)


Quoting Christopher Menzel <cmenzel@xxxxxxxx>:    (086)

> Well, I said there would be no more from me in this thread, but I feel  
> it necessary to correct and counter the more egregious errors and  
> confusions here, on the off chance that someone might be misled by them.    (087)

Yes, and I'm doing the same thing, just in order to show that 
a person cannot have any real reason for being a transfinitist.    (088)

> > Pat and Chris are claiming that finitism is somehow not enough, even  
> > though they cannot show how, and hold on to transfinitism because  
> > they have been taught into it.
> 
> Another sure sign of crackpottery.  Instead of providing coherent  
> arguments, a crackpot simply suggests that his opponents are under  
> some sort of spell -- their early education has got such a hold on  
> them that they are unable to break free.  It never seems to occur to  
> them that their opponents actually understand them and simply find  
> their arguments unsound -- because, of course, if they really *did*  
> understand, they too would see the light.  So obviously their early  
> training has led to an ossified worldview from which they are  
> incapable of escaping.    (089)

I have tried to answer in the best way I can, and I can't see any other 
reason for your transfinitism than education. And you have given no
other reason.    (090)

> > You can make the distinction to countable and uncountable infinity  
> > only after you have entered into the transfitist realm. If you have  
> > done that, you implicitly accept transfinitism. If you have not  
> > entered, it makes no sense to talk about any innumerable infinities.  
> > Transfinitism, like Islam and Christianity, is the sort of a thing  
> > that requires belief into it. Without the belief, it has no meaning.
> 
> Further crackpottery -- you are ignorant of the actual mathematics so  
> it appears to you to be no different from religious belief.  The fact,  
> of course, is that the mathematician starts with axioms and  
> definitions and derives theorems.  Belief, in anything like the  
> religious sense, has nothing to do with it.  The concept of the  
> infinite is itself rigorously defined and the existence of infinite  
> sets follows from the axioms of set theory.  That's it.  No creeds, no  
> faith, just axioms, definitions, and theorems.    (091)

Just axioms and that's it? You of course believe in the axioms of ZFC.
If you believe in the axioms of ZFC, why would you not believe as well
in the axioms of Alice in the Wonderland?     (092)

        For if one person can see it as a paradise, 
        why should not another see it as a joke
        Ludwig Wittgenstein, Lectures on the 
        Foundations of Mathematics  p. 103.    (093)

Unlike Wittgenstein, I don't see the ZFC axioms as a joke,
but I see them as a measure of what is required in order to
make discrete into continuous, as a measure of how crazy
constructions do we need in order to make the point 
continuum 'intelligible'. The axioms make transfinity exist
by stating "Hey, it exists". They state that the power set
of every set exists, but don't give any method of creating it.
They say that every set has a well-ordering, but give no way 
to do the well-ordering. And a lot more. And all this is not 
needed for good science, and because it is not needed, the 
finitist-Aristotelian view which does not suffer from these
undefined horrors, is better: it can manage to do all that 
needs to be done without any controversies.    (094)

> > Of course, L÷wenheim-Skolem theorem states that every thing that one  
> > can 'do' with an innumberable model can be done with a numerable  
> > model too.
> 
> It says no such thing.  There is in particular no concept of "doing"  
> in the theorem.  Moreover, the theorem only makes *sense* if you allow  
> the existence of transfinite sets, so you certainly have no right to  
> appeal to it in defense of your own position.    (095)

I recall that LwS says that if there is an innumerable model for some 
theory, then there is a numerable model too for that theory. But also,
that if there is a numerable model, then there is an innumerable model
too. This tells me that the difference of numerable and innumnerable 
models is very subjective. I can talk in the language of transfinitism,
even though I claim that it is not the most economical option.    (096)

> > I just wait to see in what exact way you disagree, because it seems  
> > as though you have already decided that you will disagree before  
> > even reading this. The reason for this is that you believe that  
> > transfinitism is true, and you want to defend it.
> 
> Actually, I'm rather agnostic about its truth in any ultimate sense.   
> I am more of a pragmatist about these things, and I'm quite certain  
> Pat is as well.  Properly axiomatized, set theory with its infinite  
> sets has proved incredibly fruitful and entirely unproblematic.  And  
> because the existence of infinite sets is at the root of most of  
> modern mathematics we are well-warranted in accepting the axioms of  
> set theory unless they prove contradictory, just as we are well- 
> warranted in accepting quantum mechanics, bizarre as it is.  They are  
> both examples of our best science, and believing the deliverances of  
> our best scientific theories is the very height of rationality.    (097)

