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. (01)
On Feb 13, 2008, at 2:15 AM, Avril Styrman wrote: (02)
> John, Pat, Chris,
>
> sorry for the delayed response. The roles in this discussion are
> clear. I'm claiming that finitism is better than transfinitsim
> because it is simpler, uncontroversial, and nothing more is needed.
> 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. (03)
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. (04)
> 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. (05)
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. (06)
> Of course, LöwenheimSkolem theorem states that every thing that one
> can 'do' with an innumberable model can be done with a numerable
> model too. (07)
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. (08)
>>> It is not just an ideology like some religion. It is evident that
>>> everything paradoxical should be pruned off from logic.
>
> CM:
>> Indeed. And if you could show one single paradox in contemporary
>> mathematics you might be on to something. Note this does not mean
>> producing an informal argument involving undefined terms with
>> unstated premises. It means taking the *axioms* of any branch of
>> mathematics  let's say ZF set theory plus Choice, since preZF
>> set theory was a fairly rich source of genuine paradox  and
>> *demonstrating* a contradiction, that is, producing from those
>> axioms a deduction of A and notA, for some proposition A. (Note if
>> you can do this, you've probably got a Fields Medal coming.) Until
>> you can do that, any claims to the effect that classical,
>> infinitary mathematics is paradoxical is on the same logical
>> footing as, say, Scientology.
>
> Fair enough. What about the BuraliForti paradox. All Cantorist
> sets exist as 'being', not as generating. (09)
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
muddleheaded philosophical codswallop. (010)
> So, there should also exist the greatest ordinal OMEGA, the
> universal closure of ZFC. (011)
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 BuraliForti 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? (012)
> But the axioms say there are always greater and greater ordinals. (013)
Well, it's a theorem, but yes. (014)
> This is a case of A and not A. (015)
It is exactly not such a case. It is a case of notA. Period. (016)
Note to those who are actually interested in understanding ZFC: the
day before yesterday was the 100th anniversary of the publication of
Zermelo's axiomatization of set theory  see
http://www.heise.de/newsticker/meldung/103411
if you read German; it's a pretty good popularized account. Thanks
to Pat Hayes for this tidbit. (017)
> 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. (018)
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. (019)
> 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. (020)
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 wellwarranted 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. (021)
>>> 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! (022)
A staggering display of ignorance of even the most elementary concepts
of set theory. (023)
> This means that you don't even need the axiom of choice to well
> order it :) (024)
Yes, you don't need choice to *impose* a wellordering upon it because
it is *finite*, not because it is intrinsically ordered. (025)
> You have learned the convention: omega0 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. (026)
Sorry, no, this is not a convention, it is a *provable* fact of set
theory. (027)
> 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 (028)
Unbelievable. Ok, tell you what. In ZFC, proposition 3 is provably
false. So you obviously don't accept ZFC. 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.) 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. (029)
> 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. (030)
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. (031)
> 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. (032)
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. (033)
> And this is the point: being paradoxfree. Transfinitism is the very
> swamp of contradictions, as I have shown above, (034)
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 notA (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. (035)
> 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. (036)
Uh, Aristotle was at a bit of a disadvantage, arriving on the scene
about 2500 years too soon. (037)
>>> 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. 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''. (038)
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". (039)
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. Second, ZFC is a powerful and
timetested 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. 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. 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 firstorder logic with
clear axioms and definitions so that your illdefined 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. (040)
That is definitely it for me on this topic. (041)
Chris Menzel (042)
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