Torkel Franzen on truth
Newberry
indicates that the human mind surpasses any computer.
>> ... the mistaken idea that "Gödel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulas which cannot be proved in the system, but which we can see to be true." The theorem states no such thing. As has been emphasized, in general we simply have no idea whether or not the Gödel sentence of a system is true, even in those cases when it is in fact true. What we know is that the Gödel sentence is true if and only if the system is consistent, and that much is provable in the system itself. << p. 55
>> ... there is no doubt whatever about the consistency of any of the formal systems we use in mathematics. << p. 105
>> If the axioms of ZFC are manifestly true, they are obviously consistent. << p. 105
I am not sure that I understand what Franzen is saying. Is he saying
that
a) We are absolutely certain about the truths of PA, even those PA
cannot prove
b) The consistency of PA can be proven in ZFC
c) Therefore we can write a computer program emulating ZFC that can
generate the truths of PA
d) We are absolutely certain about the truths of ZFC, even those ZFC
cannot prove
e) There is a theory X in which we can prove the consistency of ZFC
f) Therefore we can write a computer program emulating X that can
generate the truths of ZFC
g) We are not certain about the truths of X
??
Peter_Smith
> In "Gödel's theorem" Torkel Franzen disputes that the theorem
> indicates that the human mind surpasses any computer.
>
> >> ... the mistaken idea that "Gödel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulas which cannot be proved in the system, but which we can see to be true." The theorem states no such thing. As has been emphasized, in general we simply have no idea whether or not the Gödel sentence of a system is true, even in those cases when it is in fact true. What we know is that the Gödel sentence is true if and only if the system is consistent, and that much is provable in the system itself. << p. 55
> >> ... there is no doubt whatever about the consistency of any of the formal systems we use in mathematics. << p. 105
> >> If the axioms of ZFC are manifestly true, they are obviously consistent. << p. 105
>
> I am not sure that I understand what Franzen is saying.
The first quote you give, I take it is entirely clear (and correct!).
Your second quote is misleading: What TF in fact wrote was "Nothing in
Gödel's theorem in any way contradicts the view that there is no doubt
whatever about the consistency of any of the formal systems we use in
mathematics." TF isn't there endorsing the view (as your truncated
quotation suggests), he is just pointing out that Godel's theorem
doesn't refute it -- a point evidently consistent with the first
quote.
The third quote you give starts with an emphasized "If" in TF. It is a
triviality (any set of truths is consistent!).
He is not, at least in those quotations, saying any of (a) to (g).
Gc
> On 8 Nov, 06:01, Newberry <newberr...@gmail.com> wrote:
>
> > In "Gödel's theorem" Torkel Franzen disputes that the theorem
> > indicates that the human mind surpasses any computer.
>
> > >> ... the mistaken idea that "Gödel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulas which cannot be proved in the system, but which we can see to be true." The theorem states no such thing. As has been emphasized, in general we simply have no idea whether or not the Gödel sentence of a system is true, even in those cases when it is in fact true. What we know is that the Gödel sentence is true if and only if the system is consistent, and that much is provable in the system itself. << p. 55
> > >> ... there is no doubt whatever about the consistency of any of the formal systems we use in mathematics. << p. 105
> > >> If the axioms of ZFC are manifestly true, they are obviously consistent. << p. 105
>
> > I am not sure that I understand what Franzen is saying.
>
> The first quote you give, I take it is entirely clear (and correct!).
This I don`t understand:
"As has been emphasized, in general we simply have no idea whether or
not the Gödel sentence of a system is true, even in those cases when
it is in fact true."
I have thought: If we assume the consistency of PA we can proof in PA
+ con(PA) that the gödel sentence of PA being true but not-provable
(thus it follows from this that also the con(PA) is not provable from
the axioms of PA). And certainly we have an least an informally "idea"
that PA is consistent.
Gc
Oh wait. OK. Now I understand? The point is in GENERAL we don`t have
an idea if the gödel sentence is true, like in New Foundations?
Peter_Smith
Yes, I think that's what TF was after: we'll in general not know
whether T's standard Gödel's sentence is true because we'll not know
whether T is consistent. (Of course, we are usually interested in
theories T which we have pretty good reason to think are consistent,
and we are usually not interested in the other cases! TF is just
reminding us of the general situation.)
george
> I am not sure that I understand what Franzen is saying.
Don't panic; neither did he.
> Is he saying that
No.
> a) We are absolutely certain about the truths of PA,
There is no such thing as a truth of PA.
PA is an axiom-set. You prove things from it.