Show one actual engineering problem that requires infinity in some way,
and I'll be satisfied. The idea of axiomatic systems is of course good,
but let's not let the axioms be undefined and unnecessary statements.     (098)

> >>> Having a set {1,2,3, ..., n}, its order type is n
> > CM:
> >> That is an unordered set.  It has no order type.
> >
> > Of course a set such as {1,2,3} has an order type: 3,and is also  
> > ordered!
> 
> A staggering display of ignorance of even the most elementary concepts  
> of set theory.    (099)

It is all and only pettifoggery to say that {1,2,3} is not well-ordered,
and that you must use <1,2,3> to denote it is well-ordered. Why not
use the Wiener-Kuratowski method of making it well-ordered without
having to use <>? After all, if {1,2,3} exists, then also its well-
ordered (by WK) version exists. You could have been kind and uderstood
that I meant that WK-set ;)    (0100)

> > This means that you don't even need the axiom of choice to well- 
> > order it :)
> 
> Yes, you don't need choice to *impose* a well-ordering upon it because  
> it is *finite*, not because it is intrinsically ordered.
> 
> > You have learned the convention: omega-0 is the order type of an  
> > inductive set, such as the set of the natural numbers.  Along this  
> > convention, you have learned that there are infinitely many  
> > naturals, but these are all finite.
> 
> Sorry, no, this is not a convention, it is a *provable* fact of set  
> theory.    (0101)

Provable, similarly as "God exists" is provable if you first accept 
that "God is that then which nothing greater can be conceived" as an
axiom, and also believe in God yourself.     (0102)

> > 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.      (0103)

Yes, I don't accept it as something intelligible, but I can understand
that once you throw away common sense, many things become possible.    (0104)

> So here's what you can  
> do.  First, show *exactly* which axiom in the ZFC proof of 3 you think  
> is false instead of blowing philosophical smoke; explain why it  
> shouldn't be accepted.  (Of course, you'll actually have to study ZFC  
> to do this.)      (0105)

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.    (0106)

> Second, provide us with your OWN axioms and your OWN  
> definitions of "cardinality" and "infinite" so that everyone can  
> clearly see what you are talking about.  Then *prove* proposition 3 in  
> your theory formally instead of just asserting it.  Until you can do  
> that, your claims have as much semantic content as a dog's bark.    (0107)

This I can do. My axiom is this:    (0108)

"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"     (0109)

This axiom totally as objective as the axioms of ZFC.    (0110)


> > If there is a neverending amount of members, all different, and the  
> > next always greater than the first, it implies that there are also  
> > infinitely big/long members.
> 
> Not in ZFC it doesn't.  Only in your fairy tales.  Put up or clam up.   
> Do some mathematics or quit wasting everyone's time.    (0111)

Many call ZFC a fairy tale, and I think that it is a great advantage
to get rid of it, and turn towards constructivism: it exists if you
can build it step by step. This is also very much like computer 
programming, which has more to do with CS ontology than ZFC.    (0112)

> > This interpretation of complete induction is totally as objective as  
> > that which you are holding. Yet another interpretation is that  
> > induction simply stops after a vague finite limit.
> 
> The mathematics of induction in ZFC is completely rigorous and precise  
> -- though admittedly you do have to study it to realize that.  And  
> since that's the only mathematics in the neighborhood, your assertions  
> to the contrary are completely empty.    (0113)

As empty, as are the arguments of a muslim to an orthodox. There is nothing
more especial in this debate: humans argue with one another claiming that 
that in which they believe in is better than another thing.    (0114)

> > And this is the point: being paradox-free. Transfinitism is the very  
> > swamp of contradictions, as I have shown above,
> 
> You have shown no such thing.  As I said before, the ONLY way to show  
> that ZFC is inconsistent is to derive an explicit contradiction from  
> the axioms.  You got your "contradiction" above by noting (correctly)  
> that ZFC proves not-A (there is no largest ordinal) and then by simply  
> *asserting* that it "should" prove A.  What a joke.  It would be funny  
> if there were no danger of anyone taking you seriously.    (0115)

