Counterintuitions and the well-ordering theorem
Aatu Koskensilta
It is often said that the well-ordering theorem, or the existence of a
well-ordering of the reals in particular, is counterintuitive. Alas,
I've never quite fathomed what intuitions are contradicted. Perhaps
someone with keener intuition into the mysteries of sets can shed some
light on this pressing matter?
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Nam Nguyen
> It is often said that the well-ordering theorem, or the existence of a
> well-ordering of the reals in particular, is counterintuitive. Alas,
> I've never quite fathomed what intuitions are contradicted. Perhaps
> someone with keener intuition into the mysteries of sets can shed some
> light on this pressing matter?
>
Don't have a every details yet but it's possible the order be
relativistic over the (more absolute) existences of the reals,
which could give rise to paradoxes if we insist there's a "global"
absolute well order. Imho.
Aatu Koskensilta
> Don't have a every details yet but it's possible the order be
> relativistic over the (more absolute) existences of the reals, which
> could give rise to paradoxes if we insist there's a "global" absolute
> well order. Imho.
Whatever the merits of this, it appears to be a theoretical
(philosophical?) doctrine or hypothesis, not an intuition contradicted
by the well-ordering theorem.
Nam Nguyen
> Nam Nguyen <namduc...@shaw.ca> writes:
>
>> Don't have a every details yet but it's possible the order be
>> relativistic over the (more absolute) existences of the reals, which
>> could give rise to paradoxes if we insist there's a "global" absolute
>> well order. Imho.
>
> Whatever the merits of this, it appears to be a theoretical
> (philosophical?) doctrine or hypothesis, not an intuition contradicted
> by the well-ordering theorem.
>
It's only as much as philosophical as the current mathematical reasoning
based on the assumed knowledge of the naturals is. If you already
preconceived the truth of well-order of the reals isn't philosophically
based and isn' counterintuitive, well then there would be - as you said -
"mysteries of sets" you'd miss.
If you keep an open-minded "attitude", I'd further elaborate.
Rupert
> It is often said that the well-ordering theorem, or the existence of a
> well-ordering of the reals in particular, is counterintuitive. Alas,
> I've never quite fathomed what intuitions are contradicted. Perhaps
> someone with keener intuition into the mysteries of sets can shed some
> light on this pressing matter?
>
Some *consequences* of the existence of a well-ordering of the reals,
such as the Banach-Tarski paradox, are counter-intuitive.
Aatu Koskensilta
> Some *consequences* of the existence of a well-ordering of the reals,
> such as the Banach-Tarski paradox, are counter-intuitive.
Well, I would argue that no-one not already deep into set theory,
analysis, etc. has any intuitions about such matters as touched on in
the Banach-Tarski theorem -- in particular, the usual explanations of
the supposed counter-intuitiveness depend on the baffling idea that
non-measurable sets corresponds to "cuttings" in some physical
sense. That aside, I was wondering specifically about the claim that the
well-ordering theorem itself is counter-intuitive, as alluded to in
e.g. the famous quip
The axiom of choice is obviously true, the well-ordering theorem
obviously false -- and who can tell of Zorn's lemma?
Just what intuitions are contradicted by the well-ordering theorem?
Nam Nguyen
> Rupert <rupertm...@yahoo.com> writes:
>
>> Some *consequences* of the existence of a well-ordering of the reals,
>> such as the Banach-Tarski paradox, are counter-intuitive.
>
> Well, I would argue that no-one not already deep into set theory,
> analysis, etc. has any intuitions about such matters as touched on in
> the Banach-Tarski theorem -- in particular, the usual explanations of
> the supposed counter-intuitiveness depend on the baffling idea that
> non-measurable sets corresponds to "cuttings" in some physical
> sense. That aside, I was wondering specifically about the claim that the
> well-ordering theorem itself is counter-intuitive, as alluded to in
> e.g. the famous quip
>
> The axiom of choice is obviously true, the well-ordering theorem
> obviously false -- and who can tell of Zorn's lemma?
>
> Just what intuitions are contradicted by the well-ordering theorem?
>
It's counter intuitive to assert either M(Pi) < M(e), or M(e) < M(Pi),
where M(x) is the Major number of x and Pi and e are the 2 well known
transcendentals (please refer to the thread "Ttranscendental Goldbach
Conjecture"), since nobody would have a slightest intuition which one
assertion is the case.
Since the well-ordering of reals implies one assertion over the other
is true, it's therefore counterintuitive.
Put it differently, it's quite possible it's impossible to know which
inequality would hold - in all models of reals. It's therefore counter-
intuitive to talk about the well-ordering of the whole set of reals.
Herman Jurjus
> Rupert <rupertm...@yahoo.com> writes:
>
>> Some *consequences* of the existence of a well-ordering of the reals,
>> such as the Banach-Tarski paradox, are counter-intuitive.
>
> Well, I would argue that no-one not already deep into set theory,
> analysis, etc. has any intuitions about such matters as touched on in
> the Banach-Tarski theorem -- in particular, the usual explanations of
> the supposed counter-intuitiveness depend on the baffling idea that
> non-measurable sets corresponds to "cuttings" in some physical
> sense. That aside, I was wondering specifically about the claim that the
> well-ordering theorem itself is counter-intuitive, as alluded to in
> e.g. the famous quip
>
> The axiom of choice is obviously true, the well-ordering theorem
> obviously false -- and who can tell of Zorn's lemma?
>
> Just what intuitions are contradicted by the well-ordering theorem?
Imho, the quip tries to express something completely different.
It's not an expression of mathematician's intuitions about sets.
Mathematicians don't care about sets - they care about mathematics.
And a general purpose foundation for mathematics should ideally not turn
something into an indispensable truth that is, for mathematics, quite
dispensable.
Btw, personally i find Freyling's argument quite appealing. But i also
think that AD and AC are both true, so you should better not take my
opinions too seriously. (Not that there was any danger that you'd do
that, anyway.)
--
Cheers,
Herman Jurjus
Jesse F. Hughes
> Aatu Koskensilta wrote:
>> Rupert <rupertm...@yahoo.com> writes:
>>
>>> Some *consequences* of the existence of a well-ordering of the reals,
>>> such as the Banach-Tarski paradox, are counter-intuitive.
>>
>> Well, I would argue that no-one not already deep into set theory,
>> analysis, etc. has any intuitions about such matters as touched on in
>> the Banach-Tarski theorem -- in particular, the usual explanations of
>> the supposed counter-intuitiveness depend on the baffling idea that
>> non-measurable sets corresponds to "cuttings" in some physical
>> sense. That aside, I was wondering specifically about the claim that the
>> well-ordering theorem itself is counter-intuitive, as alluded to in
>> e.g. the famous quip
>>
>> The axiom of choice is obviously true, the well-ordering theorem
>> obviously false -- and who can tell of Zorn's lemma?
>>
>> Just what intuitions are contradicted by the well-ordering theorem?
>>
>
> It's counter intuitive to assert either M(Pi) < M(e), or M(e) < M(Pi),
> where M(x) is the Major number of x and Pi and e are the 2 well known
> transcendentals (please refer to the thread "Ttranscendental Goldbach
> Conjecture"), since nobody would have a slightest intuition which one
> assertion is the case.
So, your problem is with the law of excluded middle?
> Since the well-ordering of reals implies one assertion over the other
> is true, it's therefore counterintuitive.
That's not what I'd call counterintuitive. There are all sorts of
things that satisfy this. There is a nickel in my drawer, where I
can't see it. It is either heads up or tails up, but I don't have the
slightest intuition which one is the case. Do you find that situation
counterintuitive?
> Put it differently, it's quite possible it's impossible to know
> which inequality would hold - in all models of reals. It's therefore
> counter- intuitive to talk about the well-ordering of the whole set
> of reals.
Yes, I think it would be counterintuitive to speak of "the"
well-ordering, but surely people only speak of "a" well-ordering?
--
Jesse F. Hughes
"Why do the dirty villains always have to tie your hands *behind* ya?"
"That's what makes them villains."
--Adventures by Morse (old radio show)
Daryl McCullough
>Well, I would argue that no-one not already deep into set theory,
>analysis, etc. has any intuitions about such matters as touched on in
>the Banach-Tarski theorem -- in particular, the usual explanations of
>the supposed counter-intuitiveness depend on the baffling idea that
>non-measurable sets corresponds to "cuttings" in some physical
>sense.
I don't think what you are saying makes any sense. Bringing up
non-measurable sets is not a way to explain the counter-intuitiveness
of Banach-Tarski. It's a way of arguing that it *ISN'T* counter-intuitive
(because people don't have good intuitions about non-measurable sets).
The situation with Banach-Tarski to me is that (1) There is an informal,
intuitively true claim, (something along the lines of: If you
cut up a solid object into pieces, and you rearrange the pieces,
you'll get a new object that has the same volume as the original.
(2) There is an attempt to formalize the informal claim, by defining
what a "piece" might be, what "rearrange" means, etc.
(3) Then Banach-Tarski shows that the formal version is false.
You can argue that the informal claim is true, and that the
problem is that the formalization does not capture the informal
notion of "piece". But it is completely wrong to say that
the formalization depends on "baffling ideas" about non-measurable
sets. The formalization doesn't *MENTION* non-measurable sets.
Rather, the formalization just didn't specifically *RULE* *OUT*
non-measurable sets.
--
Daryl McCullough
Ithaca, NY
WM
> It is often said that the well-ordering theorem, or the existence of a
> well-ordering of the reals in particular, is counterintuitive. Alas,
> I've never quite fathomed what intuitions are contradicted. Perhaps
> someone with keener intuition into the mysteries of sets can shed some
> light on this pressing matter?
Certainly. Well-ordering is equivalent to AC. But AC ...
"So, even though, for example, the Hausdorff-Banach-Tarski paradox
has
been called the most paradoxical result of the twentieth century,
classical mathematicians have to convince themselves that it is
natural, because it is a consequence of the Axiom of Choice, which
classical mathematicians are determined to uphold, because the Axiom
of Choice is required for important theorems which classical
mathematicians regard as intuitively natural."
[Henry Flynt: IS MATHEMATICS A SCIENTIFIC DISCIPLINE? (1994)]
http://www.henryflynt.org/studies_sci/mathsci.html
Regards, WM
WM
>
> >> Don't have a every details yet but it's possible the order be
> >> relativistic over the (more absolute) existences of the reals, which
> >> could give rise to paradoxes if we insist there's a "global" absolute
> >> well order. Imho.
>
> > Whatever the merits of this, it appears to be a theoretical
> > (philosophical?) doctrine or hypothesis, not an intuition contradicted
> > by the well-ordering theorem.
>
> It's only as much as philosophical as the current mathematical reasoning
> based on the assumed knowledge of the naturals is. If you already
> preconceived the truth of well-order of the reals isn't philosophically
> based
It is based on an erroneous proof.
Zeremlo's first proof of well ordering contains the sentence: "Wäre
nun m' das erste Element von M', welches von dem entsprechenden
Elemente m'' verschieden wäre, ..." meaning: If the gamma-set M' would
differ from the gamma-set M'' by the element m' for the first
time, ... [E. Zermelo: Beweis, daß jede Menge wohlgeordnet werden
kann, Math. Ann. 59 (1904) 514-516]
That implies: Zermelo's proof shows that the well-ordered set, e.g. R,
can be brought into an unbroken linear sequence, called gamma-set M',
were no element differs from the alternative sequence gamma-set M''.
There is no first element missing in one of the sets, and every
element has a precursor (otherwise it would not be sure whether the
element without precursor was the first element that could possible be
different in M' and M'').
Hence, Zermelo proves that R can be put into an unbroken sequence,
contradicting Cantor's proof.
Regards, WM
WM
> Just what intuitions are contradicted by the well-ordering theorem?
Allegedly more numbers can be well-ordered than can be addressed or
named. What else, however, is ordering but addressing or naming
successively?
>
> --
> Aatu Koskensilta (aatu.koskensi...@uta.fi)
>
> "Wovon mann nicht sprechen kann, darüber muss man schweigen"
> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
There is no path to infinity, not even an endless one. [§ 123]
It isn't just impossible "for us men" to run through the natural
numbers one by one; it's impossible, it means nothing. […] you can’t
talk about all numbers, because there's no such thing as all numbers.
[§ 124]
Ludwig Wittgenstein, Philosophical Remarks
Regards, WM
David C. Ullrich
<aatu.kos...@uta.fi> wrote:
>
>It is often said that the well-ordering theorem, or the existence of a
>well-ordering of the reals in particular, is counterintuitive. Alas,
>I've never quite fathomed what intuitions are contradicted. Perhaps
>someone with keener intuition into the mysteries of sets can shed some
>light on this pressing matter?
Come on. Surely it's clear that we need someone with a _less_
keen intuition to explain this.
So I'll step in. It contradicts the inuitively clear fact that there
is no well-ordering of R.
Seriously. You're not going to get a clear _mathematical_
answer to your question. I don't think that anyone's suggested
that the existence of a well-ordering of R is particularly
counterintuitive _to_ someone who's actually studied set
theory. But the quip about AC being obviously true while
WOT is clearly false simply _is_ the way it seems to many
people who haven't studied these things in any depth. Not
that I can explain why that should be, but I can give
evidence that it's so:
One sees people who don't see the need for AC as an
axiom since it's clearly true.
One sees people writing books where they carefully
state that this or that theorem requires AC, giving the
impression that they want to make it clear what parts
of the theory do and what parts do not depend on AC,
but then in the same book they prove theorems that
do depend on AC without acknowledging this, hence
I conjecture without being aware of it. (Eg books
on analysis treating, say, the Hahn-Banach theorem
and also giving a "careful" proof that a countable
union of countable sets is countable - a very smart
analyst down the hall simply didn't believe me
when I told him that that's not a theorem of ZF.) It
does seem to me that AC is so intuitively clear to
people that they use it all the time without realizing that
that's what they're doing (I for one am always very nervous
claiming that I have _not_ used AC anywhere in
some argument).
Otoh one sees the same people giving the impression
that the existence of an uncountable well-ordered
set requires AC. I'm not sure I've ever seen an
_explicit_ statement to this effect, but that's
certainly the impression one gets, especially when
the author seems to be trying to avoid AC when it's
not needed but uses it for this fact.
It does seem to me that for whatever reason people
_do_ use AC without even being aware of it, and
that simply is not true of WOT. Because AC simply
_is_ clear, while WOT is not.
That's just observation of what people seem to do,
hence it seems to me evidence of what their intuitions
actually _are_ - WOT contradicts the intuition that
R cannot be well-ordered, which is why we need AC,
well known to have counterintuitive consequences,
to prove it.
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
Herman Jurjus
It always strikes me that people talk about applications of countable
choice or dependent choice as 'requiring AC'. I'd say that, when a
result can be proved with just CC and DC, it makes much more sense to
say that it does -not- require AC, even though perhaps it's not provable
from ZF alone. CC is sooo much weaker. (And much less controversial,
mathematically.)
> It
> does seem to me that AC is so intuitively clear to
> people that they use it all the time without realizing that
> that's what they're doing
Do you remember any cases where full AC was really required, instead of
just CC/DC?
> (I for one am always very nervous
> claiming that I have _not_ used AC anywhere in
> some argument).
>
> Otoh one sees the same people giving the impression
> that the existence of an uncountable well-ordered
> set requires AC. I'm not sure I've ever seen an
> _explicit_ statement to this effect, but that's
> certainly the impression one gets, especially when
> the author seems to be trying to avoid AC when it's
> not needed but uses it for this fact.
>
> It does seem to me that for whatever reason people
> _do_ use AC without even being aware of it, and
> that simply is not true of WOT. Because AC simply
> _is_ clear, while WOT is not.
>
> That's just observation of what people seem to do,
> hence it seems to me evidence of what their intuitions
> actually _are_ - WOT contradicts the intuition that
> R cannot be well-ordered, which is why we need AC,
> well known to have counterintuitive consequences,
> to prove it.
Nothing snipped; spot on!
--
Cheers,
Herman Jurjus
John Jones
> It is often said that the well-ordering theorem, or the existence of a
> well-ordering of the reals in particular, is counterintuitive. Alas,
> I've never quite fathomed what intuitions are contradicted. Perhaps
> someone with keener intuition into the mysteries of sets can shed some
> light on this pressing matter?
>
If mathematics is a universal language then it cannot syntactically
present metatheoretical statements. In which case its formalism is
restricted to the signs and symbols as they are presented on paper. In
which case again, no intuitions come into mathematics.
Nam Nguyen
> Nam Nguyen <namduc...@shaw.ca> writes:
>
>> Aatu Koskensilta wrote:
>>> Rupert <rupertm...@yahoo.com> writes:
>>>
>>>> Some *consequences* of the existence of a well-ordering of the reals,
>>>> such as the Banach-Tarski paradox, are counter-intuitive.
>>> Well, I would argue that no-one not already deep into set theory,
>>> analysis, etc. has any intuitions about such matters as touched on in
>>> the Banach-Tarski theorem -- in particular, the usual explanations of
>>> the supposed counter-intuitiveness depend on the baffling idea that
>>> non-measurable sets corresponds to "cuttings" in some physical
>>> sense. That aside, I was wondering specifically about the claim that the
>>> well-ordering theorem itself is counter-intuitive, as alluded to in
>>> e.g. the famous quip
>>>
>>> The axiom of choice is obviously true, the well-ordering theorem
>>> obviously false -- and who can tell of Zorn's lemma?
>>>
>>> Just what intuitions are contradicted by the well-ordering theorem?
