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[ontolog-forum] Finitism vs. Transfinitism

To: Pat Hayes <phayes@xxxxxxx>, "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Avril Styrman <Avril.Styrman@xxxxxxxxxxx>
Date: Tue, 4 Mar 2008 15:21:49 +0200
Message-id: <1204636909.47cd4ced2580f@xxxxxxxxxxxxxxxxxxx>
Pat,    (01)

sorry for the delayed answer, 'been busy.     (02)

Lainaus Pat Hayes <phayes@xxxxxxx>:    (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.
> Of course we do, being (as you say) finite beings 
> ourselves. But our finiteness does not prevent us 
> thinking about things larger than ourselves, so 
> please continue your thought, above. We always 
> take finitely many... of what? Or if you prefer 
> the active form, we always take finitely many 
> from what? What is this bucket, or supply, that 
> we take these integers from? Is this not an 
> infinite set? (Or do you also not believe in 
> sets?)    (04)

> >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.
> I understand all that, of course. But now show me 
> how to actually provide a foundation for all this 
> talk of 'potential'. One gets similar 
> difficulties when trying to formalize vague 
> predicates such as 'large' or 'fairly small'. 
> These ideas have never been satisfactorily, 
> rigorously, accounted for. If you feel you know 
> how to do it, by all means explain it to us, I 
> will be all ears.    (05)

Aristotle's potential infinity:    (06)

        For generally the infinite has this mode of existence: 
        one thing is always being taken after another, and 
        each thing that is taken is always finite, but always 
        different. Physics, book 3, chapter 6.    (07)

So, starting from 0, we can always take more and more.
There is the act of taking, that can be expressed in the 
form of the successor function: succ(0)=1, succ(1)=2,
and so on. But the "and so on" means, "as many as you
will want to take, or can take". There is a finite path to 
Ack(Ack(5,5),5), and to every finite number that can be 
typed down.     (08)

So, think of the series 1,2,3,... as potentially infinite.
It is, of course, not considered as a never ending totality. 
It goes on as far as is needed. Because we can only take 
finitely many, the potential infinity can be in this sense
be reduced to a vague finite border.     (09)

Turing machine is very strongly an implementation of
the idea of potential infinite. It's memory tape is 
potentially infinite.    (010)

> >To exists as "being a completed totality" is very different to
> >existing as "Generating".
> No, to exist is to exist. There aren't different 
> 'kinds' of existence. And this has nothing to do 
> with finite or infinite: its simply what one 
> means by 'exists'. I have no idea what 'to exist 
> as generating' can mean (except maybe that a 
> generating process exists, and can always be run 
> to a next stage: but even to say that last bit 
> requires me to consider infinite models of my 
> axioms...)    (011)

For 1,2,3,... to exist as generating, means that 
first 1 exists, then succ(1)=2 exists, and so on.
But, the existence of 3 requires that act of 
committing succ(2)=3. When you talk about 
Ack(Ack(5,5),5), it can be reached by committing
the succ function enough many times. But, it is 
possible to commit it only finitely many times.     (012)

> >>  3. Continuous mathematics is certainly useful, I would claim needed.
> >
> >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.
> But even if one denies that the continuum is 
> constructed from points - a topic which gave rise 
> to heated debate among mathematicians as little 
> as 150 years ago - it is hard to see how one can 
> rationally deny that it is possible to make 
> indefinitely fine discriminations of location on 
> the continuum. So if a continuum exists at all, 
> then a point continuum exists alongside it, as it 
> were.    (013)

Naturally, if we are talking about a purely mathematical
continuum, then we cannot investigate any "one and only"
continuum, because our investigation also defines
what it is. So, if we accept that some purely 
mathematical continuum exists, we cannot deny
that some person thinks that it consists of points.
We cannot deny that somebody else thinks that it 
consists of segments either. I think that it is best
to consider that it consists of nothing, but if it 
has to consists of something, then segments rather 
than points.    (014)

However, if we want to model the physical continuum,
there is not much sense to talk about magnitudes
smaller than the Planck length, or if there is, there
is always enough accuracy in the potential infinity.    (015)

> >  I do not see why,
> >because mapping the point-continuum into nature only makes movement
> >impossible. This was Zeno's reasoning
> Zeno's reasoning was simply faulty, and 
> Aristotle's response to it was inadequate.    (016)

In what sense do you mean inadequate?    (017)