I asked the two questions above, that you have to answer if you do not
think that BF is a paradox.    (0116)

> > Do you disagree with Aristotle too:
> >
> > Our account does not rob mathematicians of their science, by  
> > disproving the actual existence of the infinite in the direction of  
> > increase, in the sense of the untraversable. In point of fact they  
> > do not need the infinite and do not use it.
> 
> Uh, Aristotle was at a bit of a disadvantage, arriving on the scene  
> about 2500 years too soon.    (0117)

Many of Aristotle's ideas are remarkable and still applicable today, such 
as his potential infinity. We should not throw his good ones away, even 
he was not perfect. He e.g. though that 1) the rest of the universe 
circulates Earth, and 2) that speed and time cannot be placed into a ratio.
1) and 2) ought to be thrown away, and have been thrown away, but the 
child flew out with the washing water.    (0118)

> > The conception of arbitrary is severely rotten. It is supposed to be  
> > a number randomly selected from within the set of all natural  
> > numbers. First of all, the selection process itself is impossible.  
> > How would you do it? Arbitrary, like the whole transfinitisim, is  
> > given an existence simply by saying ''hey, it exists''.
> 
> Rubbish.  To say a set of numbers contains arbitrarily large members  
> is completely clear and precise.  It means simply that there is no  
> largest member; for any natural number in the set, there is a larger  
> one in the set.  That's it.  It has nothing to do with "random  
> selection" or any sort of "selection process".    (0119)

There is no largest member? Isn't that the same as the definition
of an inductive set? And that is nothing else that the definition
of complete induction. So you say, if there is an inductive set,
such as {1,2,3,...}, then there are arbitrary numbers in the set.     (0120)

How do you select an arbitrary number then? I see here the burying of
problems into the term arbitrary. I suppose that you are satisfied 
with "The arbitrary number just are there because I believe so".    (0121)

What do you btw need the arbitrary numbers for?    (0122)

> Here are the simple facts.  You obviously lack even an elementary  
> understanding of set theory, so you don't even know what you are  
> talking about when you argue against it.  You should remedy this if  
> you wish anyone to take you seriously.      (0123)

I understand it pretty well, certainly well enough to see why it should 
be abandoned.    (0124)

> Second, ZFC is a powerful and  
> time-tested theory in which the usual derivations of the paradoxes of  
> naive set theory are blocked and in which no contradictions have been  
> found despite 100 years of rigorous scrutiny and extensive use by  
> thousands of mathematicians.      (0125)

If you first believe in it, then it is easy to turn the controversy 
into something else. If the collection of the axioms work together
without being in clear contradiction with each other, it does not 
make it a desired theory. In a desired theory, also what the axioms
say should be clear, instead of them saying "You can do whatever you can
imagine".    (0126)

> Third, you yourself have offered no  
> mathematics, no precision, no rigor, just one vague, ungrounded and  
> dogmatic assertion after another, argued at best from unstated  
> premises and undefined terms.      (0127)

I have clearly argued that potential infinity is totally enough
for the needs of the man kind, and that it can eventually be 
reduced into finitism. You have not offered an reason for me 
to be a transfinitist. The reason for this is that there is 
no reason to be a transfinitist. Transfinitism is invented as a
subjective supplement to logic, which makes it very much 
non-logical.    (0128)

> Either find a genuine contradiction in  
> set theory and report it here (before picking up your Fields medal) or  
> formulate a mathematical theory of your own in first-order logic with  
> clear axioms and definitions so that your ill-defined assertions might  
> be cashed in clear mathematical terms.  Then provide valid proofs in  
> your system of such propositions as 3 above.  Anything less is just  
> hot air.    (0129)

I wait for your answers, since I have asked you how you feel e.g. about the 
speed of generation -question.    (0130)

> That is definitely it for me on this topic.    (0131)

Thanks Chris and Pat. Now I know a little more about how 
a transfinitist thinks, but there are still many open
questions here, so please continue.    (0132)

Avril    (0133)

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