>>>
>> It's counter intuitive to assert either M(Pi) < M(e), or M(e) < M(Pi),
>> where M(x) is the Major number of x and Pi and e are the 2 well known
>> transcendentals (please refer to the thread "Ttranscendental Goldbach
>> Conjecture"), since nobody would have a slightest intuition which one
>> assertion is the case.
>
> So, your problem is with the law of excluded middle?
No. It's not LEM at all! That it's _impossible_to know the truth value
of an arithmetic statement doesn't at all mean, say, Jesse or Nam is
allowed to believe the statement's truth value is in between true and
false.
But it does mean Jesse could assert the statement be true and Nam false
and it's _impossible_ to tell who's right or wrong.
Which means the truth or falsehood of the statement is relativistic: if
ones takes it as true, another one can _equally and logically_ take it
as false. All the while LEM is still not broken.
>
>> Since the well-ordering of reals implies one assertion over the other
>> is true, it's therefore counterintuitive.
>
> That's not what I'd call counterintuitive. There are all sorts of
> things that satisfy this.
You're mistaken. The "All sort of things" cases don't have the word
'impossible'/'unknownability' in their descriptions. My cases do! Your
side just never pays attention to the implication of those words (which
I've cited numerous times). They do bring impact on the foundation of
FOL reasoning.
> There is a nickel in my drawer, where I can't see it.
But can you or somebody else see it by walking up to the drawer and
open it? So, can you see it, _in principle_? Of course you can!
> It is either heads up or tails up, but I don't have the
> slightest intuition which one is the case. Do you find that situation
> counterintuitive?
As just mentioned, of course you have knowledge about which one is the
case _in principle_. So nothing is counterintuitive here.
In brief, your analogy is close but not quite correct. Let me give a
more precise analogy. Let's consider the statement:
(1) In each of the _infinite_ number of universes there exists a planet
with biological life during the planet's lifetime.
Can you see the truth or falsehood of (1), even _in principle_?
Of course (1) is either true or false and not in between. But there's
a genuine _impossibility_ to know which truth value be the case. And,
again, consequently the assertion that it be true or false is completely
relativistic, depending which value one would _choose_.
Closer to home, in FOL reasoning, there's an analogous situation. Whether
or not it's true the last digit of a extraordinarily large prime number P
is 1 is analogous to your "nickel in my drawer" case: we do know the truth
_in principle_ because the proof of it can be done in _finite_ steps.
But the arithmetic truth of say GC could be something else all together.
It's either true or false of course (so LEM is not broken) but it could
be impossible to know (since its "proof" could be infinite). So its truth
would be relativistic, while LEM is not broken.
>
>> Put it differently, it's quite possible it's impossible to know
>> which inequality would hold - in all models of reals. It's therefore
>> counter- intuitive to talk about the well-ordering of the whole set
>> of reals.
>
> Yes, I think it would be counterintuitive to speak of "the"
> well-ordering, but surely people only speak of "a" well-ordering?
>
You missed the point: M(Pi) < M(e) could be relativistic in any model
of reals. It's therefore counterintuitive to talk about any well-ordering
of reals when it's _impossible_ to know if M(Pi) < M(e) is true or false.
Put it differently, how could you talk about the order of numbers if it's
_impossible_ to tell which of the 2 numbers is greater in the order
ladder?
Jesse F. Hughes
> As just mentioned, of course you have knowledge about which one is the
> case _in principle_. So nothing is counterintuitive here.
>
> In brief, your analogy is close but not quite correct. Let me give a
> more precise analogy. Let's consider the statement:
>
> (1) In each of the _infinite_ number of universes there exists a planet
> with biological life during the planet's lifetime.
>
> Can you see the truth or falsehood of (1), even _in principle_?
>
> Of course (1) is either true or false and not in between. But there's
> a genuine _impossibility_ to know which truth value be the case. And,
> again, consequently the assertion that it be true or false is completely
> relativistic, depending which value one would _choose_.
No, that clause beginning "consequently" does not follow.
I'll assume, as you say, that either (1) is true or false, and we
don't know (and can't know) which. It simply doesn't follow that the
truth value of (1) depends on whether I choose to believe it or not.
Rather, either I choose correctly or I don't, but (1) is true or false
independently of my choice.
> Closer to home, in FOL reasoning, there's an analogous situation. Whether
> or not it's true the last digit of a extraordinarily large prime number P
> is 1 is analogous to your "nickel in my drawer" case: we do know the truth
> _in principle_ because the proof of it can be done in _finite_ steps.
>
> But the arithmetic truth of say GC could be something else all together.
> It's either true or false of course (so LEM is not broken) but it could
> be impossible to know (since its "proof" could be infinite). So its truth
> would be relativistic, while LEM is not broken.
Again, either GC is true of the natural numbers or it is not. This
fact doesn't depend on what we choose to believe. Your claims about
relativism are simply wrong.
--
"Memoirists like Frey and Augusten Burroughs belong to the long list of
those who should never have stopped using drugs. The drugs might have
made Frey more interesting, or they might have killed him. Either way,
American literature would have benefited." --John Dolan, www.exile.ru
Jesse F. Hughes
>>> Put it differently, it's quite possible it's impossible to know
>>> which inequality would hold - in all models of reals. It's therefore
>>> counter- intuitive to talk about the well-ordering of the whole set
>>> of reals.
>>
>> Yes, I think it would be counterintuitive to speak of "the"
>> well-ordering, but surely people only speak of "a" well-ordering?
>>
>
> You missed the point: M(Pi) < M(e) could be relativistic in any model
> of reals. It's therefore counterintuitive to talk about any well-ordering
> of reals when it's _impossible_ to know if M(Pi) < M(e) is true or false.
>
> Put it differently, how could you talk about the order of numbers if it's
> _impossible_ to tell which of the 2 numbers is greater in the order
> ladder?
It's not impossible. It just depends on which well-ordering we're
speaking of. Let M be any well-ordering of R, i.e., M:R -> |R|.
Define a new well-ordering M' by
M'(pi) = min{M(pi), M(e)}
M'(e) = max{M(pi), M(e)}
M'(x) = M(x) for all x distinct from pi and e.
Now, it is perfectly clear that M' is a well-ordering of R and
M'(pi) < M'(e). So, your claim that it is impossible to compare pi
and e in a well-ordering of R is simply false.
--
"Customers have come to SCO asking what they can do to respect and
help protect the rights of the SCO intellectual property in Linux.
SCO has created the Intellectual Property License for Linux in
response to these customers needs." -- SCO responds to needs.
David C. Ullrich
wrote:
No. Yes, that would be much better for the point I'm making.
Nam Nguyen
> Nam Nguyen <namduc...@shaw.ca> writes:
>
>> As just mentioned, of course you have knowledge about which one is the
>> case _in principle_. So nothing is counterintuitive here.
>>
>> In brief, your analogy is close but not quite correct. Let me give a
>> more precise analogy. Let's consider the statement:
>>
>> (1) In each of the _infinite_ number of universes there exists a planet
>> with biological life during the planet's lifetime.
>>
>> Can you see the truth or falsehood of (1), even _in principle_?
>>
>> Of course (1) is either true or false and not in between. But there's
>> a genuine _impossibility_ to know which truth value be the case. And,
>> again, consequently the assertion that it be true or false is completely
>> relativistic, depending which value one would _choose_.
>
> No, that clause beginning "consequently" does not follow.
>
> I'll assume, as you say, that either (1) is true or false, and we
> don't know (and can't know) which.
Right; although it should be reminded that there's a distinction between
"don't know" and "can't [possibly] know": the later logically implies
the former, but not the other way around! The case for (1) is the
"can't [possibly] know" case; and that would make a difference.
> It simply doesn't follow that the
> truth value of (1) depends on whether I choose to believe it or not.
If, given an underlying reasoning framework, a truth of a statement can't
be possibly known then the truth is _undecidable_, notwithstanding the
fact it's either true or false. And whence it's undecidable, one is
*free to _decide_* which way it be!
> Rather, either I choose correctly or I don't, but (1) is true or false
> independently of my choice.
Once something is undecidable, there's *no _correct_* choosing or deciding!
>
>> Closer to home, in FOL reasoning, there's an analogous situation. Whether
>> or not it's true the last digit of a extraordinarily large prime number P
>> is 1 is analogous to your "nickel in my drawer" case: we do know the truth
>> _in principle_ because the proof of it can be done in _finite_ steps.
>>
>> But the arithmetic truth of say GC could be something else all together.
>> It's either true or false of course (so LEM is not broken) but it could
>> be impossible to know (since its "proof" could be infinite). So its truth
>> would be relativistic, while LEM is not broken.
>
> Again, either GC is true of the natural numbers or it is not.
Right. A formula is either true or false in a model of a T, just as
the simultaneity of 2 events is either true or false in a frame of
reference. But which frame are you talking about? Similarly, which
"natural numbers" are you referring to? The one in which GC is true?
Or the one in which cGC (hence ~GC) is true?
> This fact doesn't depend on what we choose to believe.
Which _undecidable_ fact are you referring to? The "natural numbers"-fact
in which GC is true? Or the "natural numbers"-fact where it's false?
> Your claims about relativism are simply wrong.
Rather, you don't seem to understand how mathematical _impossibility_ is
equated to mathematical relativity.
Nam Nguyen
> Nam Nguyen <namduc...@shaw.ca> writes:
>
>>>> Put it differently, it's quite possible it's impossible to know
>>>> which inequality would hold - in all models of reals. It's therefore
>>>> counter- intuitive to talk about the well-ordering of the whole set
>>>> of reals.
>>> Yes, I think it would be counterintuitive to speak of "the"
>>> well-ordering, but surely people only speak of "a" well-ordering?
>>>
>> You missed the point: M(Pi) < M(e) could be relativistic in any model
>> of reals. It's therefore counterintuitive to talk about any well-ordering
>> of reals when it's _impossible_ to know if M(Pi) < M(e) is true or false.
>>
>> Put it differently, how could you talk about the order of numbers if it's
>> _impossible_ to tell which of the 2 numbers is greater in the order
>> ladder?
>
> It's not impossible. It just depends on which well-ordering we're
> speaking of. Let M be any well-ordering of R, i.e., M:R -> |R|.
> Define a new well-ordering M' by
>
>
> M'(pi) = min{M(pi), M(e)}
> M'(e) = max{M(pi), M(e)}
> M'(x) = M(x) for all x distinct from pi and e.
>
> Now, it is perfectly clear that M' is a well-ordering of R and
> M'(pi) < M'(e). So, your claim that it is impossible to compare pi
> and e in a well-ordering of R is simply false.
>
Your analogy with M'(pi) < M'(e) is incorrect. My point is very _specific_:
"M(Pi) < M(e) could be relativistic" in any given model of R, hence in any
well-ordering of R. That specificity has nothing to do with your M'(pi) < M'(e).
Rupert
We are referring to the unique model of second-order Peano arithmetic.
Jesse F. Hughes
Right.
> And whence it's undecidable, one is *free to _decide_* which way it
> be!
I don't see how that follows.
>
>> Rather, either I choose correctly or I don't, but (1) is true or false
>> independently of my choice.
>
> Once something is undecidable, there's *no _correct_* choosing or
> deciding!
You already agreed that the statement (1) is either true or false,
regardless of our choice. If (1) is true and we've chosen to believe
(1), then we've chosen correctly. In the same circumstances, if we've
chosen to believe (1) is false, then we've chosen incorrectly.
What could be clearer?
We both agree that, whichever we choose (assuming that we do indeed
choose), we will not know whether that choice is the correct choice or
not. I simply can't imagine why you think this implies there is no
correct choice. Assuming that the statement (1) is meaningful and
either true or false, then it is clear what we mean when we say that
we've chosen to believe correctly -- regardless of whether we will
ever know that this is the case.
(Note: logical positivists would presumably reason rather differently
and claim that (1) is neither true nor false, but this does not
support your claim.)
>>> Closer to home, in FOL reasoning, there's an analogous situation. Whether
>>> or not it's true the last digit of a extraordinarily large prime number P
>>> is 1 is analogous to your "nickel in my drawer" case: we do know the truth
>>> _in principle_ because the proof of it can be done in _finite_ steps.
>>>
>>> But the arithmetic truth of say GC could be something else all together.
>>> It's either true or false of course (so LEM is not broken) but it could
>>> be impossible to know (since its "proof" could be infinite). So its truth
>>> would be relativistic, while LEM is not broken.
>>
>> Again, either GC is true of the natural numbers or it is not.
>
> Right. A formula is either true or false in a model of a T, just as
> the simultaneity of 2 events is either true or false in a frame of
> reference. But which frame are you talking about? Similarly, which
> "natural numbers" are you referring to? The one in which GC is true?
> Or the one in which cGC (hence ~GC) is true?
The natural numbers. The smallest set satisfying the Peano axioms.
>
>> This fact doesn't depend on what we choose to believe.
>
> Which _undecidable_ fact are you referring to? The "natural numbers"-fact
> in which GC is true? Or the "natural numbers"-fact where it's false?
My comment refers to the fact that either GC is true or not.
>> Your claims about relativism are simply wrong.
>
> Rather, you don't seem to understand how mathematical _impossibility_ is
> equated to mathematical relativity.
Well, you're right there. I certainly don't understand this odd claim.
--
Jesse F. Hughes
"Notice how I am, see how little social crap affects me?
You are social animals. What do you know of truth?"
--James S. Harris is like Spock, man. Or Buddha. Maybe both.
Jesse F. Hughes
I guess I haven't a clue what you mean, then. What is the meaning of
M(pi)? I assumed that it was the mapping of pi under a given
well-ordering of R. But there is no canonical well-ordering, as far
as I can see, so this claim that it is unknowable is false. It
depends on the well-ordering we're considering. In particular,
starting with a given well-ordering M, I can construct a well-ordering
M' (possibly equal to M) in which M'(pi) < M'(e).
I don't see how any of this has anything to do with particular models
of R.
--
Jesse F. Hughes
"You know that view most people have of mathematicians as brilliant
people? What if they're not?" -- James S. Harris
Nam Nguyen
If one isn't free to decide which way it be then it's not undecidable.
Don't you think? What do you think "undecidable" mean?
>
>>> Rather, either I choose correctly or I don't, but (1) is true or false
>>> independently of my choice.
>> Once something is undecidable, there's *no _correct_* choosing or
>> deciding!
>
> You already agreed that the statement (1) is either true or false,
> regardless of our choice.
Where did I agree that (1) is true or false "regardless of our choice"?
> If (1) is true and we've chosen to believe
> (1), then we've chosen correctly. In the same circumstances, if we've
> chosen to believe (1) is false, then we've chosen incorrectly.
>
> What could be clearer?
You simply don't understand a clear simple fact: one simply can't
apply LEM to (1) - without a context. And the context is one's
assumption that it be true, or false. In other words, (1) is a
statement that _has no absolute truth_! There are statements
(scientific or not) that appear to have absolute truth but they don't.
"God exists" is a non-scientific example, while (2) is a scientific
one, where (2) is:
(2) "Two particular events e1 e2 happen simultaneously".
We're 100+ years after 1905 when SR was presented! So again, you should
make it clear in your mind that (1) is not a kind you could claim
it's true or false without a (relativization) context.
>
> We both agree that, whichever we choose (assuming that we do indeed
> choose), we will not know whether that choice is the correct choice or
> not. I simply can't imagine why you think this implies there is no
> correct choice.
That's because you're too quick to contemplate on what "impossibility"
means, to the foundation of reasoning. Could you tell me what correct
choice you'd have for (2)?
> Assuming that the statement (1) is meaningful and
> either true or false, then it is clear what we mean when we say that
> we've chosen to believe correctly -- regardless of whether we will
> ever know that this is the case.
Your misconception here is a meaningful statement must have an absolute
truth value. (2) is an example this is not the case.
>> But which frame are you talking about? Similarly, which
>> "natural numbers" are you referring to? The one in which GC is true?
>> Or the one in which cGC (hence ~GC) is true?
>
> The natural numbers. The smallest set satisfying the Peano axioms.
Which "smallest set satisfying the Peano axioms" are you talking about?
the one in which GC is true? Or the one in which ~GC is true?
>>> This fact doesn't depend on what we choose to believe.
>> Which _undecidable_ fact are you referring to? The "natural numbers"-fact
>> in which GC is true? Or the "natural numbers"-fact where it's false?
>
> My comment refers to the fact that either GC is true or not.
How do you even know that "absolute fact" when you're under the assumption
that it be _impossible_ to know the truth value of GC?
>
>>> Your claims about relativism are simply wrong.
>> Rather, you don't seem to understand how mathematical _impossibility_ is
>> equated to mathematical relativity.
>
> Well, you're right there. I certainly don't understand this odd claim.
I'm almost certain you know something about SR and have heard of (2) in
one form or another. Why don't you reflect on those scientific matters
for a moment, in contemplating about similar issue: relativity in
mathematical reasoning.
I'll give you a hint that I think would help you here. In the thread
"Nonfirstorderizability" on August 7, 2005, Torkel Franzen mentioned
something to the effect that there are intuitive concepts that can't
be translated into a FOL statement (hence the phrase "Nonfirstorderizability").
Now, let me ask you a question about the reverse of "Nonfirstorderizability":
would you think there is any statement than can be formulated as a FOL
formula but which can't be "modellizable" in a collection of models?
If your answer is that there isn't any, then could you show your meta
proof to back your claim? If not then you got to allow the possibility
that such "nonmodellizability" could happen to GC, cGC, ~GC, ~cGC, in the
collection of all that are perceived as arithmetic models - including
the naturals.