There are THEOREMS of PA, things that are PROVABLE
from PA. Everything else IS FALSE in AT LEAST ONE model
of PA, so there is no point in calling it a truth "of PA".
> even those PA cannot prove
If PA cannot prove it, then there is a model of PA in which it is
false,
so it is not a "truth of PA". THEORIES *don't have* "truths".
"Truth" comes from MODELS. THEORIES have THEOREMS.
> b) The consistency of PA can be proven in ZFC
Well, this is true, regardless of whether he meant it.
But even there, you have to use epsilon_0 induction.
> c) Therefore we can write a computer program emulating ZFC that can
> generate the truths of PA
No, this is false. Just because PA is consistent does NOT mean there
is a computer program that can "generate" all its "truths", especially
since there aren't any. There is a program that can recursively
enumerate
all of PA's THEOREMS, yes, but you don't need as much as ZFC to do
*that*.
That program is not that complicated (unless you want it to be
efficient).
> d) We are absolutely certain about the truths of ZFC, even those ZFC
> cannot prove
Again, ZFC, like PA, IS A THEORY, so it does NOT HAVE "truths".
> e) There is a theory X in which we can prove the consistency of ZFC
Trivially, X=the-theory-whose-only-axiom-is-"ZFC-is-consistent".
Torkel Franzen is not saying any of this (he knows better).
Your paraphrases are confused.
Newberry
> On 8 Nov, 06:01, Newberry <newberr...@gmail.com> wrote:
>
> > In "Gödel's theorem" Torkel Franzen disputes that the theorem
> > indicates that the human mind surpasses any computer.
>
> > >> ... the mistaken idea that "Gödel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulas which cannot be proved in the system, but which we can see to be true." The theorem states no such thing. As has been emphasized, in general we simply have no idea whether or not the Gödel sentence of a system is true, even in those cases when it is in fact true. What we know is that the Gödel sentence is true if and only if the system is consistent, and that much is provable in the system itself. << p. 55
> > >> ... there is no doubt whatever about the consistency of any of the formal systems we use in mathematics. << p. 105
> > >> If the axioms of ZFC are manifestly true, they are obviously consistent. << p. 105
>
> > I am not sure that I understand what Franzen is saying.
>
> The first quote you give, I take it is entirely clear (and correct!).
>
> Your second quote is misleading: What TF in fact wrote was "Nothing in
> Gödel's theorem in any way contradicts the view that there is no doubt
> whatever about the consistency of any of the formal systems we use in
> mathematics." TF isn't there endorsing the view (as your truncated
> quotation suggests), he is just pointing out that Godel's theorem
> doesn't refute it -- a point evidently consistent with the first
> quote.
TF is not saying that Gödel's theorem does not contradict the view
that the system is consistent. He says it does not contradict the view
that there is no doubt. So who is the one that does not have any
doubts?
kleptoma...@hotmail.com
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -
Consistency of the system at hand is just another arithmetical
statement, like "every prime of the form 4k+1 is a sum of two squares"
or "the sum of the divisors of the nth positive integer is less than
or equal to Hn + exp(Hn)log(Hn) where Hn is the nth harmonic number".
As far as I am aware, the epistemological issues for determining the
truth of the consistency of the theory are no different from the
issues for those statements I just mentioned.
As for doubts, doubt and certainty are human emotions. One could be
certain of con(PA) if one could prove that theorem from a list of
arithmetical axioms which one felt certain were true. The same as for
any other theorem.
george
> Consistency of the system at hand is just another arithmetical
> statement, like "every prime of the form 4k+1 is a sum of two squares"
> or "the sum of the divisors of the nth positive integer is less than
> or equal to Hn + exp(Hn)log(Hn) where Hn is the nth harmonic number".
No, it is NOT just like THOSE. THOSE are THEOREMS.
THOSE are PROVABLE from the axioms of PA and therefore
true in all models of PA. The consistency statement for PA
is not provable from/in PA.
MoeBlee
> In "Gödel's theorem" Torkel Franzen disputes that the theorem
> indicates that the human mind surpasses any computer.
>
> >> ... the mistaken idea that "Gödel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulas which cannot be proved in the system, but which we can see to be true." The theorem states no such thing. As has been emphasized, in general we simply have no idea whether or not the Gödel sentence of a system is true, even in those cases when it is in fact true. What we know is that the Gödel sentence is true if and only if the system is consistent, and that much is provable in the system itself. << p. 55
> >> ... there is no doubt whatever about the consistency of any of the formal systems we use in mathematics. << p. 105
> >> If the axioms of ZFC are manifestly true, they are obviously consistent. << p. 105
>
> I am not sure that I understand what Franzen is saying. Is he saying
> that
>
> a) We are absolutely certain about the truths of PA, even those PA
> cannot prove
What do you mean by "the truths of PA"?