> Aristotle was very good for his time, but that 
> was a very long time ago.     (018)

Some of his stuff was good only for his time, and 
perhas not even for his time: today we know that the 
Universe does not circulate Earth; we must be able 
to say 100km/h, and a lot of other little things, 
but things like the potential infinity are still 
perfectly applicable, similarly as a2+b2=c2 is 
still applicable.    (019)

> >>  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.)
> >
> >As you might know, many people think that the diagonal
> >argument proves nothing.
> I have never seen a remotely convincing 
> counter-argument to it, however. I'd be delighted 
> to see yours, if you have one.    (020)

I'll be very happy to send it to you after some editing.
There's no sense in retyping all formalism here.    (021)

> >  > >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.)
> >
> >I can in no way deny that it exists, once you have
> >typed it down.
> But you DID deny it exists, in an earlier post. 
> What then is your actual position? If there is an 
> upper bound, please tell us roughly where it is, 
> and I will undertake to immediately reply by 
> naming a number which is much larger than it.
> >It is a good criterion for the
> >existence of numbers. But, there are other criteria
> >too, such as the use of numbers.
> But we are here only discussing existence.
> >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)?
> I enjoy contemplating very large numbers. What I 
> do in private is none of your business. The point 
> at issue is, do they exist? I take it that you 
> have been denying that they do.    (022)

That they exist can be judged based on the criteria 
that we give for existence. What is required for something to 
be a number, and what does it mean that a number
exists? A number exists if     (023)

a) we can write the number down in some numeral notation    (024)

b) we can understand the magnity of the number    (025)

c) there is some sort of actual correspondence with the 
number and reality, such as 10 has correspondence because 
there are more myriads of collections of 10 things.     (026)

d) we have some other use for the number.     (027)

If none of these criteria is filled by a proponent x, then it 
is hard to imagine why somebody wants to call x a number.    (028)

Also other criteria can be given, but it is clear that 
numbers such as 7 and 8 pass all citeria clearly,
when transfinite numbers have difficulties in this.    (029)

All the numbers that have a potential applicability,
are within the borders of potential infinity, and 
potential infinity can in one sense be reduced 
into finitism.    (030)

> >Does not a never ending totality bother you?
> No. It seems kind of obvious. Imagine a 
> perspective view of parallel lines. Its even in 
> the Lord's prayer: "...for ever and ever, amen."
> >How can it be a totality, if it never ends?
> >How can it be completed, if it never ends?
> What exactly do you mean by 'completed'? I am of 
> course a finite being, but I can think of an 
> infinity as a whole.    (031)

It facilitates the use of language to say "The 
set of natural numbers". This can be used as a 
tag that is communicated between mathematicians,
when they quantify "for all naturals". But, with
a closer look, problems start appearing.    (032)

The problems are due to the fact that a never
ending should be considered as a totality. How
can each one be inspected, when the task has
no end?     (033)

        There’s no such thing as ’all numbers’, simply 
        because there are infinitely many. And because it 
        isn’t a question here of the amorphous ’all’, such 
        as occurs in ”All the apples are ripe”, where the 
        set is given by an external description: it’s a question 
        of a collection of structures, which must be given 
        precisely as such. [120] XII.126    (034)

        . . . It’s, so to speak, no business of logic how many 
        apples there are when we talk of the apples. Whereas 
        it’s different in the case of the numbers: there it 
        has an individual responsibility for each one of them. 
        [120] XII.126    (035)

> >  > >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.
> >
> >I interpret the three dots not as "ad infimum", but as
> >"ad enough".
> You referred to "the series" - your words, above. 
> Now, what was IT that you were referring to, that 
> thing called "the series"? Your answer must be 
> ONE thing.    (036)

With 1,2,3,... I denote the potentially infinite series,
that is always a finite one.    (037)

> >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.
> Independently from this conversation, I have a 
> very low opinion of most of Wittgenstein's 
> philosophy. I hesitate even to call it 
> philosophy: it smacks more of a collection of 
> jottings. The much-vaunted 'picture theory of 
> meaning' in the Tractatus is clearly absurd. I 
> rank Wittgenstein somewhere near the writings of 
> Lawrence of Arabia. As for mathematics, there 
> isn't the slightest evidence that Wittgenstein 
> understood anything about it at all.    (038)