Aatu Koskensilta
> Aatu Koskensilta wrote:
>
>> That aside, I was wondering specifically about the claim that the
>> well-ordering theorem itself is counter-intuitive, as alluded to in
>> e.g. the famous quip
>>
>> The axiom of choice is obviously true, the well-ordering theorem
>> obviously false -- and who can tell of Zorn's lemma?
>>
>> Just what intuitions are contradicted by the well-ordering theorem?
>
> Imho, the quip tries to express something completely different.
> It's not an expression of mathematician's intuitions about sets.
> Mathematicians don't care about sets - they care about mathematics.
Sure. What do you take the quip to (try to) express?
> And a general purpose foundation for mathematics should ideally not
> turn something into an indispensable truth that is, for mathematics,
> quite dispensable.
I'm not sure what you have in mind here.
> Btw, personally i find Freyling's argument quite appealing. But i also
> think that AD and AC are both true, so you should better not take my
> opinions too seriously.
What are your reasons for thinking AD true?
> (Not that there was any danger that you'd do that, anyway.)
I take nothing too seriously.
Aatu Koskensilta
> I don't think what you are saying makes any sense. Bringing up
> non-measurable sets is not a way to explain the counter-intuitiveness
> of Banach-Tarski.
I didn't say it was. I said that in order for the Banach-Tarski theorem
to strike anyone as counter-intuitive we must implicitly depend on some
identification of arbitrary and wild sets of points with physical
volumes etc.
> It's a way of arguing that it *ISN'T* counter-intuitive (because
> people don't have good intuitions about non-measurable sets).
Most people have no intuitions whatever about non-measurable sets. Of
course, in order to illustrate that mathematical constructions needn't
have anything to do with balls and pieces in the ordinary sense we can
just observe that e.g. removing all points with rational coordinates
from the unit ball doesn't correspond to any physical operation.
> The situation with Banach-Tarski to me is that (1) There is an
> informal, intuitively true claim, (something along the lines of: If
> you cut up a solid object into pieces, and you rearrange the pieces,
> you'll get a new object that has the same volume as the original. (2)
> There is an attempt to formalize the informal claim, by defining what
> a "piece" might be, what "rearrange" means, etc. (3) Then
> Banach-Tarski shows that the formal version is false.
This is not a very accurate account of the situation. There are usually
no attempts to formalise anything. Rather, some people simply mistakenly
think a mathematical result means something it doesn't mean.
Aatu Koskensilta
> Come on. Surely it's clear that we need someone with a _less_ keen
> intuition to explain this.
So it appears. Regarding the evidence you provided for the evidence of
the axiom of choice (or at least dependent choice or countable choice),
well, there is a reason the axiom of choice is an axiom and the
well-ordering theorem a theorem. (Cantor did at one point think the
well-ordering principle was a "law of thought", but later came around
the view it's something that needs to be proven.) My favourite example
of a result where it's often difficult even for people with some
experience in such matters to spot the invocation of (countable) choice
is the theorem that omega_1 is regular.
> Otoh one sees the same people giving the impression that the existence
> of an uncountable well-ordered set requires AC.
This idea is not uncommon in news...
Daryl McCullough
>
>stevend...@yahoo.com (Daryl McCullough) writes:
>
>> I don't think what you are saying makes any sense. Bringing up
>> non-measurable sets is not a way to explain the counter-intuitiveness
>> of Banach-Tarski.
>
>I didn't say it was. I said that in order for the Banach-Tarski theorem
>to strike anyone as counter-intuitive we must implicitly depend on some
>identification of arbitrary and wild sets of points with physical
>volumes etc.
Yes, and I'm saying that that is completely wrong. As I explained,
there are two claims at work: (1) a formal statement, and (2) an
informal paraphrase of that statement. The informal paraphrase is
counterintuitive. It is not counter-intuitive because someone has
implicitly accepted some identification of arbitrary and wild
sets of points with physical volumes.
You can argue that the informal paraphrase is *inaccurate*, that
it doesn't really capture the meaning of the formal statement
(or that the formal statement doesn't capture the meaning of
the informal statement).
>> The situation with Banach-Tarski to me is that (1) There is an
>> informal, intuitively true claim, (something along the lines of: If
>> you cut up a solid object into pieces, and you rearrange the pieces,
>> you'll get a new object that has the same volume as the original. (2)
>> There is an attempt to formalize the informal claim, by defining what
>> a "piece" might be, what "rearrange" means, etc. (3) Then
>> Banach-Tarski shows that the formal version is false.
>
>This is not a very accurate account of the situation.
Sure it is. There are two claims, one is informal and intuitively true,
and the other is formal and provably false. They are claimed to be
paraphrases of each other.
>There are usually no attempts to formalise anything.
Sure there is. You have to formalize what a rearrangement means,
and you have to formalize what a "piece" of a solid object is.
>Rather, some people simply mistakenly think a mathematical result
>means something it doesn't mean.
I don't know what point you are making, unless you are saying
that the order is: formal result --> inaccurate informal paraphrase,
instead of informal statement --> inaccurate formal paraphrase.
What difference does that make?
Daryl McCullough
Anyway, your claim about the counterintuitveness of the Banach
Tarski result being dependent on wild assumptions about non-measurable
sets is patently false. Maybe you don't care about whether your
observations are true or not, but that particular observation is
certainly not true. It's provably false because I am a counter-example.
The Banach Tarski theorem is counter-intuitive to ME, and I believe I
understood perfectly well that the "pieces" into which they cut a
sphere are non-measurable, and that therefore they do not correspond
to anything like a physical object.
So what you are saying is false. Not that you necessarily care.
Rupert
> stevendaryl3...@yahoo.com (Daryl McCullough) writes:
> > I don't think what you are saying makes any sense. Bringing up
> > non-measurable sets is not a way to explain the counter-intuitiveness
> > of Banach-Tarski.
>
> I didn't say it was. I said that in order for the Banach-Tarski theorem
> to strike anyone as counter-intuitive we must implicitly depend on some
> identification of arbitrary and wild sets of points with physical
> volumes etc.
>
I would think when you tell the man in the street about the Banach-
Tarski paradox, he doesn't explicitly formulate to himself some kind
of argument involving volumes, he just has the strong intuition based
on his experience with the physical world that that couldn't possibly
be the case.
David C. Ullrich
<aatu.kos...@uta.fi> wrote:
>David C. Ullrich <dull...@sprynet.com> writes:
>
>> Come on. Surely it's clear that we need someone with a _less_ keen
>> intuition to explain this.
>
>So it appears. Regarding the evidence you provided for the evidence of
>the axiom of choice (or at least dependent choice or countable choice),
>well, there is a reason the axiom of choice is an axiom and the
>well-ordering theorem a theorem. (Cantor did at one point think the
>well-ordering principle was a "law of thought", but later came around
>the view it's something that needs to be proven.) My favourite example
>of a result where it's often difficult even for people with some
>experience in such matters to spot the invocation of (countable) choice
>is the theorem that omega_1 is regular.
Ok, that's another exercise for me. What does "regular" mean here?
>> Otoh one sees the same people giving the impression that the existence
>> of an uncountable well-ordered set requires AC.
>
>This idea is not uncommon in news...
David C. Ullrich
Rupert
> On Sun, 18 Oct 2009 17:46:49 +0300, Aatu Koskensilta
>
> >David C. Ullrich <dullr...@sprynet.com> writes:
>
> >> Come on. Surely it's clear that we need someone with a _less_ keen
> >> intuition to explain this.
>
> >So it appears. Regarding the evidence you provided for the evidence of
> >the axiom of choice (or at least dependent choice or countable choice),
> >well, there is a reason the axiom of choice is an axiom and the
> >well-ordering theorem a theorem. (Cantor did at one point think the
> >well-ordering principle was a "law of thought", but later came around
> >the view it's something that needs to be proven.) My favourite example
> >of a result where it's often difficult even for people with some
> >experience in such matters to spot the invocation of (countable) choice
> >is the theorem that omega_1 is regular.
>
> Ok, that's another exercise for me. What does "regular" mean here?
>
A cardinal kappa is said to be regular if there does not exist an
increasing well-ordered sequence of ordinals, the sequence being of
length less than kappa, whose limit is kappa.
Herman Jurjus
> Herman Jurjus <hjm...@hetnet.nl> writes:
>
>> Aatu Koskensilta wrote:
>>
>>> That aside, I was wondering specifically about the claim that the
>>> well-ordering theorem itself is counter-intuitive, as alluded to in
>>> e.g. the famous quip
>>>
>>> The axiom of choice is obviously true, the well-ordering theorem
>>> obviously false -- and who can tell of Zorn's lemma?
>>>
>>> Just what intuitions are contradicted by the well-ordering theorem?
>> Imho, the quip tries to express something completely different.
>> It's not an expression of mathematician's intuitions about sets.
>> Mathematicians don't care about sets - they care about mathematics.
>
> Sure. What do you take the quip to (try to) express?
>
>> And a general purpose foundation for mathematics should ideally not
>> turn something into an indispensable truth that is, for mathematics,
>> quite dispensable.
>
> I'm not sure what you have in mind here.
Well, it was my attempt to describe what the quip tried to express, but
i think David Ullrich said it better: the quip itself -is- the best
possible expression of the intuition.
>> Btw, personally i find Freyling's argument quite appealing. But i also
>> think that AD and AC are both true, so you should better not take my
>> opinions too seriously.
>
> What are your reasons for thinking AD true?
That's a good question, and i'm still in the process of analyzing my
intuitions in order to come up with something more tangible.
I'm not sure whether anyone would be interested in more details, but
since you asked for some, here comes.
Simplistic subjective blather:
If i forget for a moment that ZFC+AD is inconsistent, and i start with a
clean sheet, so to say, and i give an honest account of what i mean with
all the notions involved (N, set, game, sequence, etc.) then it becomes
rather uncontroversial to me that either player 1 can win, or he
can't, i.e. player 2 has some defense. The degree of 'reliability' that
this has (for me) is not less than that of the power set axiom or AC
(rather the opposite).
Mathematical/pragmatic blather (short version):
If AD fails, it seems to become pointless to accept a f.o.m. in which
the LEM is an absolute truth for infinite sets.
BTW, i only have these intuitions about AD as restricted to N^N. But AD
on N^N is already in direct contradiction with the existence of a
non-trivial ultra-filter on P(N), so this combination of intuitions is
already weird enough.
Just for what it's worth:
Analyzing my intuitions has lead me so far in the direction of all kinds
of semantics that explain the difference between potentially and
actually infinite. The challenge (still open) is to come up with a
system that has both types of infinity, and which makes AD true when the
game-histories are allowed to be potentially infinite sequences, and
false when they have to be actually infinite sequences.
The weird thing is: i keep getting systems for which the 'potentially
infinite part' also allows non-trivial ultrafilters on P(N) to exist.
(I.e. the systems don't really resolve the clash of intuitions.)
>> (Not that there was any danger that you'd do that, anyway.)
>
> I take nothing too seriously.
Same here, and by all means let's keep it that way.
--
Cheers,
Herman Jurjus
David C. Ullrich
Thanks - I recalled this a little later.
Daryl McCullough
>Simplistic subjective blather:
>If i forget for a moment that ZFC+AD is inconsistent, and i start with a
>clean sheet, so to say, and i give an honest account of what i mean with
>all the notions involved (N, set, game, sequence, etc.) then it becomes
>rather uncontroversial to me that either player 1 can win, or he
>can't, i.e. player 2 has some defense. The degree of 'reliability' that
>this has (for me) is not less than that of the power set axiom or AC
>(rather the opposite).
I know that you're not offering that as a mathematical argument, but I
would like to understand your intuitions here. "Either player 1 has a
winning strategy, or player 2 has a defense" needs a little more argument
to be compelling, because of games like "Rock, paper, scissors" where
no strategy is guaranteed to win. What's the argument against the
possibility that (1) for every strategy for the first player, there
is a defense for the second player, and (2) for every defense for the
second player, there is a strategy for the first player that beats it.
I don't see any reason to believe that this CAN'T be the case. So it's
not clear to me that this has anything to do with the law of excluded
middle.
Bill Taylor
> Simplistic subjective blather:
> If i forget for a moment that ZFC+AD is inconsistent,
> and i start with a clean sheet, so to say, and i give an honest
> account of what i mean with all the notions involved
> (N, set, game, sequence, etc.) then it becomes rather
> uncontroversial to me that either player 1 can win, or he can't,
> i.e. player 2 has some defense. The degree of 'reliability' that
> this has (for me) is not less than that of the power set axiom
> or AC (rather the opposite).
I agree with all this.
Daryl McCullough responds:
> I know that you're not offering that as a mathematical argument,
> but I would like to understand your intuitions here.
> "Either player 1 has a winning strategy, or 2 has a defense"
> needs a little more argument to be compelling, because of games
> like "Rock, paper, scissors" with no strategy guaranteed to win.
OUCH! Daryl, this is not up to your usual high debating standards.
RockPS, like almost any game of incomplete information, has no
"guaranteed winning strategy", as you say, but AD games are
not of this type.
The typical AD game is written *as if* it were a game of
simultaneous choice with incomplete information, but it is
not really so, OC - it is a perfectly well defined game of
length omega where all moves are made with complete information.
Such a game *definitely* should have a winning astrategy for
one or other player!
To repeat the general game, for those unfamiliar:-
CONSECUTIVE DESCRIPTION
=======================
Player one chooses a natural number; player two (aware of
the first choice) chooses another natural number; player one
(aware of both preceding choices) chooses a third natural
number, and so on. The game thus produces a clearly defined
sequence of natural numbers. If this sequence is a member of
some pre-assigned set A, then player one wins. If this sequence
is not a member of A, then player 2 wins.
Thus the game is essentially defined/described by the set A,
a subset of N^N.
SIMULTANEOUS DESCRIPTION
========================
The two players simultaneously choose a "full strategy";
which consists of a function from finite sequences of integers,
to integers. Player 1's "first move" is given by f_1(empty),
player 2's "first move" is given by f_2(f_1(empty)), player 1's
"2nd move" by f_1(f_1(empty), f_2(f_1(empty))), and so on.
The winner is then determined as in the first case.
-------------------------------------------------------
It is surely clear that these two descriptions are isomorphic.
Indeed, the compementarity involved is faintly reminiscent
of the wave(simultaneous)/particle(consecutive) description
of quantum matter! :-)
The first description makes it "intuitively clear" that
the game should always be "determined", that is, have
a winning strategy for one or the other player, as at
every step it is a game of complete information.
It is a theorem of game theory that if the game is sure
to terminate before some particular number of moves, then it is
determined. It is a theorem of set theory that all Baire sets A
lead to a determined game. It seems very natural to a game
player that ANY set A should lead to a determined game.
Alas, it is a theorem of ZFC set theory that there are
undetermined games, that is, undetermined sets A.
(Of course, as is usual with such matters, no explicit
example can ever possibly be produced!)
It is known to be consistent with ZF, and even ZF + CC,
that all such sets A are determined.
To me, as a regular game player, it is AT LEAST as obvious
that every game should be determined, as it is obvious
that every countable cross-product should have a member, (CC);
and both are a GREAT DEAL more intuitively obvious than
that every set whatsoever should have a choice function.
> What's the argument against the possibility that
> (1) for every strategy for the first player,
> there is a defense for the second player, and
> (2) for every defense for the second player,
> there is a strategy for the first player that beats it.
These are intuitively contradictory.
> I don't see any reason to believe that this CAN'T be the case.
Viewing the simultaneous description, that may seem plausible;
it is plausable either way - i.e. that 1&2 could both be true,
or that they cannot. BUT, viewing the consecutive description,
it is just plain impossible, as there would be a sequence that
both players would win with. Intuitively.
This is assuming the informal intuitive equivalence of
"choosing a strategy once and for all" with
"choosing each move as it comes according to what's seen already".
This is an informal equivalence, so not susceptible of proof
or disproof, but in view of the isomorphism of the two game
descriptions, it seems (to me) to be unimpeachable!
I think the conflict between AD and (full) AC is one of
the starkest in math - much more convincing than
the Banach decomposition paradox.
-- Withering William
** They travel as waves but arrive as particles.
Herman Jurjus
Full agreement.
--
Cheers,
Herman Jurjus
Butch Malahide
wrote:
Can you explain why determinacy is intuitively clear when each move is
choosing a natural number, but no longer clear when each move is
choosing a countable ordinal? or a set of real numbers?
What does your intuition say about the following game? First, White
chooses a set X or real numbers; then Black chooses a real number x.
Black wins if either X is empty or x is in X; White wins if X is
nonempty and x is not in X. I'm sure you will agree, Bill, that
neither player has a winning strategy. What gives?
Daryl McCullough
>OUCH! Daryl, this is not up to your usual high debating standards.
>RockPS, like almost any game of incomplete information, has no
>"guaranteed winning strategy", as you say, but AD games are
>not of this type.
I disagree. The sort of games that are discussed in AD
*are* games of incomplete information. Each player
decides his strategy *before* seeing any moves by the
other player. I should say, rather, there is no disadvantage
to deciding your strategy ahead of time.
A strategy, remember, is a rule for how you *respond*
to moves by the other player. Any possible way that you
could take into account the information about the
actual moves the other player makes can be folded into
the strategy you adopt at the very beginning.
>The typical AD game is written *as if* it were a game of
>simultaneous choice with incomplete information, but it is
>not really so, OC - it is a perfectly well defined game of
>length omega where all moves are made with complete information.
You don't have complete information, because you don't
know what strategy your opponent is going to follow. In
a *finite* game, this lack of information is eliminable,
because you can assume that your opponent will play
optimally, and for finite games there always *is* a notion
of an optimal strategy. The existence of an optimal
strategy is a *theorem* of finite games, you have to
prove it, you can't just assume it.