There are theorems of PA, and they are true in any model in which the
axioms of PA are true. And, for each model for the langauge of PA,
there is the set of truths in that model; specifically, there is the
set of truths in the standard model for the language of first order
PA.
> b) The consistency of PA can be proven in ZFC
ZFC proves first order PA is consistent. I would think Franzen agrees.
> c) Therefore we can write a computer program emulating ZFC that can
> generate the truths of PA
What is a "program emulating ZFC"? And, again, you say, "truths of
PA".
If you mean the theorems of first order PA, and if by "generate" you
mean recursively enumerate, then, yes, there is a recursive
enumeration of the theorems of first order PA.
If you mean the sentences true in the standard model of the language
of first order PA, then it follows from the incompleteness theorem
that there is no recursive enumeration of the set of sentences true in
the standard model of the language of first order PA.
> d) We are absolutely certain about the truths of ZFC, even those ZFC
> cannot prove
What do you mean by a "truth of ZFC"?
If you mean the theorems of ZFC, then note that "a theorem of ZFC that
cannot be proven in ZFC" is an oxymoron. (And there is no sentence S
of any language such that there is no theory that proves S.)
So only you can say what you mean by "the truths of ZFC".
> e) There is a theory X in which we can prove the consistency of ZFC
For any theory T (even an inconsistent T) there exists theories that
prove the consistency of T. That's trivial. "Theory X proves the
consistency of theory T" is not necessarily a very "substantive"
claim.
> f) Therefore we can write a computer program emulating X that can
> generate the truths of ZFC
Again, what are "the truths of ZFC"?
However, since ZFC is a recursively axiomatized theory, there is a
recursive enumeration of the theorems of ZFC.
> g) We are not certain about the truths of X
> ??
Again, for a theory X, what do you mean by "the truths of X"?
MoeBlee
george
> certain of con(PA) if one could prove that theorem from a list of
> arithmetical axioms which one felt certain were true. The same as for
> any other theorem.
Not really. The whole point about theorems is that truth
doesn't even MATTER for them. Even if the axioms are false,
the theorems ARE STILL provable from them.
MoeBlee
> On Nov 8, 12:26 am, Peter_Smith <ps...@cam.ac.uk> wrote:
> > > >> ... there is no doubt whatever about the consistency of any of the formal systems we use in mathematics. << p. 105 [Newberry's quote of Franzen]
> > Your second quote is misleading: What TF in fact wrote was "Nothing in
> > Gödel's theorem in any way contradicts the view that there is no doubt
> > whatever about the consistency of any of the formal systems we use in
> > mathematics." TF isn't there endorsing the view (as your truncated
> > quotation suggests), he is just pointing out that Godel's theorem
> > doesn't refute it -- a point evidently consistent with the first
> > quote.
>
> TF is not saying that Gödel's theorem does not contradict the view
> that the system is consistent. He says it does not contradict the view
> that there is no doubt. So who is the one that does not have any
> doubts?
To say "there are no doubts" may be understood as a casual way of
saying "there is no reasonable basis for doubt" as opposed to an
unnecessarily extremely literalistic interpretation that there does
not exist in ceratin people the psychological experience of doubt
about the constinency of certain formal theories. Franzen's book is
written at a very informal level and it is grossly missing the point
to split hairs about a non-technical use of such expressions as "there
is no doubt", just as when in, say, a debate, someone says, "So there
is no doubt whatever that the proposed amendment is too costly", it is
not meant literally that there are not people who experience doubt
whether the the amdendment is too costly.
And, then, with that more reasonable sense ('a reasonable basis for
doubt' as opposed to a sweeping claim as to what psychological
experiences people have), still Franzen in that particular passage did
not say that there are not reasonable bases for doubt but rather that
the incompleteness theorem itself does not provide any such reasonable
bases.
MoeBlee
kleptoma...@hotmail.com
Woah. OK con(PA) is different in that it can be proven in PA. But be
careful, the second of those two statements is (equivalent to) the
RIEMANN HYPOTHESIS, and it is an important unsolved problem in
mathematics. We don't know if it is provable in PA or not (wow, usenet
bickering is so much fun!). Actually I wasn't specifically talking
about con(PA), even if it may have seemed like it. What I was trying
to say was that consistency statements in general, though they are
intimately related to Godel's theorem, have no particular
epistemological relevance compared to other arithmetical statements.