I agree that reading e.g. Tractatus is simply
horrible. He could as well have spelled it out
unmystically. The only part of his stuff that 
I find compelling is the very critique of 
transfinitism, and a couple of other ideas. 
But, since I find that compelling, why would not 
some other person find some other part of his 
work complelling? Anyhow, if it was a know  
strategy for him to appear mystical, it has 
worked perfectly for him since he is so much
read.    (039)

I added an two excerpts in the end of this mail 
that can facilitate understanding his attitude.      (040)

> >I can very well maintain that the limit is 0 in the
> >domain where the needed accuracy is 100 digits.
> You did not previously refer to limits in a 
> domain, you simply said the limit of the series. 
> Is this notion well-defined, in your view, or 
> must it depend upon the 'domain'? If so, we need 
> a mathematical theory of these domains.    (041)

If we talk about finite model theory, we get driven 
into the same questions again: what is the biggest
finite model, or how many individuals are in the 
biggest finite model? You can maintain that there 
are as big finite models as there are natural numbers. 
I can appeal again to the potential infinity: there are 
as big finite models as we define. The idea of the
Finitist set theory is that you first give the number of
ur-elements, and the maximum rank, and generate
all sets with these restrictions. For example, if there
are two sets and the max rank is 1, there are sets
{a}, {b}, {a,b}.    (042)

The same question is also that how many finite 
model theories we can axiomatize.    (043)

> >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
> How then do you distinguish between rationals and 
> irrationals? Or would you prefer that we had no 
> such distinction?    (044)

I could use the term 'irrational', but only to denote the 
process of generating finite rationals that are called e.g. 
pi and the square root of 2.    (045)

> >You just cannot overlook the Burali-Forti paradox and say it is
> >nonsense.
> I don't say that; but I do say that it is not a 
> paradox. It is one of a number of illustrations 
> of the fact, now well-understood, that 
> mathematical formalisms can only capture 
> mathematical intuitions expressed within the 
> system, rather than outside the system. Thus the 
> central question for a foundational theory (such 
> as ZFC) is, can all of mathematics be conducted 
> within this system? Which is a pragmatic question 
> as much as a philosophical one.
> >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.
> OK so far.
> >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.
> OK...
> >This is the paradox
> ?? ...what paradox? Why did you think there would 
> be a largest ordinal? You of all people should be 
> content with this result, I would have thought. 
> ZFC proves that there is no largest ordinal. This 
> might be thought surprising, but it is not 
> paradoxical. It might be viewed as a critique of 
> ZFC, indeed it was so viewed at one time; but it 
> is still not paradoxical.
> >, and it does not help to call it a
> >theorem that there is no greatest set/ordinal.
> Its not a question of what you call it: it IS a 
> theorem of ZFC. That is simply a fact.
>     (046)

If the hierarchy exists a being a completed 
totality, then it has to have a size. However,
the size cannot be stated with the formalism
of the system itself.    (047)

> >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.
> >
> >1) Can you see the analogy with hierarchy of transfinity and
> >the natural numbers?
> Yes, of course. But it is a superficial analogy, 
> and mathematical foundations are not made of 
> analogies.    (048)

I can see similar analogies all over ZFC. E.g. the cardinality 
of a set {a,b} is 2, because it has as many members as the set 
2. IF, we commit to the idea that there is some sort of a 
hierarchy of transfinities, then I find it a very important 
question that how many levels of transfinity there are. Of 
course, there are only finitely many different types of higher 
transfinity, as explained in [3].    (049)

In [3], on the top of the hierarchy, and I'm talking 
about the very top, there is "1=0". That is, a 
contradiction. So, it can be considered that 1=0 is 
the final limit of all transfinity, and it cannot be 
reached, because contradiction cannot be reached.    (050)

However, in the case of normal limit processes, like 
0.1, 0.01, 0.001, .... where the limit is 0, the general 
(I think wrong) idea is that the limit is reached. To 
'reach' a never ending is, to me, as contradictory as 
1=0. Of course, it is only the idea that the series 
0.1, 0.01, 0.001, ... gets so 'close' to 0, that it is 
natural to think that it also reaches 0, whereas it is 
harder to think that the limit 1=0 is somehow reached.     (051)

One might as well maintain that the universal closure of ZFC
with the additional axioms, as a class, can be identified 
with the ordinal OMEGA, and then continue to OMEGA+1,
OMEGA+2, and so on. And then, one could introduce more 
and more types meta-collections, until one runs out of 
symbols. Then the question is, at what point did the 
hierarchy loose all meaning? And this is again a 
question of opinion.     (052)