The existence of a winning strategy on the part of
one player or the other is provable for finite games
using the fact that finite games are *well-founded*.
You prove by induction (possibly transfinite induction)
that one player or the other has a winning strategy.
A finite game can be described as a well-founded tree,
with the arcs labeled by moves. Once you take your
first move, you have reduced the situation to a simpler
game (set-theoretically, the tree has a lower rank
in the set-theoretic universe). By induction on the
rank of the tree, you can prove that every tree
has a winning strategy for one player or the other.
But infinite games correspond to non-well-founded trees.
You can't get started to prove by induction that there
is a winning strategy.
>Such a game *definitely* should have a winning astrategy for
>one or other player!
Why? Here's a more mathematical description of the game:
The players take turns picking natural numbers. This
produces an infinite sequence x_1 x_2 .... There is a
set W of infinite sequences of natural numbers that count
as a "win" for the first player.
At each move, the player is determining his next move
from the finite sequence of naturals played so far. So
a strategy is a function from finite sequences of naturals
to naturals. Let play(f,g) be the infinite sequence resulting
from the first player following strategy f and the second
player following strategy g.
The game has a winning strategy for the first player if:
exists f, forall g, play(f,g) in W
The game has a winning strategy for the second player if:
exists g, forall f, play(f,g) not in W
These two statements are *not* negations of each other.
There is no reason necessarily to assume that one must
be true or the other must be true.
But *why* does a game of complete information imply
that there is a winning strategy for one player or
the other? That's a *theorem* that must be proved.
>> I don't see any reason to believe that this CAN'T be the case.
>
>Viewing the simultaneous description, that may seem plausible;
>it is plausable either way - i.e. that 1&2 could both be true,
>or that they cannot. BUT, viewing the consecutive description,
>it is just plain impossible, as there would be a sequence that
>both players would win with. Intuitively.
I don't see that.
>This is assuming the informal intuitive equivalence of
>
>"choosing a strategy once and for all" with
>"choosing each move as it comes according to what's seen already".
>
>This is an informal equivalence, so not susceptible of proof
>or disproof, but in view of the isomorphism of the two game
>descriptions, it seems (to me) to be unimpeachable!
I think you can prove that there is no disadvantage to choosing
a strategy once and for all, as opposed to one move at a time.
Daryl McCullough
>> I know that you're not offering that as a mathematical argument,
>> but I would like to understand your intuitions here.
>> "Either player 1 has a winning strategy, or 2 has a defense"
>> needs a little more argument to be compelling, because of games
>> like "Rock, paper, scissors" with no strategy guaranteed to win.
>
>OUCH! Daryl, this is not up to your usual high debating standards.
>RockPS, like almost any game of incomplete information, has no
>"guaranteed winning strategy", as you say, but AD games are
>not of this type.
Actually, could you explain in what sense chess or checkers are
games of complete information, and rock/paper/scissors is *not*?
In all cases, each player must make his move in ignorance of
what move his opponent will make. There is no "hidden state"
(such as the hidden playing cards in a game of cards), unless
you count the intentions of your opponent as hidden state, in
which case, all games are games of incomplete information.
Herman Jurjus
> Bill Taylor says...
>> The first description makes it "intuitively clear" that
>> the game should always be "determined", that is, have
>> a winning strategy for one or the other player, as at
>> every step it is a game of complete information.
>
> But *why* does a game of complete information imply
> that there is a winning strategy for one player or
> the other? That's a *theorem* that must be proved.
Perhaps the point is not so much that it is completely self-evident that
AD is true, but that our reasons for believing AC (or power set) are at
least as superficial and shaky, if not more so.
(Personally, i can only make sense of sets larger than P(N) when i go
into 'sloppy reasoning mode'. And in this sloppy reasoning mode, AD is
at least as plausible as AC.)
--
Cheers,
Herman Jurjus
Aatu Koskensilta
> Perhaps the point is not so much that it is completely self-evident
> that AD is true, but that our reasons for believing AC (or power
> set) are at least as superficial and shaky, if not more so.
What is evident to some might be completely opaque to others. Often
all we can say about such things is that many people do in fact find
this or that evident. This is certainly the case with the axiom of
choice, but not with determinacy. I can think of only one instance in
the literature of anyone arguing that determinacy can be seen to be
true on basis of informal reflection (this reflection having to do
with "infinitely intelligent" players and what not). In contrast,
choice seems to strike many as evident, both in the sense people
explicitly state so, and in the sense that arguments invoking (often
implicitly) choice are regarded as convincing and compelling. (The
latter observation is probably the more interesting one.) Our ideas
about what is or is not evident are of course not arbitrary, and we
can bring to bear considerations of less subjective or relative
character. For example, we may note that choice is an innocent
principle in a quasi-Hilbertian sense, in that it doesn't have any
"concrete" consequences, while determinacy is a (moral) large cardinal
axiom, of staggering consistency strength, and in particular implies
all the arithmetical consequences of "there are infinitely many Woodin
cardinals". Such considerations are by no means conclusive -- if
someone is sufficiently impressed by the evidence of determinacy they
will naturally regard this as powerful evidence for (at least the
arithmetical consequences of) infinitely many Woodin cardinals! Again,
it is an empirical observation that no set theorist seems to take this
view.
I see I have a sizable backlog of messages to address on sci.logic, so
I'll limit general blather about evidence to these haphazard remarks.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, dar�ber muss man schweigen"
Herman Jurjus
> Herman Jurjus <hjm...@hetnet.nl> writes:
>
>> Perhaps the point is not so much that it is completely self-evident
>> that AD is true, but that our reasons for believing AC (or power
>> set) are at least as superficial and shaky, if not more so.
>
> What is evident to some might be completely opaque to others. Often
> all we can say about such things is that many people do in fact find
> this or that evident. This is certainly the case with the axiom of
> choice, but not with determinacy. I can think of only one instance in
> the literature of anyone arguing that determinacy can be seen to be
> true on basis of informal reflection (this reflection having to do
> with "infinitely intelligent" players and what not). In contrast,
> choice seems to strike many as evident, both in the sense people
> explicitly state so
Might i humbly suggest that that is explainable by the fact that many
people are confronted with choice in the earliest stages of their
studies, and not with AD?
And what would have happened if ZFC+AD had not been inconsistent? Or if
the inconsistency had not been discovered in the 1950s, but, say, in 2005?
> , and in the sense that arguments invoking (often
> implicitly) choice are regarded as convincing and compelling. (The
> latter observation is probably the more interesting one.)
That's funny. I'd rather say that the interest in ZF-without-AC
is evidence for the contrary, namely the fact that practically all
arguments crucially requiring (full) AC give the impression of being not
only not convincing at all, but of being particularly unenlightening,
like the invoking of a magic stick.
It does often come in handy, so i do understand it's /popular/. But
'convincing and compelling'?
> Our ideas
> about what is or is not evident are of course not arbitrary, and we
> can bring to bear considerations of less subjective or relative
> character. For example, we may note that choice is an innocent
> principle in a quasi-Hilbertian sense, in that it doesn't have any
> "concrete" consequences, while determinacy is a (moral) large cardinal
> axiom, of staggering consistency strength, and in particular implies
> all the arithmetical consequences of "there are infinitely many Woodin
> cardinals".
How many people who accept AC (and reject AD) know the above, and for
how many is/was it an argument? Should it be?
I would like to stress, though, that i didn't defend AD at the cost of
AC, but instead said that my intuitions regarding sets make AD and AC
/both/ come out as true. AC being acceptable a.o. on the grounds that
you mention above: it's arithmetically conservative.
> Such considerations are by no means conclusive -- if
> someone is sufficiently impressed by the evidence of determinacy they
> will naturally regard this as powerful evidence for (at least the
> arithmetical consequences of) infinitely many Woodin cardinals! Again,
> it is an empirical observation that no set theorist seems to take this
> view.
Fwiw: the way i understand AD to be true is not incompatible with this
empirical observation.
Note, though, that the sample in this thread is not in accordance with
your empirical observation (four participants, two find AD compelling, a
third gives the strong impression that he simply hasn't given it a good
thought, yet.)
> I see I have a sizable backlog of messages to address on sci.logic, so
Apparently you get more sci.logic traffic than we do. What's your
newsserver?
> I'll limit general blather about evidence to these haphazard remarks.
Yah; it doesn't seem to make sense to continue much further with this.
--
Cheers,
Herman Jurjus
Herman Jurjus
> Bill Taylor says...
>
>>> I know that you're not offering that as a mathematical argument,
>>> but I would like to understand your intuitions here.
>>> "Either player 1 has a winning strategy, or 2 has a defense"
>>> needs a little more argument to be compelling, because of games
>>> like "Rock, paper, scissors" with no strategy guaranteed to win.
>> OUCH! Daryl, this is not up to your usual high debating standards.
>> RockPS, like almost any game of incomplete information, has no
>> "guaranteed winning strategy", as you say, but AD games are
>> not of this type.
>
> Actually, could you explain in what sense chess or checkers are
> games of complete information, and rock/paper/scissors is *not*?
Now it's my turn to say 'OUCH!'
Can you explain in what way chess or checkers are /not/ games of
complete information?
> In all cases, each player must make his move in ignorance of
> what move his opponent will make. There is no "hidden state"
> (such as the hidden playing cards in a game of cards), unless
> you count the intentions of your opponent as hidden state, in
> which case, all games are games of incomplete information.
Only if you mis-characterize them as 'both players simultaneously choose
a strategy, and keep following these'. But that's not what happens when
two people play chess.
It really /does/ make a big difference whether a game is of complete
information or not. And it's not a coincidence that the (finite) games
of complete information are all determined. Likewise, it's not a
coincidence that it's non-trivial to find concrete (i.e. ZF definable)
counterexamples against AD.
Look, there may be any amount of valid reasons to be suspicious of AD,
or to dislike it, or whatever. But your objection is not one of them, imho.
--
Cheers,
Herman Jurjus
Herman Jurjus
> Can you explain why determinacy is intuitively clear when each move is
> choosing a natural number, but no longer clear when each move is
> choosing a countable ordinal? or a set of real numbers?
>
> What does your intuition say about the following game? First, White
> chooses a set X or real numbers; then Black chooses a real number x.
> Black wins if either X is empty or x is in X; White wins if X is
> nonempty and x is not in X. I'm sure you will agree, Bill, that
> neither player has a winning strategy. What gives?
Huh? Black clearly has a winning strategy, no?
--
Cheers,
Herman Jurjus
Daryl McCullough
>
>Daryl McCullough wrote:
>> Bill Taylor says...
>>
>>>> I know that you're not offering that as a mathematical argument,
>>>> but I would like to understand your intuitions here.
>>>> "Either player 1 has a winning strategy, or 2 has a defense"
>>>> needs a little more argument to be compelling, because of games
>>>> like "Rock, paper, scissors" with no strategy guaranteed to win.
>>> OUCH! Daryl, this is not up to your usual high debating standards.
>>> RockPS, like almost any game of incomplete information, has no
>>> "guaranteed winning strategy", as you say, but AD games are
>>> not of this type.
>>
>> Actually, could you explain in what sense chess or checkers are
>> games of complete information, and rock/paper/scissors is *not*?
>
>Now it's my turn to say 'OUCH!'
>Can you explain in what way chess or checkers are /not/ games of
>complete information?
I just explained what I meant. In ALL games, the unknown is what
your opponent will do. That's true for chess, checkers,
rock/paper/scissors. In the case of games of incomplete
information, there is ADDITIONAL information to be had
(the contents of hidden cards, in a card game, for example)
BESIDES your opponent's future choices.
>> In all cases, each player must make his move in ignorance of
>> what move his opponent will make. There is no "hidden state"
>> (such as the hidden playing cards in a game of cards), unless
>> you count the intentions of your opponent as hidden state, in
>> which case, all games are games of incomplete information.
>
>Only if you mis-characterize them as 'both players simultaneously choose
>a strategy, and keep following these'. But that's not what happens when
>two people play chess.
Mathematically, it doesn't make any difference whether you choose
a strategy all at once, or choose it move by move, taking into account
what your opponent played. There is, of course, a *practical* difference,
in that the computational difficulty of thinking many moves ahead means
that in practice, you adopt a short-term strategy, and then revise
it in light of new information. But surely you agree that it makes
no mathematical difference, right? The short-term strategy modified
in light of developments is mathematically equivalent to a particular
long-term strategy.
Anyway, if you are talking about the plausibility of determinism,
it certainly isn't people's experience that chess has a winning
strategy. It might be mathematically provable, but it's not a
fact that is self-evident, and it's not a fact that comes into
play much in actual games.
>It really /does/ make a big difference whether a game is of complete
>information or not.
You haven't defined what "a game of complete/incomplete information"
means. The Wikipedia definition says: "Complete information is a term
used in economics and game theory to describe an economic situation or
game in which knowledge about other market participants or players is
available to all participants. Every player knows the payoffs and
strategies available to other players."
It seems to me that by that definition, rock/paper/scissors IS a
game of complete information. The difference with a game like
chess is not whether its complete information or not, but whether
the moves are alternating or simultaneous.
>Look, there may be any amount of valid reasons to be suspicious of AD,
>or to dislike it, or whatever. But your objection is not one of them, imho.
Well, certainly you would agree that IF the infinite game is played
in the following way, then it is the same situation as rock/paper/scissors:
Each player chooses a strategy, which is a deterministic function
from finite sequences of naturals to finite sequences of naturals.
Then we compute an infinite sequence by interleaving applications
of the two strategies. If the result is in the given set, the first
player wins, otherwise, the second player wins.
With this description, there is no reason to believe that there is
a winning strategy for either player, any more than there is for
rock/paper/scissors. You might not like to think of it in terms
of choosing a strategy, rather than choosing naturals one at a
time, but I can't see, mathematically, how it could make any
difference. Maybe you can explain that?
Daryl McCullough
>Might i humbly suggest that that is explainable by the fact that many
>people are confronted with choice in the earliest stages of their
>studies, and not with AD?
>
>And what would have happened if ZFC+AD had not been inconsistent? Or if
>the inconsistency had not been discovered in the 1950s, but, say, in 2005?
I don't understand why you think that "all games are determined" is
intuitively true. It's not *obvious* that chess or checkers has a
winning strategy; it's *provable*. To prove it, you have to use the
fact that they are finite-length games (actually, the possibility of
ties or cycles makes chess not precisely a finite sequential game, but
you can change the question to: is there some strategy that can guarantee
avoiding a loss?) What reason is there for believing that the principle
applies to games for which is not provable?
Butch Malahide
Yes and no.
A strategy for Black (as I understand it, but maybe you mean something
different) is a function f which assigns, to each possible White move
(set of reals) X, a real number f(X) as Black's reply to X. To say
that f is a *winning* strategy for Black means that, whenever X is a
nonempty set of real numbers, f(x) is an element of X. In other words,
a winning strategy for Black is nothing more nor less than a choice
function for the nonempty subsets of the real line; the existence of
such a fuction is, of course, equivalent to the existence of a well-
ordering of the real line.
To answer your question: I agree that Black clearly has a winning
strategy, because the axiom of choice is true. However, anyone who
believes that AD is true (and so there is no well-ordering of the
reals) will have to admit that *this* game is undetermined. While us
AC-believers can go all the way and assume that the whole universe is
well-ordered, anyone claiming that AD is "intuitively obvious" can
*not* (consistently) assume determinacy for all games large and small,
and should be asked to explain why the intuition only applies to games
of limited size.
Herman Jurjus
> Well, certainly you would agree that IF the infinite game is played
> in the following way, then it is the same situation as rock/paper/scissors:
>
> Each player chooses a strategy, which is a deterministic function
> from finite sequences of naturals to finite sequences of naturals.
> Then we compute an infinite sequence by interleaving applications
> of the two strategies. If the result is in the given set, the first
> player wins, otherwise, the second player wins.
>
> With this description, there is no reason to believe that there is
> a winning strategy for either player, any more than there is for
> rock/paper/scissors. You might not like to think of it in terms
> of choosing a strategy, rather than choosing naturals one at a
> time, but I can't see, mathematically, how it could make any
> difference. Maybe you can explain that?
So we have two descriptions, which you claim to be equivalent.
(I'm not so sure that the two are conceptually the same, but let's leave
that, for now.)
With one of these descriptions, a certain conclusion is not
self-evident, with the other it is (at least for me; you don't agree
with the latter).
For me, the equivalence between the situations would lead me to accept
the conclusion in the second case as well. You, on the other hand, seem
to reject the conclusion in the first case, on the grounds that it is
not self-evident in the second case.
I cannot look inside your brain, but could it be that, so far, you
simply /didn't/ intuitively evaluate the first description for yourself
at all? I.e. that you just replaced it in your mind with the second
description, and judged the situation based on that second description,
while further totally ignoring the first description?
[Because, in retrospect, that's what /i/ always did when thinking about
AD, in the past.]
--
Cheers,
Herman Jurjus
Herman Jurjus
Well, don't look at me; i said right from the start that both AD and AC
are evidently true.
--
Cheers,
Herman Jurjus
Butch Malahide
>
> Well, don't look at me; i said right from the start that both AD and AC
> are evidently true.
Not much use, then, in trying to use reductio ad absurdum on you,
then, is there? Actually, my comment was directed to Bill Taylor, who
is going to ignore it because he doesn't have a good answer.
Daryl McCullough
>I cannot look inside your brain, but could it be that, so far, you
>simply /didn't/ intuitively evaluate the first description for yourself
>at all? I.e. that you just replaced it in your mind with the second
>description, and judged the situation based on that second description,
>while further totally ignoring the first description?