As for PA, the fact that con(PA) is not derivable from the axioms of
PA is interesting, but it has no different epistemological status to
all the other statements which are not provable in PA.
kleptoma...@hotmail.com
How could you possibly disagree with that statement? Maybe I could
substitute "the same as for being certain of the truth of any other
theorem" for "The same as for any other theorem." Happy now?
Newberry
Here is another quote from TF:
>> We do of course know the Gödel sentence of, for example PA, to be true since we know PA to be consistent. << p. 117
TF does make the point that the incompleteness theorem does not
contradict the view that we know PA/ZFC to be be consistent with
absolute certainty. But he also does endorse the view that we know PA
to be consistent with absolute certainty (since the axioms are
manifestly true.)
He clearly believes that the we know PA/ZFC to be consistent with the
same certainty as any mathematical theorem i.e. we can prove G. So the
question arises how we can construct a machine that can do the same.
obviously not by emulating PA/ZFC.
Newberry
Do you mean that PA is PROBABLY consistent?
Peter_Smith
Read "pretty good reason" to mean "at least pretty good reason, maybe
conclusive reason". As it happens I think there are conclusive reasons
to believe PA consistent.
aatu.kos...@xortec.fi
> Read "pretty good reason" to mean "at least pretty good reason, maybe
> conclusive reason". As it happens I think there are conclusive reasons
> to believe PA consistent.
Yes, PA is obviously consistent.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
abo
> As it happens I think there are conclusive reasons
> to believe PA consistent.
And:
On Nov 9, 1:17 pm, aatu.koskensi...@xortec.fi wrote:
>
> Yes, PA is obviously consistent.
Conclusive! Obvious! Who could doubt what one learned as a young boy
in Sunday school?
Herman Jurjus
> On 9 Oct, 10:20, Peter_Smith wrote:
>> Read "pretty good reason" to mean "at least pretty good reason, maybe
>> conclusive reason". As it happens I think there are conclusive reasons
>> to believe PA consistent.
>
> Yes, PA is obviously consistent.
So what?
--
Cheers,
Herman Jurjus
george
> He clearly believes that the we know PA/ZFC
There is no such thing as PA/ZFC.
PA is one thing. ZFC is another.
> to be consistent with the
> same certainty as any mathematical theorem
The theorem that PA is consistent is provable in ZFC.
It is NOT provable from/in PA. That ZFC is consistent has never
been proven. It is certainly not provable from ZFC. Bothering to
try to prove it in anything stronger is pointless; you would just have
the same question about whether that stronger theory was vs. wasn't
consistent.
> i.e. we can prove G.
"G" is NOT *one* thing. There is a DIFFERENT G for every
(sufficiently
rich) recursive axiom-set. You canNOT prove G(PA) in PA.
You canNOT prove G(ZFC) in ZFC.
> So the question arises how we can construct a machine that can do the same.
> obviously not by emulating PA/ZFC.
First-order logic is complete. It has inference rules. You just
construct a machine
that applies the inference rules repeatedly. There are some
treatments with as few as
one rule. And there *is* a way of doing this that you could think of
as "emulating ZFC".
Or emulating anything else for that matter. The point being that the
axioms from which
you are going to derive this theory are just one more INPUT to the
machine.
george
> Yes, PA is obviously consistent.
Come on.
The Gentzen proof is not obviously understandable.
N doesn't obviously exist at all.
george
> > Yes, PA is obviously consistent.
On Nov 9, 8:00 am, abo <dkfjd...@yahoo.com> wrote:
> Conclusive! Obvious! Who could doubt what one learned as a young boy
> in Sunday school?
I rated that 5 stars.
Which I really don't like having to do for people with whom I've had
bitter arguments.
george
So N is a model of it, so G is true, so a whole buncha stuff.
Pick one. I don't think AK will care as much as Newberry will.
Daryl McCullough
>He clearly believes that the we know PA/ZFC to be consistent with the
>same certainty as any mathematical theorem i.e. we can prove G. So the
>question arises how we can construct a machine that can do the same.
>obviously not by emulating PA/ZFC.
Well, here's an attempt at describing an informal metatheory that
captures a lot of human metatheoretic reasoning:
1. Every axiom of ZFC is true.
2. For every statement Phi in the language of ZFC,
Phi <-> Phi is true.
3. If T is any theory in the language of ZFC, and every
axiom of T is true, then every theorem of T is true.
This informal theory can prove Con(ZFC) and
Con(ZFC + Con(ZFC)), etc. And it's all perfectly
mechanical; you can write a program to work out
all the consequences of rules 1-3.
Of course, we can give a name to this new theory:
Let ZFC_1 = the collection of all statements in
the language of ZFC that follow from rules 1-3.