Anyhow, I understand that the proponents of the higher 
transfinite hierarchy claim to know, up to some degree, 
what there can be between 0 and 1=0. And all that 
exists there exists as a completed totality. Also the class 
OMEGA exists as a completed totality. And this is where I still
see the BF paradox. Take omega-0. It contains all natural
numbers. Omega-0 is the very first transfinity, and can be 
seen as the _reached_ limit of the series 1,2,3,... Similarly,
OMEGA can be seen as the reached limit of all ordinals.
If it is reached, does it not mean that 1=0 is reached?    (053)

But, assume that some utility had got out from the reserch
of transfinity, some utility that can be used in some 
engineering application, similarly as there is use for 
imaginary numbers even though they sound suspicious. In this 
case, the same utility could have be reached also by 
maintaining that there was nothing infinite in the first 
place. The "higher infinite" has a hype-value, which makes
it interesting to many, similarly as agent technology had
a hype value.    (054)

> >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.
> Why? All I need say is that the elements all 
> exist but their totality does not (or if it does, 
> is not of the same kind that they are, eg is a 
> 'proper class'). Things do not "go on" in 
> mathematics. There need be no 'generator' : 
> indeed, the very notion hardly seems meaningful 
> in the very-large-ordinal domain.    (055)

So, the elements do exist as a totality, yes, but 
there is no highest ordinal. Exactly the same can be 
said about the natural numbers. A clear analogy,
that only proves that a system that has been created 
to govern infinity, faces itself the impossibility 
of having a never-ending as a totality.    (056)

The notion of class is introduced here as a medium to 
get rid of the controversy. Just build a bigger junk 
yard where you put the previous one.    (057)

> But this talk of 'speed 
> of generation' is certainly not from Brouwer, and 
> to me only emphasizes the coddish nature of the 
> wallop. Even if it made sense, what does it 
> matter how 'fast' the integers 'grow'?    (058)

The idea of the generation does not, happily, 
depend on intuitionism.    (059)

I read a good article of intuitionism [2], that clarifies 
the situation. The generation in intuitionism has 
nothing to do with the speed of generation. It is only 
about the laws that guide the generation of e.g. the 
points in the interval [0 1], such as the law that
"divide it continuously with 2" gives the point 0. 
However:    (060)

"...Brouwer allows the species of all (constructible)
infinite sets of natural numbers and accepts Cantor's 
proof that this set cannot stand in one-one
correspondence with the natural numbers, But he will
deny that we have thus created a new cardinal number,
greater than aleph-0."    (061)

This is just another interpretation of complete induction.
Another is Peirce's idea is that there are as many points 
in the interval [0 1] is is the cardinality of the greatest 
conceivable transfinity. I spelled this only to show the
subjective nature of complete induction.    (062)

> >You give a too friendly interpretaion of ZFC. It is in the nature
> >of ZFC that all that exists, exists as a completed totality.
> You are trying to force a philosophical view onto 
> a formal theory, and complaining that it does not 
> fit. Chris is saying that if it doesn't fit, 
> forget the philosophy and stick to the formalism. 
> That after all is what the formalism was designed 
> for in the first place.    (063)

I'm trying to sort out just which parts of the theory 
are above logic.    (064)

> >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.
> As I have long suspected, having attended several 
> of their meetings, it seems that Wittgensteinian 
> scholarship is a form of religious fundamentalism.    (065)

Also Islam has a resemblance with mathematics: Coran
is the set of axioms like ZFC, and Sunnah is one 
interpretation of the system.    (066)

> >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.
> >
> >     Anything invented as a supplement to logic
> >     must of course be nonsense
> >     -Wittgenstein, Philosophical Remarks XII.129.
> >
> >I explained another subjective interpretaion of complete
> >induction to Chris below.
> >
> You have not answered my challenge. Convice me 
> that what you say above is true: that "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."
> It seems to me that the infinite set of numerals 
> {1,2,3,...} is an obvious counterexample. This is 
> a never-ending collection of members, each bigger 
> than the one before it, and each of them is 
> obviously finite. What mistake have I made?    (067)

You impose one interpretation of complete induction.
I impose another, but only in order to show that the 
interpretations are subjective.    (068)