You're right, that I substituted an alternative description that I
thought was more mathematically tractable to reason about. So I'll
try to stay to the first description: First player
selects a natural, then the second player selects a natural in
response, then back to the first player, etc. But I don't understand
how this description intuitively suggests that one player or the
other has a winning strategy.
In a finite game, we can always in theory work backwards from
a winning position. Let W_0 be the set of winning positions
for the first player. Back up two moves and let W_1 be the
set of positions such that there is a move by the first player
such that no matter what countermove the second player makes,
he will end up in a position in W_0. Keep backing up to form
W_0, W_1, W_2, etc.
Now, let W = the union of all the W_i. This is the set of all
positions such that it is possible for the first player to force
a win, starting in that position. If the starting position is included
in set W, then the first player can force a win. If not, then the
second player can *avoid* the positions in W. In other words, the
second player can avoid losing. But in a finite game, if the
second player avoids losing for long enough, then he wins.
This argument doesn't work for infinite games, because
there is no way to "back up" from a winning position.
Aatu Koskensilta
> Look, there may be any amount of valid reasons to be suspicious of AD,
> or to dislike it, or whatever. But your objection is not one of them,
> imho.
A very simple reason to be suspicious of AD is that it amounts to a
strong form of quantifier switch that is in general not valid. (I really
couldn't make anything of your comments about the law of excluded
middle, which isn't involved.) I'm utterly baffled by the idea that AD
might be as evident as choice -- although of course it's a perfectly
understandable position that both are horribly and equally dubious --
since for me choice is simply a mathematical expression of the idea that
sets are arbitrary collections, an idea pleasing to the intellect,
allowing me to make mathematical use of various intuitively palatable
informal notions, conceptions, ideas, principles, while determinacy is
an assertion about sets of reals about which I really have no intuitions
whatever, certainly not merely on basis of my (mathematically very firm)
grasp of the conceptual picture involved. (This is of course a very
boring observation, since it amounts to merely stating I conceive of
sets in such a manner as to make choice a triviality. The only interest
in it in this context is that regarding AD we can't make the analogous
boring observation. The observation holds for all the other usual axioms
-- and various small large cardinal axioms -- as well, with the possible
exception of replacement. Those who declare that choice, powerset,
etc. aren't evident aren't presumably objecting to this, but to the
conception involved itself, finding it "vague", "indeterminate",
"theological", and so on. I base this on my experience that e.g. ardent
and doctrinaire intuitionist are just as capable as I am of explaining
why e.g. separation, infinity, and so on, naturally flow from the image
of the world of sets provided by the narrative of the cumulative
hierarchy. G�ran Sundholm once put it to me, that "There's too much
slack in the classical 'meaning explanation' for set theoretic talk".)
Your stance, that both choice and determinacy are true, calls for some
further elucidation -- these principles are after all contradictory! The
usual set theoretic "intuition" is that determinacy holds for some
restricted class of sets, the obvious choice for this class being
identified with L(R) very soon after the introduction of the axiom --
which Mycielski didn't propose as a truth about sets, but precisely as a
principle that might be fruitfully studied, and hold, in context of some
subuniverse of sets -- and this is precisely what is borne out in the
study of large cardinals. Indeed, I think /both/ large large cardinals
and (quasi-projective) determinacy derive their plausibility in a large
extent from the various and systematic results about interconnections
between these fields. (Kanamori and Moschovakis are excellent sources on
this. And, Maddy's articles on set theoretic axioms haven't lost any of
their currency by being less current owing to the passage of time.)
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
Aatu Koskensilta
> Well, don't look at me; i said right from the start that both AD and
> AC are evidently true.
What do you mean by this? On the face of it, it makes about as much
sense as thunderously declaiming that there are infinitely many twin
primes but there are only finitely many twin primes. Perhaps you hold
some doctrine that set theoretic talk, in contrast to talk about
naturals, say, is to be interpreted in some not-at-face-value manner,
allowing for your stance, which, on a flat-footed, common-sense view, is
rather odd?
Aatu Koskensilta
My apologies for butting in, but what was it that you asked? Bill is a
sheep-shagging fence-sitter, and a notorious proponent of an apparently
irremediably vaguely formulated form of "definitionalism" when it comes
to sets. This with all respect to Bill, who will no doubt take it all in
good humour.
Aatu Koskensilta
> Aatu Koskensilta wrote:
>> In contrast, choice seems to strike many as evident, both in the
>> sense people explicitly state so
>
> Might i humbly suggest that that is explainable by the fact that many
> people are confronted with choice in the earliest stages of their
> studies, and not with AD?
There's no question that many people explicitly state they find choice
evident is because choice is a very commonly mentioned, invoked and
treated a principle. I wonder, though, whether you meant to suggest
people find choice evident on reflection and do not similarly find
determinacy evident simply because they're exposed to choice earlier in
their studies? This is a highly dubious claim. Set theorists have found,
to various degrees, principles such as Vopenka's principle, the
existence of a measurable, etc. compelling, on all sorts of grounds, but
these people, who should know about this stuff, have never claimed to
find determinacy similarly compelling, evident, clear, although many
have of course offered the opinion that they think it's likely a correct
principle for L(R).
> And what would have happened if ZFC+AD had not been inconsistent? Or if
> the inconsistency had not been discovered in the 1950s, but, say, in 2005?
I'm not sure what's the relevance of this hypothetical.
>> , and in the sense that arguments invoking (often
>> implicitly) choice are regarded as convincing and compelling. (The
>> latter observation is probably the more interesting one.)
>
> That's funny. I'd rather say that the interest in ZF-without-AC is
> evidence for the contrary, namely the fact that practically all
> arguments crucially requiring (full) AC give the impression of being
> not only not convincing at all, but of being particularly
> unenlightening, like the invoking of a magic stick.
This is a pipe-dream. There's no widespread interest in ZF without
choice, and most mathematicians are simply unable to even recognise
invocations of choice -- as already mentioned, I doubt anyone who hasn't
made it their business to study independence results in set theory
realises the result that omega_1 has uncountable cofinality requires
countable choice. Further, even though appeals to choice are
occasionally noted, we do, by and large, accept as true a result if
presented with a proof using (implicitly or explicitly) choice. (David's
message on this, which accords with my experience, was very much to the
point. Here we must of course stress, as I think you did, that usually
implicit uses of choice reduce back, in so far as logical necessity is
considered, to countable or dependent choice -- this is not surprising,
since in ordinary mathematics we rarely meet stuff where these two
principles wouldn't be sufficient. What we are make of this is an
erudite matter, since a logical analysis, establishing whether this or
that result logically requires choice, can't answer questions about
whether the principle, and which form of the principle, was in fact
"implicitly" used by this or that author; we need rather try and figure
out what sort of modes of reasoning, ideas, plan of attack, guided the
author in arriving at the proof -- I presented just this observation to
Shapiro, when he gave a talk on "self-evidence" of various (set
theoretic) axioms, and observed in particular it's obscure why we should
see in this or that paper an "implicit" appeal to choice, and not
dependent choice, or, say, global choice.)
> It does often come in handy, so i do understand it's /popular/. But
> 'convincing and compelling'?
Yes, convincing and compelling. I have done some (very limited)
empirical research on this, presenting choiceful proofs to logically
innocent mathematicians and asking them if they find anything
amiss. No-one objected to the constructions involving choice (even
choice beyond dependent choice). I've had similar results with proofs
using replacement. Usually, the lab rats were surprised when I explained
the proofs involve (classically, and nowadays almost solely
historically) controversial set theoretic principles. This goes with my
long-held thesis, that the justification of the axioms of set theory
does not come from the conceptual or philosophical analysis of the
cumulative hierarchy -- this analysis or picture is simply an
/explication/, not a justification -- but rather from the simple
observation (found already in Peano) that these axioms simply say the
set theoretic universe is closed under the operations mathematicians
routinely (and without any worries or doubts) apply in their work
(beginning with Peano, Cantor, Dedekind, etc.).
>> Our ideas about what is or is not evident are of course not
>> arbitrary, and we can bring to bear considerations of less subjective
>> or relative character. For example, we may note that choice is an
>> innocent principle in a quasi-Hilbertian sense, in that it doesn't
>> have any "concrete" consequences, while determinacy is a (moral)
>> large cardinal axiom, of staggering consistency strength, and in
>> particular implies all the arithmetical consequences of "there are
>> infinitely many Woodin cardinals".
>
> How many people who accept AC (and reject AD) know the above, and for
> how many is/was it an argument? Should it be?
Very few, probably, if we understand by "people who accept AC"
mathematicians who would not shy from invoking (often unwittingly)
choice in their work. I'll now trot out an irrelevant hobby-horse. Set
theorists are in the habit of talking about "equiconsistency results"
etc. This is totally silly, since the results are /always/ stronger,
establishing arithmetical conservativeness, strong reflection principles
for set theoretic statements, and so on, on general grounds; we know,
from our logical studies, even if it's not very often explicitly stated,
that forcing, inner models, and all the methods, with the exception of
G�delian stuff, we have at our disposal for proving independence
results, equiconsistency results, and such like, always respect
arithmetical truths, truth below a rank, and so forth and so on. There's
not much hope of changing this ingrained terminology, but (again in my
experience) it does actually lead to some confusion in the mind of the
innocent student, and is something that, while appreciate by everyone
who knows their set theory, is often missed by the less savvy outsider.
> I would like to stress, though, that i didn't defend AD at the cost of
> AC, but instead said that my intuitions regarding sets make AD and AC
> /both/ come out as true. AC being acceptable a.o. on the grounds that
> you mention above: it's arithmetically conservative.
There is a very natural idea, that stuff in arithmetic, in finitary
combinatorics, statements about objects in (intuitionistically
acceptable, say) inductively defined classes in a general sense, are
more determinate, objective, unambigous, than what we find in, say,
higher set theory, by which we may here understand just the application
of the "set-of" -operation a few times over the naturals, the reals,
function spaces, groups, etc. It is surprisingly difficult to elucidate
this natural attitude or idea, and explain in some detail just how talk
of arbitrary sets of naturals is "more vague", "more indeterminate",
"more theological" than talk of, say, arbitrarily large naturals. All
these notions necessarily rely on our faculties of mathematical
imagination, and unless we adopt some theses about these faculties, or
their reach to some objective realm of mathematical objects, or their
transcendental conditions, or whatever, it's difficult to see why, on
any other than purely pragmatic grounds, the arithmetical
conservativeness of choice should be an argument in its support. (Here
by pragmatic grounds I have in mind just the purely mathematical
observation that if e.g. a proof of Goldbach's conjecture should rely on
choice, this reliance is necessarily inessential.) (Another complaint
against set theorists: invariance under forcing, for the theory of
hereditarily countable sets, for the next iteration, and so on, is often
talked about in terms of "completeness" -- if we have enough large
cardinals the theory of this or that mathematically important collection
of objects is complete, in the sense that we can't disturb it by moving
to a generic extension of the universe. This is a mathematically
fruitful idea, and also very appealing, from an intuitive point of view,
on which view mathematical statements we can't make true or false,
willy-nilly, by model-theoretic means which don't really have any
mathematically or conceptually obvious choice for "correctness", are
somehow determinate, objective, what have you, in a contingent sort of
way. But those of who are philosophically inclined, in the warped sense
I am, naturally ask, Just why is invariance under forcing conceptually
significant?)
> Fwiw: the way i understand AD to be true is not incompatible with this
> empirical observation.
I'm afraid I don't understand your line of thought here. Does it not
give you pause that AD implies the consistency (and all the arithmetical
consequences, and more) of the large large cardinal axiom that there be
infinitely many Woodin cardinals? Do you regard this fact as an argument
for the existence (or arithmetical soundness) of an infinity of Woodin
cardinals (perhaps, to borrow a phrase from Kanamori, "in the clarity of
an inner model")?
> Note, though, that the sample in this thread is not in accordance with
> your empirical observation (four participants, two find AD compelling,
> a third gives the strong impression that he simply hasn't given it a
> good thought, yet.)
My opinions and intuitions on these matters are of very little interest,
and so are, sadly, those of the others who have said their say here --
we are not set theorists, and not a representative sample of
mathematicians, and so a brute fact is that our opinions carry very
little weight. I may be presumptuous in assuming that I've probably gone
through more large cardinal arcana (and descriptive set theory arcana)
than the other participants, but, alas, even this doesn't really entitle
me to have any particularly informed opinion or "intuition" on the
evidence of AD, Woodin cardinals, what not. Thus my coy insistence on
deferring to what people in general, and what set theorists in
particular, have to say about AD and AC, and what is implicit in their
thinking and mathematical work.
>> I see I have a sizable backlog of messages to address on sci.logic, so
>
> Apparently you get more sci.logic traffic than we do. What's your
> newsserver?
I have a mobile Internet connection through DNA, and apparently it
provides a newsserver courtesy of Saunalahti (or Elisa). I really must
answer a few of Daryl's posts, having detected a hint of exasperation in
some of his replies, and having committed to the eternal archives of
Usenet a few false claims, which claims Daryl found objectionable, owing
to which state of affairs I should publicly flagellate myself. I seem to recall
there were a few posts by Nam, too, that require my urgent attention --
in particular a baffling argument, according to which there's something
wrong (and "non-syntactical") in the perfectly fine finitist proof of
the consistency of Robinson arithmetic in Shoenfield. As you can surely
appreciate, these matters Usenetical can't possibly be glossed over.
> Yah; it doesn't seem to make sense to continue much further with this.
Indeed. I trust my continuing much further with this pleases or vexes
you no end.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
Butch Malahide
> Butch Malahide <fred.gal...@gmail.com> writes:
> > On Oct 27, 11:17 am, Herman Jurjus <hjm...@hetnet.nl> wrote:
>
> >> Well, don't look at me; i said right from the start that both AD and
> >> AC are evidently true.
>
> > Not much use, then, in trying to use reductio ad absurdum on you,
> > then, is there? Actually, my comment was directed to Bill Taylor, who
> > is going to ignore it because he doesn't have a good answer.
>
> My apologies for butting in, but what was it that you asked? Bill is a
> sheep-shagging fence-sitter, and a notorious proponent of an apparently
> irremediably vaguely formulated form of "definitionalism" when it comes
> to sets. This with all respect to Bill, who will no doubt take it all in
> good humour.
Observing that, unlike choice, which can hold "all the way up",
determinacy can only hold for games of limited size, I wanted Bill (or
someone who claims that AD is "intuitive") to explain, if it is
"intuitively obvious" that determinacy holds for games where a play is
a sequence of natural numbers (or real numbers), why the same
intuition does not lead (falsely) to determinacy of games where a play
is a sequence of countable ordinals (or sets of real numbers)?
Aatu Koskensilta
> Observing that, unlike choice, which can hold "all the way up",
> determinacy can only hold for games of limited size, I wanted Bill (or
> someone who claims that AD is "intuitive") to explain, if it is
> "intuitively obvious" that determinacy holds for games where a play is
> a sequence of natural numbers (or real numbers), why the same
> intuition does not lead (falsely) to determinacy of games where a play
> is a sequence of countable ordinals (or sets of real numbers)?
Well, I couldn't really make any of the arguments for determinacy that
were presented. But your observation is related to the only instance of
anyone in the literature proposing an (informal) argument for the truth
of determinacy. The idea, as far as I've managed to make out, is that we
have two "infinitely clever" players, who know all facts about (sets of)
reals, and so should be able to use this their infinite wisdom to win
any winnable game. This idea (which, I think, was, probably whimsically,
put forth by Mycielski and Steinhaus) is obviously wrong-headed since it
obviously implies /all/ games, whatever their nature, and lofty location
in the set theoretic hierarchy, are determined. In all fairness to the
AD proponents in this thread, it should be observed that Rheinhardt's
arguments for the "ultimate large cardinal axiom", that there be a
non-trivial elementary embedding of the set theoretic universe in
itself, were pure waffle of similarly exacting exactitude. (Maddy puts
this somewhat more diplomatically in her /Believing the Axioms/
papers...)
Butch Malahide
> Butch Malahide <fred.gal...@gmail.com> writes:
> > Observing that, unlike choice, which can hold "all the way up",
> > determinacy can only hold for games of limited size, I wanted Bill (or
> > someone who claims that AD is "intuitive") to explain, if it is
> > "intuitively obvious" that determinacy holds for games where a play is
> > a sequence of natural numbers (or real numbers), why the same
> > intuition does not lead (falsely) to determinacy of games where a play
> > is a sequence of countable ordinals (or sets of real numbers)?
>
> Well, I couldn't really make any of the arguments for determinacy that
> were presented. But your observation is related to the only instance of
> anyone in the literature proposing an (informal) argument for the truth
> of determinacy. The idea, as far as I've managed to make out, is that we
> have two "infinitely clever" players, who know all facts about (sets of)
> reals, and so should be able to use this their infinite wisdom to win
> any winnable game. This idea (which, I think, was, probably whimsically,
> put forth by Mycielski and Steinhaus) is obviously wrong-headed since it
> obviously implies /all/ games, whatever their nature, and lofty location
> in the set theoretic hierarchy, are determined. In all fairness to the
> AD proponents in this thread, it should be observed that Rheinhardt's
> arguments for the "ultimate large cardinal axiom", that there be a
> non-trivial elementary embedding of the set theoretic universe in
> itself, were pure waffle of similarly exacting exactitude. (Maddy puts
> this somewhat more diplomatically in her /Believing the Axioms/
> papers...)
You mean William Reinhardt, not "Rheinhardt". Not a spelling flame,
just trying to be helpful in case anyone reading this hasn't heard of
Reinhardt before and wants to look up his work. Thanks for your reply.