Then we can come up with yet another theory by
modifying rule1:
1'. Every axiom of ZFC_1 is true.
Then we could let ZFC_2 be the set of all consequences
of rules 1', 2, and 3. etc.
--
Daryl McCullough
Ithaca, NY
LauLuna
wrote:
> Newberry says...
>
> >He clearly believes that the we know PA/ZFC to be consistent with the
> >same certainty as any mathematical theorem i.e. we can prove G. So the
> >question arises how we can construct a machine that can do the same.
> >obviously not by emulating PA/ZFC.
>
> Well, here's an attempt at describing an informal metatheory that
> captures a lot of human metatheoretic reasoning:
>
> 1. Every axiom of ZFC is true.
>
> 2. For every statement Phi in the language of ZFC,
> Phi <-> Phi is true.
>
> 3. If T is any theory in the language of ZFC, and every
> axiom of T is true, then every theorem of T is true.
>
> This informal theory can prove Con(ZFC) and
> Con(ZFC + Con(ZFC)), etc. And it's all perfectly
> mechanical; you can write a program to work out
> all the consequences of rules 1-3.
>
> Of course, we can give a name to this new theory:
>
> Let ZFC_1 = the collection of all statements in
> the language of ZFC that follow from rules 1-3.
So defined, ZFC_1 is not the informal theory you described, since
there is no predicate in the language of ZFC expressing the truth
predicate for ZFC sentences, by Tarski's indefinability theorem.
Regards
Newberry
> On 9 Oct, 10:20, Peter_Smith wrote:
>
> > Read "pretty good reason" to mean "at least pretty good reason, maybe
> > conclusive reason". As it happens I think there are conclusive reasons
> > to believe PA consistent.
>
> Yes, PA is obviously consistent.
Con(T) --> T(G) --> G
We have just proven G. Now there are two possibilities
a) The human mind surpasses any machine
b) The human mind does not surpass a machine
TF is in favor of b.
In the first case the human mind can see the axioms of PA as
manifestly true, which a machine cannot. If b is the case then I
wonder how we can construct a machine that can generate all the truth
of PA.
Newberry
Are ZFC_1, ZFC_2 etc. consistent? How do we know that they are? Can a
machine generate theories ZFC_1, ZFC_2 etc?
Peter_Smith
> b) The human mind does not surpass a machine
>
> TF is in favor of b.
>
> If b is the case then I
> wonder how we can construct a machine that can generate all the truth
> of PA.
Why should any machine be able to generate all the truths of (the
language of) PA? After all, we can't do that either.
Newberry
How can we construct a machine that can generate all the truth of PA
that we can?
Daryl McCullough
They are consistent if ZFC is true.
>How do we know that they are?
I can't say for sure I do know that they are,
but some people might.
>Can a machine generate theories ZFC_1, ZFC_2 etc?
Sure.
Daryl McCullough
>> 1. Every axiom of ZFC is true.
>>
>> 2. For every statement Phi in the language of ZFC,
>> Phi <-> Phi is true.
>>
>> 3. If T is any theory in the language of ZFC, and every
>> axiom of T is true, then every theorem of T is true.
>>
>> This informal theory can prove Con(ZFC) and
>> Con(ZFC + Con(ZFC)), etc. And it's all perfectly
>> mechanical; you can write a program to work out
>> all the consequences of rules 1-3.
>>
>> Of course, we can give a name to this new theory:
>>
>> Let ZFC_1 = the collection of all statements in
>> the language of ZFC that follow from rules 1-3.
>
>So defined, ZFC_1 is not the informal theory you described, since
>there is no predicate in the language of ZFC expressing the truth
>predicate for ZFC sentences, by Tarski's indefinability theorem.
That's why I said "all statements in the language
of ZFC" rather than "all statements".
Peter_Smith
> How can we construct a machine that can generate all the truth of PA
> that we can?
Well, who knows which truths *those* are?
Newberry
>
> > Read "pretty good reason" to mean "at least pretty good reason, maybe
> > conclusive reason". As it happens I think there are conclusive reasons
> > to believe PA consistent.
>
> Yes, PA is obviously consistent.
>
Peter_Smith
There is no conflict at all between what I said (something TF held
too), and that latter quote. To hold that PA is clearly consistent is
quite compatible with holding that, with some arbitrarily thrown-
together extension of Q, we won't in the general case know whether it
is consistent, and hence won't know whether its canonical Gödel
sentence is true.
LauLuna
Good question.
In 'Inexhaustibility' TF poses the following question:
It seems that whenever human logico-mathematical reason (HLMR) sees as
evident a set of axioms, it also sees as evident the proposition that
those axioms are consistent (which is a kind of reflection principle).