> >  > > 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.
> Given as choice between ZFC and all of 
> 20th-century mathematics, on the one hand, and 
> proposition 3. above, on the other, I 
> unhesitatingly choose the former. I would likely 
> do that even if 3. seemed intuitively true: but 
> as it seems obviously, in-your-face, blatantly 
> false, the choice really is a no-brainer.    (069)

I gave the axiom that makes 3. intelligible:    (070)

"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"    (071)

This axiom totally as objective as the interpretation of complete
induction in ZFC.    (072)

> >  > >, 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.
> >
> >Take the Aristotelian view,
> OK, go ahead. Tell me the Aristotelian flaw in the above reasoning.    (073)

The same old potential infinity maintains that you can take 
as much as you want, but you'll never get into the end
of a never ending process.    (074)

> >  > >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'.
> >
> >So, once you have specified some size n, there is always
> >a larger size.
> Right.
> >  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?
> The word 'arbitrary' is not a descriptive 
> adjective: it has adverbial force. "Pick an 
> arbitrary number" means "Arbitrarily pick a 
> number". No actual number is 'arbitrary', any 
> more than any particular person is 'average' or 
> 'typical'    (075)

But how do you pick a number arbitrarily? I'm arguing 
that the picking process is impossible, and therefore
there is no sense to talk about arbitrary numbers.
There is only sense to talk about a random selection
from within finite specified bounds, such as the 
interval [1 333].     (076)

We can get strange results with arbitrary numbers,
such as, if you select randomly omega-0 arbitrary 
numbers, and put them in a basket, and then select 
one more arbitrary number X, the probability that X 
was already in the basket is 0.     (077)

> >By saying that arbitrary numbers are unnatural, I mean
> >that they are practically very close to transifinite
> >numbers.
> You have to explain it mathematically. What does 'practically close'
> mean?    (078)

If you somehow 'have' an arbitrary number at your disposal, you can 
ask how big is it. If I select a number from the set {1, 2, 3}, the 
average is 2. When I select from {1,2,3,...,100} the average is 50. 
When I select from the set of all cantorist naturals, I cannot even
say how big is the average. Assume that the number is picked between
0 and omega_0. The average should be something like (0+omega_0)/2.
This way, an arbitrary number is practically very close to transfinite.
If we follow normal arithmetics, infinity divided by two is infinity.    (079)

> >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.
> >
> >This holds for it: type a number down in decimal notation.
> Why decimal? Why not allow floating-point 
> notation? Silicon chips understand it, after all. 
> Why not allow Conway's arrow-chain notation?    (080)

Any notation is fine.     (081)

> >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
> What do you mean by this? I claim I do understand it. How will you refute
> me?    (082)

You could as well said HUGE instead of Ack(Ack(Ack(5,5),5),5).    (083)

> >>  >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?)
> >
> >You can imagine a textual string Ackerman(Ackerman(5 5) 5),
> >but you can imagine njnbaöbnjaöjdöbfk as well.
> Im not imagining the text string: I'm reading the 
> text string and imagining the number. I do this, 
> if you wish to know the details, by using a 
> special number which is the current best guess of 
> the ratio between the diameter of the known 
> universe and the Plank length - so I have a 
> vision of a 'packed-full universe' - and then 
> imagining a super-universe which has one of those 
> in each cell, and iterating from there.    (084)

But isn't Ackerman(Ackerman(5 5) 5) clearly too much for this ratio,
that is anyway less than 10 in the power of 1000?     (085)

> >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.
> But we can never know what this will be. And in 
> any case, Im against any proposal to reduce 
> mathematics to currently applied mathematics. The 
> history of science suggests that this would be a 
> very bad strategy indeed.    (086)

In any case, all that is in the area of potential infinity.    (087)

> >As said before, finitism gives a different answer to
> >''what are real numbers?'' than Dedekind-cuts.
> I imagine it would have to, yes. So, what answer DOES it give??    (088)

Personally, I'd use these number classes:    (089)

Naturals: as many as are needed    (090)

Rationals: ratios of naturals, as long as needed    (091)

Reals: a tag for rational serieses that can be 
continued as long as needed. The symbols such as
pi can be used.     (092)

> >So, can you by now settle with the potential infinity?
> Nope, as I have no idea what it is. And also 
> because it seems to me that I have a reasonably 
> robust intuition of actual infinity, and I can 
> work happily with it, and have no reason to 
> change my ways.    (093)