Aatu Koskensilta
> Aatu Koskensilta wrote:
>
>> Our ideas about what is or is not evident are of course not
>> arbitrary, and we can bring to bear considerations of less subjective
>> or relative character. For example, we may note that choice is an
>> innocent principle in a quasi-Hilbertian sense, in that it doesn't
>> have any "concrete" consequences, while determinacy is a (moral)
>> large cardinal axiom, of staggering consistency strength, and in
>> particular implies all the arithmetical consequences of "there are
>> infinitely many Woodin cardinals".
>
> How many people who accept AC (and reject AD) know the above, and for
> how many is/was it an argument? Should it be?
I forgot to expound in detail (ha-ha!) on your last question. What I
wrote above is a mouthful, and I'm unsure just what you're asking should
or should not be an argument, and for what. I already indirectly
expressed the opinion that the arithmetical conservativeness -- the
conservativity results can be extended a few levels up in the analytic
hierarchy, using Shoenfield's absoluteness lemma and a few more refined
devices -- of choice isn't a good argument for choice. This is mainly
because mathematicians virtually never appeal to this fact in their
proofs (and, as you so shrewdly imply, are very rarely even aware of the
fact) or in their accounts of their mathematical dealings with sets. In
light of this it would be odd to regard the conservativity as an
argument for choice, at least in any sense of "argument" that has
anything to do with actually convincing people.[1] AD's strength is, on
the other hand, something I'd expect anyone who's even superficially
acquainted with these matters to be fully aware of. And it's simply
impossible to find any mathematician unwittingly relying on AD to prove
this or that. My comment above was, to a large extent, subjunctive [my
sense of English says this is a very odd choice of word, but it seems to
be in accord with at least pretentious academic usage] in that I would
simply surmise that if someone were, by their keen intuition, by "a sort
of persuasion", by being hit on the head with a brick, immediately
struck with the obvious truth of determinacy, the observation that the
arithmetical soundness of an infinity of large large cardinals follows
would probably give them pause. (After all, most people who regard
choice as evident, acceptable, obvious, something it's interesting to
study the consequences of, are comforted by the (commonly half-digested)
knowledge it's relatively consistent. And most people are, often for no
deep reason, but simply for not having been exposed to such matters,
suspicious of large cardinals, let alone large large cardinals. This
comment I again base on the observation that people often go on about
such matters in a most naive and innocent manner. as you yourself have
no doubt witnesses in news, and in other venues.)
Footnotes:
[1] There was, several years ago, an amusing thread on the FOM mailing
list, in which Stephen Simpson would insist that something may well be
of "general intellectual interest" even if, in fact, it is of absolutely
no interest to anyone. Needless to say, there's no need to take this
sort of "objectivism" very seriously. Arguments are directed at people,
evidence is something we experience or fail to experience, stuff is
interesting to the extent we actually find it interesting... This
doesn't of course preclude the possibility that something that at first
strikes us as boring is later found to be very interesting, its
potentially opaque connection to what we take interest in having been
made more apparent; that we come to recognise something as evident after
it's explained to us it's just another way of saying something we
already unhesitantly accept, a trivial corollary thereof, etc.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon mann nicht sprechen kann, dar�ber muss man schweigen"
Aatu Koskensilta
> You mean William Reinhardt, not "Rheinhardt". Not a spelling flame,
> just trying to be helpful in case anyone reading this hasn't heard of
> Reinhardt before and wants to look up his work.
Right you are, of course, it's "Reinhardt"! My mistake is baffling and
inexcusable -- composing the reply I did in fact consult Part II of
_Axiomatic Set Theory_ /Proceedings of Symposia in Pure Mathematics/,
vol. XIII Reinhardt article. In the two volumes (the publication
separated by a few years, if I'm not mistaken), we find a true
cornucopia of wonderful papers, including, taking out a few random
titles from Part I:
/Sets constructible using L-kappa-kappa/ by Chang [A very short paper,
consisting almost entirely of definitions. But we all know sometimes
the right definitions are the crux of the matter! Armed with the right
definitions, the right concepts, we practically immediately, for free,
obtain strong generalisations of familiar results.]
/Comments on the foundations of set theory/ [A very silly paper,
demonstrating very well the well-known truism that even mathematicians
who have done seminal work in foundations can blather boringly, piling
a platitude on a platitude, and on the rare moment of making a
substantial assertion -- being guilty of the sin of utmost, idiotic
philosophical naivete. But, then, this might well be said of some
passages in G�del...]
/Sets, semisets, models/ by Petr Hajek [This paper is almost as
unreadable as the book. Great fun for those who have a penchant for
formalism.]
/Primitive recursive set functions/ by Jensen and Karp [Most later
stuff basically goes back to this paper, in so far as essential
definitions are concerned, and which includes almost all the modern
simplifications.]
/Observations on popular discussions on foundations/ by Kreisel
[Kreisel never disappoints those of us who like perceptive and
well-deserved potshots at philosophically naive people -- in this
instance at Cohen and Robinson; here we also find the famous
observation that second-order set theory decides the continuum
hypothesis, plus a few memorable Kreiselisms...]
/On the logical complexity of several axioms of set theory/ by Levy [No
need to say anything about this, ]
/On some consequences of the axiom of determinateness/ by Mycielski
[When did the axiom become that of "determinacy"? In many places where
Kechris's paper on dependent choice holding in L(R) assuming AD is
mentioned we find determinacy spelled "determinancy". What's with this
orthographic chaos?]
/Ordinal definability/ by Myhill and Scott [What's the G�del piece
where this notion was first introduced, only to be completely ignored
for years?]
/Unramified forcing/ by Shoenfield [Still one of the best expositions
of forcing, but possibly not for the faint of heart who would like,
say, some motivation for the notion of generic set.]
/Consistency of GCH with the existence of a measurable cardinal/ [Off
with Cohen's reliance on constructibility!]
/Real-valued measurable cardinals/ by Solovay [Everyone who's heard
about measurable cardinals should naturally wonder about /real-valued/
measures... Much more stuff in the paper than the title indicates.]
/Transfinite sequences of axiom systems for set theory/ by Sward [By
all accounts a rather inconsequential paper. Included here only because
it has a (somewhat strenuous) connection with my own research (carried
out years ago, in a mental institution)!]
Of the times (that is, the times a few years before the symposion) James
E. Baumgartner writes in his review of Kunen:
Once upon a time, not so very long ago, logicians hardly ever wrote
anything down. Wonderful results were being obtained almost weekly,
and no one wanted to miss out on the next theorem by spending the
time to write up the last one. Fortunately there was a Center where
these results were collected and organized, but even for the graduate
students at the Center life was hard. They had no textbooks for
elementary courses, and for advanced courses they were forced to rely
on handwritten proof outlines, which were usually illegible and
incomplete; handwritten seminar notes, which were usually wrong; and
Ph.D. dissertations, which were usually out of date. Nevertheless,
they prospered.
Now the Center I have in mind was Berkeley and the time was the early
and middle 1960's, but similar situations have surely occurred many
times before. In this case, to the good fortune both of the graduate
students and of the logicians not lucky enough to be in California,
all the wonderful results were eventually written down. But it took a
surprisingly long time.
Ah, to be young again, and at a different age, at a different
place. Though, in all likelihood, I would have been just extremely
depressed by everyone else around being so extremely gifted and
clever. (A reaction Dana Scott, who in this context should be credited
as a co-creator of the (elegant but practically useless) Boolean-valued
formulation of forcing, tells us he had at the time. This piece of
trivia might, in some small way, be of some consolations to us mere
mortals, who probably don't ever get to prove things as exciting as that
the reals may be a countable union of countable sets, or that collapsing
felicitously chosen cardinals introduces all sorts of interesting
structure in the continuum and sets thereof...)
Herman Jurjus
> if someone were, by their keen intuition, by "a sort
> of persuasion", by being hit on the head with a brick, immediately
> struck with the obvious truth of determinacy, the observation that the
> arithmetical soundness of an infinity of large large cardinals follows
> would probably give them pause.
If we accept both AD and AC as true, then we obviously have to give up
some of the other assumptions, either some of the other axioms of ZF, or
some of the inference rules of FOL, or some silent assumption that we're
not even aware of yet.
Once we've identified and eliminated it/them, who says that all
consequences of ZF+AD will remain provable?
--
Cheers,
Herman Jurjus
Herman Jurjus
> Your stance, that both choice and determinacy are true, calls for some
> further elucidation -- these principles are after all contradictory!
I'm well aware of that. But i can't help my intuitions, you know.
> The
> usual set theoretic "intuition" is that determinacy holds for some
> restricted class of sets
Right. Making a counterexample to AD is terribly difficult.
You have to make a subset of N^N that is very badly behaved.
So badly behaved in fact that (for me) the point is reached where
it's no longer safe to claim that
"for every sequence of naturals (being produced step-by-step, in the
course of time), the end result is either in the set W or in its complement"
> , the obvious choice for this class being
> identified with L(R) very soon after the introduction of the axiom --
> which Mycielski didn't propose as a truth about sets, but precisely as a
> principle that might be fruitfully studied, and hold, in context of some
> subuniverse of sets -- and this is precisely what is borne out in the
> study of large cardinals.
Yes. The current way to deal with the matter is very sensible.
And my 'stance' is not meant as a criticism or rejection of that
practice, in any way. It's just a toy for me on rainy days, to see how
far it can lead us. (And the fact that nobody else seems to be pursuing
this, is exactly the reason why i do.)
--
Cheers,
Herman Jurjus
Herman Jurjus
> Herman Jurjus says...
>
>> I cannot look inside your brain, but could it be that, so far, you
>> simply /didn't/ intuitively evaluate the first description for yourself
>> at all? I.e. that you just replaced it in your mind with the second
>> description, and judged the situation based on that second description,
>> while further totally ignoring the first description?
>
> You're right, that I substituted an alternative description that I
> thought was more mathematically tractable to reason about. So I'll
> try to stay to the first description: First player
> selects a natural, then the second player selects a natural in
> response, then back to the first player, etc. But I don't understand
> how this description intuitively suggests that one player or the
> other has a winning strategy.
Well, if i had a /proof/ similar to yours, that would be very odd, of
course.
But: either player 1 has a winning strategy or he hasn't.
Now what does it mean for player 1 to not have a winning strategy?
I'd say that amounts to 'player 2 has some way to prevent player 1 from
winning'.
The only remaining possibility is that perhaps this defense is not a
/winning/ strategy for player 2. How could that happen? Well, if after
infinitely many moves it's not certain that player 2 has really won, for
example because it's undetermined whether the game is won by player 1 or
by player 2, but that is ruled out by assumption.
Yes, it's shaky. But is it /more/ shaky than what we get into our heads
when we try to convince ourselves of AC or (especially) the power set
axiom?
A propos chess and checkers: to me it /is/ immediately clear that either
white has a winning strategy, or black has one that makes at least a
draw, etc. It's nice that we can also prove it, but that's not really
needed to see it's true.
It's a bit like with the Jordan curve theorem: it's nice that we can
prove it, but had our definitions been such that it had come out as
false, we would only have concluded that our definitions needed
revision, not that the Jordan curve theorem is false.
--
Cheers,
Herman Jurjus
Herman Jurjus
>
> Indeed. I trust my continuing much further with this pleases or vexes
> you no end.
I was already wondering; for someone who takes nothing too seriously,
you sure seem to spend much energy on this.
--
Cheers,
Herman Jurjus
Aatu Koskensilta
> Aatu Koskensilta wrote:
>
>> I trust my continuing much further with this pleases or vexes you no
>> end.
>
> I was already wondering; for someone who takes nothing too seriously,
> you sure seem to spend much energy on this.
Why shouldn't we spend extraordinary amounts of energy on things we
don't take too seriously? Surely it's a dismal, truly teeth-gnashing
inducing idea, a revolting notion fit only to be rejected in a violent
explosion of metaphorical vomit, that we only spend considerable length
of time on stuff that we take very very seriously. That off my chest, I
also find that I actually learn something of these electronic exchanges,
coming out of them illuminated with new wisdom -- at least in case of
exchanges with the likes of you, Daryl, Moeblee, David, ...; not
necessarily anything of mathematical nature, but something of how
people, myself included, react to this or that mode of argumentation,
this or that way of putting this or that, this or that line of thought,
this or that level of formality, gathering in the process valuable
information about whether this or that way of putting an idea is
generally intelligible, gaining for myself many a curious factoids about
this or that English idiom and its use and abuse, what have you. Or this
or that. Bits and pieces, follies and human insight. That's what I reap
from these virtual encounters.
It is also my hope that, in spite of my sometimes needlessly aggressive
debating style, and peculiar and failed attempts at humour, those with
whom I battle wits leave these Usenetical battlefields slightly
improved, with a perspective on life just a whit expanded, or warped,
from what it was before.
It is customary in many newsgroups to include in otherwise off-topic
posts a nugget of topicality. Here goes: Sigma-1 soundness implies Pi-2
soundness. But exactly on what conditions on the provability predicate?
'Tis a problem I leave you to ponder.
Aatu Koskensilta
> But: either player 1 has a winning strategy or he hasn't.
> Now what does it mean for player 1 to not have a winning strategy?
>
> I'd say that amounts to 'player 2 has some way to prevent player 1 from
> winning'.
>
> The only remaining possibility is that perhaps this defense is not a
> /winning/ strategy for player 2.
I think it's a good idea to explicitly introduce quantifiers, and the
quantifier switch in play, here. We are considering games where two
players choose integers in turn, the first player winning if the
resulting sequence is in a given subset A of the Baire space (the set of
infinite sequences of integers). Now, determinacy asserts that
For all strategies for player 1, there is a strategy for player 2, such
that the resulting play does not end with player 1 winning.
is equivalent to
There is a strategy for player 2, such that for all strategies for
player 1, the resulting play ends with player 2 winning.
Why should this be obvious? There are certainly games for which this
quantifier switch isn't valid (and I'm still bewildered by your
suggestion this is an instance of failure of the law of excluded
middle), so any argument or explanation for the evidence of AD must
depend in an essential way on the games being subsets of the Baire space
of descriptive set theory. But I can't see anything in your explanation
and arguments that wouldn't apply to just any infinite games, whence, in
part, my bafflement at your finding AD evident. (You have qualified your
stance on AD and AC in a post I haven't yet replied to, which I will
soon do, with an overly long and pointless rant, but this -- to my mind,
and apparently also to Butch's mind -- very important objection does not
hinge on the issues touched there.)
> A propos chess and checkers: to me it /is/ immediately clear that
> either white has a winning strategy, or black has one that makes at
> least a draw, etc. It's nice that we can also prove it, but that's not
> really needed to see it's true.
This is a nice example of divergent intuitions! My intuitions tell me
nothing whatever about the existence of winning strategies for chess or
checkers. (And all I know about such matters is based on vague and hazy
recollections from game theory texts. I have played chess about five
times in my life, the plays consisting of my moving the pieces
essentially at random, the opponent declaring at some arbitrary, or so
it seemed to me, point that I'd lost. I'm also utterly tone deaf, and
have no aptitude for crossword puzzles; and mentally adding two
two-digit numbers takes about half a minute for me. I'm very proud of
and pleased with all these shortcomings, mainly because it irritates me
no end people think mathematicians should invariably be musically
talented, good at chess, etc.)
Herman Jurjus
> Herman Jurjus <hjm...@hetnet.nl> writes:
>
>> But: either player 1 has a winning strategy or he hasn't.
>> Now what does it mean for player 1 to not have a winning strategy?
>>
>> I'd say that amounts to 'player 2 has some way to prevent player 1 from
>> winning'.
>>
>> The only remaining possibility is that perhaps this defense is not a
>> /winning/ strategy for player 2.
>
> I think it's a good idea to explicitly introduce quantifiers, and the
> quantifier switch in play, here. We are considering games where two
> players choose integers in turn, the first player winning if the
> resulting sequence is in a given subset A of the Baire space (the set of
> infinite sequences of integers). Now, determinacy asserts that
>
> For all strategies for player 1, there is a strategy for player 2, such
> that the resulting play does not end with player 1 winning.
>
> is equivalent to
>
> There is a strategy for player 2, such that for all strategies for
> player 1, the resulting play ends with player 2 winning.
>
> Why should this be obvious?
Who says it is?
You seem to make the same mental move as Daryl: you rephrase the game as
a simultaneous choice of strategies by the two players. With that
description, the conclusion (determinacy) is indeed not obvious.
This is what i wrote to Daryl about this:
] So we have two descriptions, which you claim to be equivalent.
] (I'm not so sure that the two are conceptually the same, but let's
] leave that, for now.)
]
] With one of these descriptions, a certain conclusion is not
] self-evident, with the other it is (at least for me; you don't
] agree with the latter).
] For me, the equivalence between the situations would lead me to
] accept the conclusion in the second case as well. You, on the other
] hand, seem to reject the conclusion in the first case, on the grounds
] that it is not self-evident in the second case.
]
] I cannot look inside your brain, but could it be that, so far, you
] simply /didn't/ intuitively evaluate the first description for
] yourself at all? I.e. that you just replaced it in your mind with the
] second description, and judged the situation based on that second
] description, while further totally ignoring the first description?
]
] [Because, in retrospect, that's what /i/ always did when thinking
] about AD, in the past.]
> There are certainly games for which this
> quantifier switch isn't valid (and I'm still bewildered by your
> suggestion this is an instance of failure of the law of excluded
> middle), so any argument or explanation for the evidence of AD must
> depend in an essential way on the games being subsets of the Baire space
> of descriptive set theory. But I can't see anything in your explanation
> and arguments that wouldn't apply to just any infinite games, whence, in
> part, my bafflement at your finding AD evident.