But, if HLMR is consistent and sufficiently rich, that proposition
does not always follow from those axioms (by Gödel's second theorem).
So, if there is an initial and sufficiently rich set of logico-
mathematical truths that must be included in HLMR and HMLR is closed
under that kind of reflection principle, there is no algorithm
representing human logico-mathematical reason.
As I interpret TF, he denies the conclusion by alleging
1. It might happen that there is no such thing as a definite HLMR.
2. Even if HLMR exists, human finiteness precludes the possibility
that it is closed under that reflection principle: humans will
hesitate as things grow increasingly involved.
TF's position (very akin indeed to Hofstadter's) seems questionable to
me because it fails to recognize the existence of an ideal legality in
human reason, that is different from what humans can actually perform,
and that he, TF, is implicitly invoking while reasoning.
Nevertheless, I think TF's arguments show clearly why Lucas's and
Penrose's arguments fail. They both are assuming implicitly that:
A. HLMR is a definite object
B. HLMR is closed under some reflection principle(s).
Clearly, A and B does not follow from Gödel's theorem.
Regards
Newberry
This is the part I did not understand. And this is how I interpreted
it:
a) We are absolutely certain about the truths of PA, even those PA
cannot prove
b) The consistency of PA can be proven in ZFC
c) Therefore we can write a computer program emulating ZFC that can
generate the truths of PA
d) We are absolutely certain about the truths of ZFC, even those ZFC
cannot prove
e) There is a theory X in which we can prove the consistency of ZFC
f) Therefore we can write a computer program emulating X that can
generate the truths of ZFC
g) We are not certain about the truths of X
Maybe I read him wrong?
> Nevertheless, I think TF's arguments show clearly why Lucas's and
> Penrose's arguments fail. They both are assuming implicitly that:
>
> A. HLMR is a definite object
> B. HLMR is closed under some reflection principle(s).
>
> Clearly, A and B does not follow from Gödel's theorem.
>
> Regards- Hide quoted text -
>
> - Show quoted text -
Peter_Smith
> a) We are absolutely certain about the truths of PA, even those PA
> cannot prove
Eh??? Who on earth is committing themselves to that silly claim???
Suppose L_1 is the language of first-order arithmetic, then there are
various classes of truths of L_1.
1. There are the truths for which we can actually give a proof in
first-order PA.
2. There are truths (like the arithmetization of Goodstein's
theorem, or like Con(PA)) which are provably not provable in PA, but
for which we have proofs in other, richer, theories -- like suitable
fragments of set theory.
3. There are other truths which it is not known whether or not they
are provable in first-order PA, though we have do proofs in other
richer theories.
4. And no doubt there are other truths for which we have no kind of
proof (as yet: or may be there could be no humanly surveyable proof).
Plainly these different sorts of truths have different epistemic
status! We might be certain of the truth of the propositions in the
first three classes, given we accept the relevant proofs. But we
certainly not certain about the truths in the fourth class (even if we
strongly suspect that some propositions like Goldbach's conjecture do
indeed fall in the class of truth-but-not-yet proved).
Newberry
So let's confine ourselves to PA for now. We can prove that it is
consistent, that is we have proven G. How did we manage to do that
without running into a contradiction? We did not simply add Con(T) as
another axiom, we proved it. I suppose we proved it in some metatheory
M. How do we know that M is consistent?
Newberry
> On 11 Nov, 16:31, Newberry <newberr...@gmail.com> wrote:
>
> > a) We are absolutely certain about the truths of PA, even those PA
> > cannot prove
>
> Eh??? Who on earth is committing themselves to that silly claim???
>
> Suppose L_1 is the language of first-order arithmetic, then there are
> various classes of truths of L_1.
>
> 1. There are the truths for which we can actually give a proof in
> first-order PA.
> 2. There are truths (like the arithmetization of Goodstein's
> theorem, or like Con(PA)) which are provably not provable in PA, but
> for which we have proofs in other, richer, theories -- like suitable
> fragments of set theory.
There is also 5. the truths of which we are absolutely certain
although they are provably unprovable in PA, like Con(T). The
certainty comes from seeing that the axioms of PA as manifestly true.
Perhaps 5 is identical with 2. How do we know that these richer
theories, such as a suitable fragment of ZFC, are consistent?
Peter_Smith
Depends what M is: it if is a suitable set theory, by getting your
head around the idea of the structure of the iterative hierarchy.