What about now? I know that you can now see that there is 
a very understandable controversy in considering a never 
ending as a totality.    (094)

> >  > >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.
> >
> >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,...
> You don't seem to have understood my point. It 
> *is* more than 1,2,3, because it grows faster at 
> an ever-increasing rate, so it meets the boundary 
> sooner.     (095)

You cannot think that anything reaches anything that 
is infinitely far away, with any conceivable finite 
speed. The two serieses    (096)

1 ,2 ,3 ,...
10,20,30,...     (097)

do not reach any end, never, and neither do    (098)

0.1,  0.2,  0.3, ...
0.01, 0.02, 0.03, ...    (099)

Consider another LW:s quote:    (0100)

        There is no path to infinity,
        not even an endless one.        
        [120] XII.123    (0101)

No series can reach something that does not 
exist: end. There can be no sense in saying that
something that does not exits is reached by series 
A faster than series B. Of course, we can say that
the series A grows faster than series B, and that 
the Ackermann series dominates any other series
after some finite number of steps.    (0102)

In one sense, I can understand that if you consider 
that the series 0.1,  0.2,  0.3, ... is 
transfinitely long, it does, in a way, reach 0, 
but you can also understand that no matter how 
many times you take a step, it never reaches 0. 
If can reach 0 only if we consider the process to 
be completed in 'one stroke'. And this is the same 
as to just say "it reaches 0 because I want to
think so".    (0103)

> >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'.
> Which of the ZFC axioms do you find so crazy? 
> They all seem quite obvious to me, except 
> possibly Choice, which we all now know to be 
> take-it-or-leave-it, and the axiom of Foundation, 
> which we now know to be optional.    (0104)

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. 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.    (0105)

I have clearly argued that potential infinity is totally 
enough for the needs of the man kind. If we go over it, 
we have to give some subjective interpretations, which 
are not necessary, and these subjective interpretations 
should no be included in logic.    (0106)

Avril    (0107)

[2] Carl Posy: Intuitionism and Philosophy, 2005
[3] Akihiro Kanamori: The Higher Infinite, 1991
[120] Ludwig Wittgenstein: Philosophical Remarks.
[121] Ludwig Wittgenstein: Philosophical Investigations,
[122] Cora Diamond (editor): Wittgenstein’s Lectures on the Foundations
ofMathematics.    (0108)

********************************************************************    (0109)

I have written down all these thoughts as remarks, short paragraphs, of
which there is sometimes a fairly long chain about the same subject, while I
sometimes make a sudden change, jumping from one topic to another. -It was
my intention at first to bring all this together in a book whose form I
pictured differently at different times. But the essential thing was that
the thoughts should proceed from one subject to another in a natural order
and without brakes. After several unsuccessful attempts to weld my results
together into such a whole, I realized that I should never succeed. The best
I could write would never be more than philosophical remarks; my thoughts
were soon crippled if I tried to force them on in any single direction
against their natural inclination. -And this was, of course, connected with
the very nature of the investigation. [121] p.ix,    (0110)

I am proposing to talk about the foundations of mathematics. An important
problem arises from the subject itself: How can I -or anyone who is not a
mathematician- talk about this? What right has a philosopher to talk about
mathematics? One might say: From what I have learned at school-my knowledge
of elementary mathematics- I know something about what can be done in the
higher branches of the subject. I can as a philosopher know that Professor
Hardy can never get such-andsuch a result or must get such-and-such a
result. I can foresee something he must arrive at.-In fact, people who
have talked about the foundations of mathematics have constantly been
tempted to make prophecies-going ahead of what has already been done. As if
they had a telescope with which they can’t possibly reach the moon, but can
see what is ahead of the mathematician who is flying there. That is not what
I am going to do at all. In fact, I am going to avoid it at all costs; it
will be most important not to interfere with the mathematicians. I must not
make a calculation and say, ”That’s the result; not what Turing says it is.”
Suppose it ever did happen-it would have nothing to do with the foundations
of mathematics. Again, one might think that I am going to give you, not new
calculations but a new interpretation of these calculations. But I am not
going to do that either. I am going to talk about the interpretation of
mathematical symbols, but I will not give a new interpretation.
Mathematicians tend to think that interpretations of mathematical symbols
are a lot of jaw-some kind of gas which surrounds the real process, the
essential mathematical kernel. [122] p.13.    (0111)

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