Let me assure you that i'm just as baffled about it as you are.
> (You have qualified your
> stance on AD and AC in a post I haven't yet replied to, which I will
> soon do, with an overly long and pointless rant, but this -- to my mind,
> and apparently also to Butch's mind -- very important objection does not
> hinge on the issues touched there.)
>
>> A propos chess and checkers: to me it /is/ immediately clear that
>> either white has a winning strategy, or black has one that makes at
>> least a draw, etc. It's nice that we can also prove it, but that's not
>> really needed to see it's true.
>
> This is a nice example of divergent intuitions! My intuitions tell me
> nothing whatever about the existence of winning strategies for chess or
> checkers.
Now /that/ is baffling. You of all people?
> (And all I know about such matters is based on vague and hazy
> recollections from game theory texts. I have played chess about five
> times in my life, the plays consisting of my moving the pieces
> essentially at random, the opponent declaring at some arbitrary, or so
> it seemed to me, point that I'd lost. I'm also utterly tone deaf, and
> have no aptitude for crossword puzzles; and mentally adding two
> two-digit numbers takes about half a minute for me. I'm very proud of
> and pleased with all these shortcomings, mainly because it irritates me
> no end people think mathematicians should invariably be musically
> talented, good at chess, etc.)
You're not also color-blind, by any chance?
--
Cheers,
Herman Jurjus
Rupert
wrote:
I must say I agree. When I first read about AD I thought it was an
interesting hypothesis, but I never had any feeling that it was
intuitively plausible. On the other hand as soon as I encountered AC I
was completely convinced that it was true.
Aatu Koskensilta
> When I first read about AD I thought it was an interesting hypothesis,
> but I never had any feeling that it was intuitively plausible. On the
> other hand as soon as I encountered AC I was completely convinced that
> it was true.
Our agreement on these matters is most touching. Let's hug! Before we do
that, I'll divulge the following piece of information about my personal
history: when I first encountered determinacy, it immediately struck me
as obviously false.
I soon learned determinacy was intended to apply to sets that behave,
sets that are in some intelligible manner built of basic, familiar and
cozy base by means of (transfinitely iterated) operations that make
(comfortable, familiar, cozy, mathematical) sense. Having learned this,
AD (as restricted to such sets) appeared to me as a rather plausible
hypothesis -- after all, since the sets are built up according to some
sort of construction we can grok, on which we have a sort of "concrete"
-- in so far as it makes sense to speak of concreteness in this context
-- grasp on, it's not such a stretch to suppose this structuredness may
be harnessed in constructing the required strategies. Here, by
"plausible" I mean of course that I wouldn't have been terribly
surprised if it turned out this is in fact the case. (And as it turns
out, that /is/ in fact case, although I'm not particularly happy with
the conceptual state of large large cardinal machinations.) In
contrast, choice, replacement, powerset, etc. were all principles I
would have thought very odd to call "plausible" -- they simply describe,
in the form of assertions, what I usually do with sets that fall under
my purview.
Aatu Koskensilta
> If we accept both AD and AC as true, then we obviously have to give up
> some of the other assumptions, either some of the other axioms of ZF,
> or some of the inference rules of FOL, or some silent assumption that
> we're not even aware of yet. Once we've identified and eliminated
> it/them, who says that all consequences of ZF+AD will remain provable?
If we follow this line of thought, the important question is, What are
we talking about? In all my blather in this thread, I have spoken of
sets in the world of sets as envisaged in the classic cumulative
hierarchy story. If we can't take this background for granted, our first
order of business is to try to work our what do we take for granted,
what sort of sets are we talking about, and so on. As you note, it is
not at all obvious whether standard results about determinacy, choice,
large cardinals, are in the least relevant.
So perhaps you will provide for us a fuller account of your conception
of this world of sets where we find, so I've gathered, both potentially
and actually infinite sequences, these sequences mingling in some sort
of a synthesis of these and those intuitions? (I'm not going to edit
that sentence any further, mainly because I'm suffering of severe sleep
deprivation, but it really was not supposed to be insulting in any
way. Colourful, perhaps, but not insulting.)
Aatu Koskensilta
> I'm well aware of that. But i can't help my intuitions, you know.
Myself, I'm completely uninterested in most of my intuitions. On the
vast majority of things, even in mathematics, I'm a completely
uninformed layman, and consequently I don't see any good reason to pay
any attention to my opinions, intuitions or ideas.
> Right. Making a counterexample to AD is terribly difficult.
Using choice it's very easy. (I'm sure you know the construction, so I
won't bother sketching it here.) The natural question is thus: in what
sense is it difficult?
> You have to make a subset of N^N that is very badly behaved. So badly
> behaved in fact that (for me) the point is reached where it's no
> longer safe to claim that "for every sequence of naturals (being
> produced step-by-step, in the course of time), the end result is
> either in the set W or in its complement"
"Safe" how? If we want to talk about these things in any interesting
fashion, we must be clear on what we're saying. It is a classical
triviality that a sequence is in W or in its complement. So, taking you
at your word, we must have some non-classical conception in mind. It
must be spelled out before any further progress is possible.
> And my 'stance' is not meant as a criticism or rejection of that
> practice, in any way. It's just a toy for me on rainy days, to see how
> far it can lead us. (And the fact that nobody else seems to be
> pursuing this, is exactly the reason why i do.)
Well, bully for you! (I hate smileys, but have acquired a peculiar
fondness for entirely uncalled for parenthetical explanations and
clarifications, which fondness leads me to solemnly proclaim I didn't
intend the "bully for you" sarcastically.)
Herman Jurjus
Bad news, Bill. We just went from 50% to 33% in the popularity polls.
--
Cheers,
Haunted Herman
Aatu Koskensilta
> Aatu Koskensilta wrote:
>
>> Why should this be obvious?
>
> Who says it is?
Naturally, when you said you find determinacy evident (or as evident as
choice) I took you to mean that determinacy as usually understood is
evident. It appears I was mistaken in this.
> You seem to make the same mental move as Daryl: you rephrase the game
> as a simultaneous choice of strategies by the two players. With that
> description, the conclusion (determinacy) is indeed not obvious.
As obliquely stated above, it appears that what you mean by determinacy
is not what I -- and set theorists in general -- mean by
determinacy. And as already said, if we are to understand determinacy in
terms of some alternative conception of set, sequences, what have you,
this conception needs be explained. (And, mainly owing to nothing but my
laziness, and by now rather severe lack of sleep, I haven't digested
your alternative determinacy interpretation as it's present in your
previous posts to Daryl -- it is thus possible my incessant requests for
further explanation and elucidation are unreasonable and petulant.)
>> This is a nice example of divergent intuitions! My intuitions tell me
>> nothing whatever about the existence of winning strategies for chess
>> or checkers.
>
> Now /that/ is baffling. You of all people?
Why does it surprise that I of all people should lack the relevant
intuitions?
>> (And all I know about such matters is based on vague and hazy
>> recollections from game theory texts. I have played chess about five
>> times in my life, the plays consisting of my moving the pieces
>> essentially at random, the opponent declaring at some arbitrary, or so
>> it seemed to me, point that I'd lost. I'm also utterly tone deaf, and
>> have no aptitude for crossword puzzles; and mentally adding two
>> two-digit numbers takes about half a minute for me. I'm very proud of
>> and pleased with all these shortcomings, mainly because it irritates me
>> no end people think mathematicians should invariably be musically
>> talented, good at chess, etc.)
>
> You're not also color-blind, by any chance?
No. But as it happens, my flatmate is colour-blind. He is in many
regards a rather singular individual. I don't think he has any
mathematical or philosophical aspirations, though.
Herman Jurjus
> So perhaps you will provide for us a fuller account of your conception
> of this world of sets where we find, so I've gathered, both potentially
> and actually infinite sequences, these sequences mingling in some sort
> of a synthesis of these and those intuitions?
Perhaps later, some time.
--
Cheers,
Herman Jurjus
Daryl McCullough
>But: either player 1 has a winning strategy or he hasn't.
>Now what does it mean for player 1 to not have a winning strategy?
>
>I'd say that amounts to 'player 2 has some way to prevent player 1 from
>winning'.
But that isn't an accurate paraphrase. That's the reason I
brought up rock/paper/scissors. I know it's a different *kind*
of game, but it is a game of sorts, and it is definitely
*not* the case that "the first player has no winning strategy"
implies that the second player has a way to prevent player 1
from winning. Also, the paraphrase is clearly not equivalent
in games involving chance.
So the claim "Player 1 has no winning strategy" is only
equivalent to "Player 2 has some way to prevent player 1
from winning" for certain kinds of games. What kinds of
games are those? Don't you need to show that a particular
game is one of those kinds of games?
The most we can say without further analysis is that if player
1 doesn't have a winning strategy, then player 2 might luck into
winning. I know you would say that there is no chance in this
game, but there is nondeterminism in the *future* moves made by
the other player. So the most you can say off the bat
is that if player 1 has no winning strategy, then there
is a strategy for player 2 that *might* win, provided that
player 1 makes the wrong choices in the future.
>The only remaining possibility is that perhaps this defense is not a
>/winning/ strategy for player 2. How could that happen?
Well, it's a cooperative game. For player 1 to win, player 2
must cooperate (whether intentionally or not), and for player 1
to *lose*, player 2 must cooperate (whether intentionally or not).
No strategy by itself is necessarily a winning or losing strategy
independent of what the other player does.
>Yes, it's shaky. But is it /more/ shaky than what we get into our heads
>when we try to convince ourselves of AC or (especially) the power set
>axiom?
Since I don't share your intuition about AD, they are less shaky
to *me*.
>A propos chess and checkers: to me it /is/ immediately clear that either
>white has a winning strategy, or black has one that makes at least a
>draw, etc. It's nice that we can also prove it, but that's not really
>needed to see it's true.
Are you sure that your intuition doesn't rely on the fact that
if white wins, he wins in a finite number of moves?
>It's a bit like with the Jordan curve theorem: it's nice that we can
>prove it, but had our definitions been such that it had come out as
>false, we would only have concluded that our definitions needed
>revision, not that the Jordan curve theorem is false.
Yes, I agree completely in this case. I feel that if it is false,
then it means we have defined "continuous curve" incorrectly. As
a matter of fact, I think it is provably *false* for some natural
ways of formulating continuity.
Herman Jurjus
> Herman Jurjus says...
>
>> But: either player 1 has a winning strategy or he hasn't.
>> Now what does it mean for player 1 to not have a winning strategy?
>>
>> I'd say that amounts to 'player 2 has some way to prevent player 1 from
>> winning'.
>
> But that isn't an accurate paraphrase.
I agree that that's a shady part. And of course there's not much more i
can say. ZFC+AD is inconsistent, so the burden is on me to analyze my
intuitions further before saying anything more.
> That's the reason I
> brought up rock/paper/scissors.
Ah. You think that that analogy is good, but i think it isn't.
If it were, it should be piece-a-cake to come up with a counterexample
of AD. But it isn't - it involves making very complex sets using AC, and
(afaik) there are no counterexamples to AD where the winning set is
ZFC-definable.
--
Cheers,
Herman Jurjus
Herman Jurjus
> There are certainly games for which this
> quantifier switch isn't valid (and I'm still bewildered by your
> suggestion this is an instance of failure of the law of excluded
> middle), so any argument or explanation for the evidence of AD must
> depend in an essential way on the games being subsets of the Baire space
> of descriptive set theory.
(The following won't convince Aatu, but i post it anyhow, for other
readers, who perhaps are also puzzled why LEM was brought up.)
Indeed, someone who considers AD to be true seems to be excluding 'too
badly behaved' sets from set theory.
Alternatively, however, he could hold that these complex sets do exist,
but LEM fails for them. I.e. that
"forall x in w^w [ x in A or x not-in A ]"
is not true for them (true for Borel sets, etc., but not for all sets in
general). By dropping LEM for non-Borel sets in this way, things
suddenly start to make much more sense to me. Because you can still have
the 'either player 1 can win, or player 2 can prevent 1 from winning' -
but the defense is not necessarily a winning strategy for player 2 - the
game is no longer one that satisfies the condition that 'at the end of
the game, either player 1 has won or he has not'.
(Anyway - just for what it's worth.)
--
Cheers,
Herman Jurjus
Daryl McCullough
>A very simple reason to be suspicious of AD is that it amounts to a
>strong form of quantifier switch that is in general not valid.
That's what I was arguing.
>I really couldn't make anything of your comments about the law of excluded
>middle, which isn't involved.
Well, I think I can understand why it *seems* like excluded middle
might be involved.
Let W be a set of sequences of naturals. To say that the first
player can force a win is to say, informally, that
there is a move by the first player such that for any move
of the second player, there is a countermove by the first
player such that, blah, blah. We can introduce a notion of
"infinitely many quantifiers" to express this:
Ex_1 Ax_2 Ex_3 ... [x_1, x_2, ...] in W
I haven't said what this notation means, precisely, but you
should have a fuzzy notion of what it might mean. Using the
same ... notation, we can express "the second player can
force a win" as:
Ax_1 Ex_2 Ax_3 ... [x_1, x_2, ...] not in W
These two statements *appear* to be negations of each other,
if we sloppily assume that the De Morgan's laws apply to
infinitely many quantifiers. So, the law of excluded middle
seems to say that one or the other must be the case.
Of course, this depends on giving a precise semantics to
infinitely many quantifiers, which I haven't done.
To me, to say that something is intuitively true, but not
provable, means that there actually is a sloppy proof, which
possibly glosses over distinctions and subtleties.
There is another point to be made about strategies and
quantifiers. As Butch pointed out, even for a *single*
alternation of quantifiers, the equivalence of
Ax_1 Ex_2 [x_1, x_2] not in W
and
Ef Ax_1 [x_1, f(x_1)] not in W
depends on the axiom of choice, in general. So if AD
involves *denying* the axiom of choice, then we shouldn't
accept alternations of quantifiers as meaning the same
thing as the existence of strategies (represented as
functions). So maybe there is a meaning of the infinite
quantifier case
Ax_1 Ex_2 ... [x_1, x_2, ...] not in W
that is *not* equivalent (without choice) to "there is
a winning strategy (in the sense of function) for the
second player".
Herman Jurjus
> Rupert <rupertm...@yahoo.com> writes:
>
>> When I first read about AD I thought it was an interesting hypothesis,
>> but I never had any feeling that it was intuitively plausible. On the
>> other hand as soon as I encountered AC I was completely convinced that
>> it was true.
>
> Our agreement on these matters is most touching. Let's hug! Before we do
> that, I'll divulge the following piece of information about my personal
> history: when I first encountered determinacy, it immediately struck me
> as obviously false.
Lo and behold: that was also my first reaction, many years ago.
Must be a coincidence.
--
Cheers,
Herman Jurjus
Daryl McCullough
>
>Daryl McCullough wrote:
>> Herman Jurjus says...
>>
>>> But: either player 1 has a winning strategy or he hasn't.
>>> Now what does it mean for player 1 to not have a winning strategy?
>>>
>>> I'd say that amounts to 'player 2 has some way to prevent player 1 from
>>> winning'.
>>
>> But that isn't an accurate paraphrase.
>
>I agree that that's a shady part. And of course there's not much more i
>can say. ZFC+AD is inconsistent, so the burden is on me to analyze my
>intuitions further before saying anything more.
>
>> That's the reason I
>> brought up rock/paper/scissors.
>
>Ah. You think that that analogy is good, but i think it isn't.
It's a completely airtight argument that the *lack* of a strategy
for one player does not imply (without additional assumptions) the
existence of a strategy for the other player. What are those
additional assumptions?
>If it were, it should be piece-a-cake to come up with a counterexample
>of AD.
No, that doesn't follow.
It's not an argument that AD is *false*. It shows
that the argument *for* it uses an invalid principle:
"If there is no winning strategy for the first player,
then there must be a winning strategy for the second player".
That's true for some types of games, and false for other
types of games. So it can't be used as a general principle.
Herman Jurjus
> Herman Jurjus says...
>> Daryl McCullough wrote:
>>> Herman Jurjus says...
>>>
>>>> But: either player 1 has a winning strategy or he hasn't.
>>>> Now what does it mean for player 1 to not have a winning strategy?
>>>>
>>>> I'd say that amounts to 'player 2 has some way to prevent player 1 from
>>>> winning'.
>>> But that isn't an accurate paraphrase.
>> I agree that that's a shady part. And of course there's not much more i
>> can say. ZFC+AD is inconsistent, so the burden is on me to analyze my
>> intuitions further before saying anything more.
>>
>>> That's the reason I
>>> brought up rock/paper/scissors.
>> Ah. You think that that analogy is good, but i think it isn't.
>
> It's a completely airtight argument that the *lack* of a strategy
> for one player does not imply (without additional assumptions) the
> existence of a strategy for the other player. What are those
> additional assumptions?
No hidden moves (where this includes 'adjourned' moves and simultaneous
moves), no chance-moves.
--
Cheers,
Herman Jurjus
Herman Jurjus
Allow me to rephrase that: games with consecutive moves by both players
(and no dice), like in a board game.
And the 'extra intuition' that leads to accepting AD seems to amount to
a form of 'extreme realism' in imagining that you can actually perform
such infinite games. (I have no better way to express it, at this moment.)
--
Cheers,
Herman Jurjus
Daryl McCullough
>
>Herman Jurjus wrote:
>> Daryl McCullough wrote:
>>> It's a completely airtight argument that the *lack* of a strategy
>>> for one player does not imply (without additional assumptions) the
>>> existence of a strategy for the other player. What are those
>>> additional assumptions?