(I know this might sound odd coming from someone whose day-job is as a
philosopher, but frankly, I do find "how we know?" questions are as
entirely boring applied to maths as applied to claims about medium-
sized dry goods. Scepticism either way is just uninteresting.)
abo
Of course it's uninteresting to you. As a rule people like the way
that they go and are not interested in reflecting on why they go that
way. At the least, there's no profit in it. Aunt Bessie goes to the
supermarket every Thursday, and she does not take kindly any
suggestions that it might be possible to go on Friday.
Peter_Smith
> On Nov 11, 8:31 pm, Peter_Smith <ps...@cam.ac.uk> wrote:
>
> > On 11 Nov, 17:46, Newberry <newberr...@gmail.com> wrote:
> > > How do we know that M is consistent?
>
> > Depends what M is: it if is a suitable set theory, by getting your
> > head around the idea of the structure of the iterative hierarchy.
>
> > (I know this might sound odd coming from someone whose day-job is as a
> > philosopher, but frankly, I do find "how we know?" questions are as
> > entirely boring applied to maths as applied to claims about medium-
> > sized dry goods. Scepticism either way is just uninteresting.)
>
> Of course it's uninteresting to you. As a rule people like the way
> that they go and are not interested in reflecting on why they go that
> way.
I'm interested in reflecting ... when given good reason to do so.
abo
Ah, well "good reason" is a term which, because of its subjectivity,
does not advance matters at all. Aunt Bessie has good reason to go
to the supermarket every Thursday; she has always done so.
Peter_Smith
Why is "good reason" subjective? And as for what Aunt Bessie has to do
with the question whether we have good reason to doubt, e.g., the
truth of PA, I'm completely stumped!
abo
What is "good reason" to you may not be "good reason" to someone
else. For instance, a hard-core theist would hold that there is not
"good reason" to discuss the existence of God.
> And as for what Aunt Bessie has to do
> with the question whether we have good reason to doubt, e.g., the
> truth of PA, I'm completely stumped!
Well, Aunt Bessie has good reason to go shopping every Thursday; other
people don't. Yet you ask (I presume with a straight face) why "good
reason" is subjective or what Aunt Bessie has to do with it.
Peter_Smith
> On Nov 11, 10:23 pm, Peter_Smith <ps...@cam.ac.uk> wrote:
>
>
>
> > Why is "good reason" subjective?
>
> What is "good reason" to you may not be "good reason" to someone
> else. For instance, a hard-core theist would hold that there is not
> "good reason" to discuss the existence of God.
Bad argument. The hard core theist might hold that, but that doesn't
obviously ential that they are right to do so.
> > And as for what Aunt Bessie has to do
> > with the question whether we have good reason to doubt, e.g., the
> > truth of PA, I'm completely stumped!
>
> Well, Aunt Bessie has good reason to go shopping every Thursday; other
> people don't. Yet you ask (I presume with a straight face) why "good
> reason" is subjective or what Aunt Bessie has to do with it.
Bad argument. The fact that A has a good reason to do castle, and B
has a good reason to not to do castle doesn't make either reason
"subjective". A and B's situation in the game may be different, and it
could -- for all that has been said -- be an objective matter that
someone in A's position has a good reason to castle and someone in B's
situation has a good reason not to castle. Mutatis mutandis for
Bessie.
Peter_Smith
> Bad argument. The fact that A has a good reason to do castle, and B
> has a good reason to not to do castle doesn't make either reason
> "subjective". A and B's situation in the game may be different, and it
> could -- for all that has been said -- be an objective matter that
> someone in A's position has a good reason to castle and someone in B's
> situation has a good reason not to castle. Mutatis mutandis for
> Bessie.
Apologies -- that's very careless editing! "to do castle" should read,
of course "to castle".
abo
>
> > On Nov 11, 10:23 pm, Peter_Smith <ps...@cam.ac.uk> wrote:
>
> > > Why is "good reason" subjective?
>
> > What is "good reason" to you may not be "good reason" to someone
> > else. For instance, a hard-core theist would hold that there is not
> > "good reason" to discuss the existence of God.
>
> Bad argument. The hard core theist might hold that, but that doesn't
> obviously ential that they are right to do so.
Yes, obviously you can objectify 'good reason' so that it just means
"good reason to Peter Smith." Then the hard-core theist is of course
wrong; his is not a good reason. Yet, what he means by the words
'good reason' is different from what you mean by the words 'good
reason.' And in the end, both your words do the same work, in that
he is not willing to reflect unless there is "good reason" just as you
are not.
>
> > > And as for what Aunt Bessie has to do
> > > with the question whether we have good reason to doubt, e.g., the
> > > truth of PA, I'm completely stumped!