>>
>> No hidden moves (where this includes 'adjourned' moves and simultaneous
>> moves), no chance-moves.
>
>Allow me to rephrase that: games with consecutive moves by both players
>(and no dice), like in a board game.
But what is the intuition behind believing that those conditions
should be sufficient?
>And the 'extra intuition' that leads to accepting AD seems to amount to
>a form of 'extreme realism' in imagining that you can actually perform
>such infinite games. (I have no better way to express it, at this moment.)
I don't see how this extreme realism is relevant here. The question is
not whether the game can be carried out; let's assume that the first
move takes 1 second, the second move 1/2 second, etc., so the whole
game is over in 2 seconds (and we know who won). So given an arbitrary
pair of strategies f and g, they can easily figure out whether f beats
g or not (when the first player follows f and the second player follows
g). Furthermore, let's assume that given any strategy f, they
can find a strategy g that beats f (if one exists), and given a
strategy g, they can find a strategy f that beats g (if one exists).
My problem is not imagining that such godlike beings might exist,
but I don't see, even granting such godlike powers, that there is
any reason to believe that one or the other will come up with a
winning strategy. If there is a winning strategy, we can assume
that they will find it, but why should there be such a strategy?
Once again, I ask you to consider how such godlike beings would
find the winning strategy for a finite game such as chess:
Define recursively a sequence W_i of sets of positions as
follows:
W_0 = the set of all positions that are automatic
wins for the first player (in chess, the check-mated positions
for black).
For n odd and greater than 0, W_n = the set of all positions such that
any move by the second player results in a position in W_{n-1}.
For n even and greater than 0, W_n = the set of all positions such that
there exists a move by the first player resulting in a
position in W_{n-1}.
Having generated such a sequence, we can say: If the starting
position is in some W_i, then the first player has a winning
strategy: make any move so that the resulting position is in W_{i-1}.
If the starting position is not in any of the W_i, then the
second player has a winning strategy: make any move so that
the resulting position is *not* in any of the W_i.
Without going through such a argument in favor of "one player or
the other has a winning strategy", I don't see why you would believe
it.
Herman Jurjus
> In article <hc9hjl$45m$1...@news.eternal-september.org>, Herman Jurjus says...
>> Herman Jurjus wrote:
>>> Daryl McCullough wrote:
>
>>>> It's a completely airtight argument that the *lack* of a strategy
>>>> for one player does not imply (without additional assumptions) the
>>>> existence of a strategy for the other player. What are those
>>>> additional assumptions?
>>> No hidden moves (where this includes 'adjourned' moves and simultaneous
>>> moves), no chance-moves.
>> Allow me to rephrase that: games with consecutive moves by both players
>> (and no dice), like in a board game.
>
> But what is the intuition behind believing that those conditions
> should be sufficient?
> Without going through such a argument in favor of "one player or
> the other has a winning strategy", I don't see why you would believe
> it.
I can't say it better, at this stage, than i already did. Sorry.
(Can you /explain/ /why/ you think the Jordan curve should definitely be
true? I can't.)
--
Cheers,
Herman Jurjus
Bill Taylor
> Actually, my comment was directed to Bill Taylor, who
> is going to ignore it because he doesn't have a good answer.
Oi oi oi! Temper, temper.
Patience my dear boy.
I am often slow to respond in these threads, because
(1) I like to take a printout of the daily posts home, so as to
peruse it more leisurely there, and
(2) I don't like dashing into print before having time to consider
the matter properly, and allow immediate intemperate
would-be retorts time to simmer down. Unlike some here.
So may of my articles will thus re-cover ground that has more
recently been addressed; annnoying for some, but OTOH by
delaying, I also reduce the chance of this happening.
However, to get back to your query, I have a response,
and will write it out right after this admonition. :)
-- Brimming-over Bill
Bill Taylor
On Oct 26, 8:50 pm, Butch Malahide <fred.gal...@gmail.com> wrote:
>Can you explain why determinacy is intuitively clear when each move
>is choosing a natural number, but no longer clear when each move is
>choosing a countable ordinal? or a set of real numbers?
This comment is too confusing for me - no countable ordinals or
reals have been mentioned up to now. However, this one...
>What does your intuition say about the following game? First, White
>chooses a set X or real numbers; then Black chooses a realnumber x.
>Black wins if either X is empty or x is in X; White wins if X is
>nonempty and x is not in X.
...is a very good example indeed, and highlights a chink in
the alleged isomorphism between the consecutive and
simultaneous descriptions, and also the meaning of Choice itself.
It is a very good query indeed.
Regarding the latter, Choice is only a logical/set-theoretical
problem when there are infinitely many to be made. However,
beginners in this area, having noted a(n alleged) problem with
Choice, often reduce it to even just a *single* choice; and ask,
"but how can you choose an element from *any* arbitrary (nonempty)
set anyway? - doesn't that mean being able to choose one from
*every* set - the main problem?"
Well, we know they are wrong,though the confusion is understandable.
It's the old "any/every" problem again - being able to choose
an element from ANY nonempty set doesn't mean being able to choose
an element from EVERY nonempty set. The learner hasn't yet cottoned
on to the fact that *logic alone* allows one to choose an element
from ANY nonempty set - the epithet "nonempty" alone guarantees
that one can (almost magically, to the learner) do this.
Yes, it's weird. As von Neumann said:
"Young man, in math you don't get to
UNDERSTAND things, you just get used to them!"
....
What we have above, is the same principle applied to these games.
And thus highlighting a difference (at least an intuitive one)
between the consecutive and the simultaneous views of these games.
Viewed consecutively, we have, at any moment, the single-choice
task, which is unproblematical. But viewed simultaneously, one
has (to be prepared) to make infinitely many choices all at once.
Which is the "more intuitive" interpretation of what the game player
has to do? Obviously, mileage is going to vary on this one!
But kudos to Butch for winkling out this problem.
...
Another view, is to ask, what does it mean, intuitively, when
the game rules state, informally, that player 1 has to announce
a set of reals, for player 2 to choose a single member from;
in Butch's game?
What is it, to "announce" a set.
Here is the heart of the (intuitive) problem.
Surely it can't be allowed to just say nasty things like
"{2,3} if Goldbach and {4,5,6} if not". That's clearly cheating,
surely? OC that particular announcement could be met by player 2
making a similarly conditional reply. But I expect it's easy enough
to make the cases sufficiently nasty to prevent this.
So it must be that "announcing" a set of reals, in some way actually
*states* what are some particular elements in it. And as soon as this
is done, Butch's example evaporates. Because an "announcement"
that specifies some actual explicit reals, is going to specify
a first example in some way, and player 2 just goes with that.
And thus we are back to considerations of Definitionalism,
which Aatu hates me talking about. But there it is.
....
Well Butch, there's my response. You may well think it a pile
of self-serving question-avoiding meretricious rubbish,
but I don't think so - I think there is something definite
to address there, insofar as there ever is when intuitive ideas
and language are used to introduce mathematical examples.
Great query, though.
-- Battling Bill
Bill Taylor
> Bill is a sheep-shagging fence-sitter,
Well now, I can hardly be both AT THE SAME TIME, can I!?
Not unless we have some veeeeery strange sheep indeed!
Kiwis don't mind sheep-shagging jokes, though I'm not too sure
about the Welsh, who also get them. We particularly chuckle when
Aussies try to make them, because we know they're just envious.
Their sheep are miserable, scrawny, scruffy brownish critters
that any respectable sheep-shagger wouldn't look twice at!
Whereas OURS are lovely, plump, fluffy-white spotlessly
clean lovable beasties.
And we retort, "What do you call an Australian with a sheep
under one arm and a goat under the other?"
Answer:- "Bi-sexual!"
> and a notorious proponent of an apparently
> irremediably vaguely formulated form of "definitionalism"
> when it comes to sets.
No and yes. Even your much-admired and admirable Maddy has
serious remarks about "definabilism", so it's hardly as muddy
as you'd like to make out. And no, it's not irremediably vague,
in fact we can make it (as I have done here before) fairly precise,
and indeed have a paper in the pipeline about it.
> This with all respect to Bill, who will no doubt take it all ingood humour.
Of course! Whyever not?
Just as you will take it in good humour when I allude to you
as a luminally-challenged terminally depressive alcoholic from
the far north with typically overly-terse unhelpful Scandinavian
laconic replies to some questions; not but that you've been
giving a LOT of extensive replies recently, thanks for that.
And not that, technically speaking, you're really Scandinavian,
but Fenno-Scandinavian. Though those who can tell the difference
between Finns and other Scands, without names on them, are probably
almost as rare as foreigners who can tell an Aussie & a Kiwi apart!
Good-natured joshing is fine, it's when the ad hominems are
used to further one's debating points that nastiness creeps in,
as it does here, with others, from time to time....
-- Borealic Bill (NOT)
Butch Malahide
wrote:
My apologies, Bill. That was a pretty good answer. Not perfect, but no
worse than my question was. You are right, that I need to cultivate
the virtue of patience. I assume that, in due course, after
considering the matter properly, you will award prizes to the co-
winners of the POC tournament. :-) Perhaps you are waiting for one of
them to die, so you only have to give one prize?
Daryl McCullough
>What we have above, is the same principle applied to these games.
>And thus highlighting a difference (at least an intuitive one)
>between the consecutive and the simultaneous views of these games.
Okay, I agree with that. But let's explore further.
For simplicity, lets take a two-step game: Player 1 make a move m_1,
then Player 2 make a move m_2. If Phi(m_1,m_2) then Player 1
wins, and otherwise Player 2 wins.in if ~Phi(m_1,m_2). So how would we
formulate the claim: Player 2 can always win? It seems to me
that there are two different formulations:
1. forall m_1, exists m_2, ~Phi(m_1, m_2)
2. exists g, forall m_1, ~Phi(m_1, g(m_1))
The first is the "consecutive" view, while the second is
the "simultaneous" view. Assuming the axiom of choice, it
doesn't make any difference. But if we *don't* want to
assume the axiom of choice, then they are conceptually
different.
But now, let's go to *infinite* games. Without adopting the
"simultaneous" view, how do you even *formulate* the claim
that the second player can always win?
If you formulate it as "there exists a strategy *function*
for the second player such that forall strategy functions
for the first player, player 2 can win", then you have
used the "simultaneous" view. But if you reject the
simultaneous view, how do you even formulate it, mathematically?
To me, if you express the axiom of determinacy in terms of
functions---Either there exists a winning strategy function
for the first player, or there exists a winning strategy
function for the second player---then you are implicitly
adopting the simultaneous view, which only is sensible if
you assume the axiom of choice, which contradicts the axiom
of determinacy.
Daryl McCullough
>Can you explain why determinacy is intuitively clear when each move is
>choosing a natural number, but no longer clear when each move is
>choosing a countable ordinal? or a set of real numbers?
>
>What does your intuition say about the following game? First, White
>chooses a set X or real numbers; then Black chooses a real number x.
>Black wins if either X is empty or x is in X; White wins if X is
>nonempty and x is not in X. I'm sure you will agree, Bill, that
>neither player has a winning strategy. What gives?
Okay, I've thought about it, and I realize that there is an
answer to your question that makes AD seem a little less
hypocritical (if that's the word for it).
Rather than formulating "winning strategy" in terms of functions,
we can formulate it in terms of sets of sequences.
Define a "quasistrategy" Q to be a set of finite sequences such
that whenever s is in Q, then
Ax Ey s^[x,y] in Q
(where s^[x,y] means s, followed by x, followed by y).
The idea is that the elements of Q represent "good" positions
for one player of the game when it is the other player's turn.
To be good, it must be the case that no matter what your opponent
does, it is possible for you to put the game back into a good
position.
If Q is a quasistrategy, then define the limit of Q to be
the set of infinite sequences s such that every initial segment
of s is in Q (or can be extended to an element of Q). In terms
of "good" positions, s is in limit(Q) if s represents a game
in which the player puts the game into a good position at every
opportunity.
Now, if W is a set of infinite sequences, then we say that
Q is a winning quasistrategy for W for the first player if
(1) limit(Q) is a subset of W, and
(2) Ex [x] in Q
In other words, the first player can put the game into a
"good" position, and if he keeps it in a "good" position,
then he will win (produce an infinite sequence in W).
Q is a winning quasistrategy for W for the second player if
(1) limit(Q) is disjoint from W, and
(2) [] is in Q
Now, we can formulate the axiom of determinacy in terms of
quasi-strategies as:
For every set W of infinite sequences, there is a quasi-strategy
Q such that Q is a winning strategy for W for the first player
or the second player.
With this formulation, there is no restriction to W containing
only infinite sequences of *naturals*. You can let the "elements"
of the sequences be anything at all.
For your case, we can let the elements of W be sets of reals.
W = { [x_1, x_2, ...] such that x_2 is a singleton subset of x_1 }
Then there is a winning quasi-strategy for the second player:
Q = { [x_1, x_2, ..., x_n] | x_2 is a singleton subset of x_1 }
That's kind of a boring notion of "strategy" in this case, but
it fits the definition.
Daryl McCullough
>And not that, technically speaking, you're really Scandinavian,
>but Fenno-Scandinavian. Though those who can tell the difference
>between Finns and other Scands, without names on them, are probably
>almost as rare as foreigners who can tell an Aussie & a Kiwi apart!
Maybe it's not foolproof, but many of the Finns that I have known
have eyes that look vaguely Asian. I don't know what the actual
term is. There are also people of Irish ancestory who have that
kind of eyes (maybe that's due to marauding Finnish Vikings?)
Bill Taylor
> > explicitly state so
>
> Might i humbly suggest that that is explainable by the fact that many
> people are confronted with choice in the earliest stages of their
> studies, and not with AD?
Another contributing factor being that almost all students are
introduced to it, motivated to it, by thoughts of making an infinite
sequence of choices. This convinces them of the common sense
of Countable Choice, which is a far less questionable assumption
than Wholesale Choice. Then the latter is slipped in more or
less surreptitiously alongside the former.
-- Wide-awake Willy
Bill Taylor
> that practically all
> arguments crucially requiring (full) AC give the impression of being not
> only not convincing at all, but of being particularly unenlightening,
> like the invoking of a magic stick.
Exactly so! Especially when it comes in the form of,
"Let's well-order everything we've got, then etc etc"
- that's when it *really* looks incredibly dodgy cheating!
> AC being acceptable a.o. on the grounds that
> you mention above: it's arithmetically conservative.
Indeed, that is a powerful fact that suggests we might adopt it,
though it also suggest it "isn't really saying anything" in some
sense.
Also, there is the point that *any* statement known to be independent
of ZFC may as well be adopted as not, as anything proved by
using it cannot be "wrong", or at least cannot be disproved.
So AC, CH, AD, Suslin's hypothesis etc could all be adopted from
time to time, even though some cannot be adopted simutaneously!
-- Broad-minded Bill
Bill Taylor
> > Actually, could you explain in what sense chess or checkers are
> > games of complete information, and rock/paper/scissors is *not*?
>
> Now it's my turn to say 'OUCH!'
Indeed!
> Look, there may be any amount of valid reasons to be suspicious of AD,
> or to dislike it, or whatever. But your objection is not one of them, imho.
Quite so. Daryl's point is just plain wrong. More accurately, the
term
"complete information" is a standard term in Game Theory,
and Daryl is using it wrongly. And as a standard term,
any (finite) game with complete info automatically has either
a) a winning strategy for player 1; XOR
b) a winning straegy for player 2, XOR
c) a drawing strategy for both.
-- Board-gaming Bill
Bill Taylor
> I don't understand why you think that "all games are determined" is
> intuitively true.
You gave the answer yourself, in terms of infinite-depth quantifiers.
> It's not *obvious* that chess or checkers has a
> winning strategy; it's *provable*.
It IS. (That is, a winning strategy or a drawing strategy for both.)
This just plain OBVIOUS to any game player.
It was obvious to me even before I started high school.
>To prove it, you have to use the
> fact that they are finite-length games
Yes, the proof requires finitude, but the intuition does NOT.
Again, consider your own i-d quantifiers!
> What reason is there for believing that the principle
> applies to games for which is not provable?
I think you are asking - whose intuitions about infinite-depth
games is more reliable. Is it the man in the street,
or the hardened game-player, or the hardened math-logician?
Obviously mileage will vary!
-- Board-gaming Bill
Bill Taylor
but an observation of an amusing counter-reaction that I'm starting
to notice more and more. (It has nothing to do with the thread.)
Originally, grammar was taught strictly, nominative/accusative and all
that,
and school-marms would ("correctly") pounce on any kiddy saying
"Me and Jane were early today!"
After years, decades, cohorts of this, we were faced with widespread
HYPER-CORRECTION, where people would say
"he gave it to Jane and I", or "a girl like I", and so on.
But now, after generations of hypercorrection, we are starting
to get a counter-reaction, whereby (some) people are not only
fastidiously avoiding hyper-correction, but indulging in
HYPO-CORRECTION, in the hope that it might be "correct",
and in fact going back to "wrong" again.
Here is the latest example....
> While us AC-believers can go all the way and...
What a beauty!
-- Wordpicky William
Bill Taylor
> So we have two [equivalent] descriptions
>
> With one of these descriptions, a certain conclusion is not
> self-evident, with the other it is ( [+ caveats] )
> For me, the equivalence between the situations would lead me to accept
> the conclusion in the second case as well. You, on the other hand, seem
> to reject the conclusion in the first case, on the grounds that it is
> not self-evident in the second case.
I think this is a very perceptive observation, and a neat way of
putting it.
-- Beaming Bill