>
> > Well, Aunt Bessie has good reason to go shopping every Thursday; other
> > people don't. Yet you ask (I presume with a straight face) why "good
> > reason" is subjective or what Aunt Bessie has to do with it.
>
> Bad argument.
Where was the argument?
> The fact that A has a good reason to do castle, and B
> has a good reason to not to do castle doesn't make either reason
> "subjective".
> A and B's situation in the game may be different, and it
> could -- for all that has been said -- be an objective matter that
> someone in A's position has a good reason to castle and someone in B's
> situation has a good reason not to castle. Mutatis mutandis for
> Bessie.
Not at all. I explained to you that Aunt Bessie's only reason for
going shopping on Thursday was that she had always done so. She might
take that as a good reason, but her grandson, who wants to take her to
the museum and can only take Thursday off work, would not. (And I
would presume you would not either, even though your epistemological
views apparently tend it that direction.)
Peter_Smith
> On Nov 11, 10:44 pm, Peter_Smith <ps...@cam.ac.uk> wrote:
>
> > On 11 Nov, 21:29, abo <dkfjd...@yahoo.com> wrote:
>
> > > On Nov 11, 10:23 pm, Peter_Smith <ps...@cam.ac.uk> wrote:
>
> > > > Why is "good reason" subjective?
>
> > > What is "good reason" to you may not be "good reason" to someone
> > > else. For instance, a hard-core theist would hold that there is not
> > > "good reason" to discuss the existence of God.
>
> > Bad argument. The hard core theist might hold that, but that doesn't
> > obviously ential that they are right to do so.
>
> Yes, obviously you can objectify 'good reason' so that it just means
> "good reason to Peter Smith."
Sigh. I was of course doing no such thing.
Newberry
> On 11 Nov, 17:46, Newberry <newberr...@gmail.com> wrote:
>
>
>
>
>
> > On Nov 11, 12:29 am, Peter_Smith <ps...@cam.ac.uk> wrote:
>
> > > On 11 Nov, 03:05, Newberry <newberr...@gmail.com> wrote:
>
> > > > On Nov 9, 4:17 am, aatu.koskensi...@xortec.fi wrote:> On 9 Oct, 10:20, Peter_Smith wrote:
>
> > > > > > Read "pretty good reason" to mean "at least pretty good reason, maybe
> > > > > > conclusive reason". As it happens I think there are conclusive reasons
> > > > > > to believe PA consistent.
>
> > > > > Yes, PA is obviously consistent.
>
> > > > OK, how do we reconcile it with this?
>
> > > > >> ... the mistaken idea that "Gödel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulas which cannot be proved in the system, but which we can see to be true." The theorem states no such thing. As has been emphasized, in general we simply have no idea whether or not the Gödel sentence of a system is true, even in those cases when it is in fact true. What we know is that the Gödel sentence is true if and only if the system is consistent, and that much is provable in the system itself. << p. 55
>
> > > There is no conflict at all between what I said (something TF held
> > > too), and that latter quote. To hold that PA is clearly consistent is
> > > quite compatible with holding that, with some arbitrarily thrown-
> > > together extension of Q, we won't in the general case know whether it
> > > is consistent, and hence won't know whether its canonical Gödel
> > > sentence is true.
>
> > So let's confine ourselves to PA for now. We can prove that it is
> > consistent, that is we have proven G. How did we manage to do that
> > without running into a contradiction? We did not simply add Con(T) as
> > another axiom, we proved it. I suppose we proved it in some metatheory
> > M. How do we know that M is consistent?
>
> Depends what M is: it if is a suitable set theory, by getting your
> head around the idea of the structure of the iterative hierarchy.
Let's assume then that M is a suitable set theory and that it is
consistent. How do we prove that M is consistent? In a metatheory M_2?
How do we know that M_2 is consistent? Which hierarchy did you have in
mind, PA, M, M_2, M_3? If I get my head around this hierarchy M-omega
does it mean that I am using a meta-meta-theory N?
> (I know this might sound odd coming from someone whose day-job is as a
> philosopher, but frankly, I do find "how we know?" questions are as
> entirely boring applied to maths as applied to claims about medium-
> sized dry goods. Scepticism either way is just uninteresting.)-
I have not heard this one yet. I do not even quite understand what you
are trying to say here. So let's just stick to the subjectmatter.
Aatu Koskensilta
> On Nov 8, 1:01 am, Newberry <newberr...@gmail.com> wrote:
>> I am not sure that I understand what Franzen is saying.
>
> Don't panic; neither did he.
Given that the quoted passage is perfectly clear it seems you're suggesting
Franzén managed to write something eminently comprehensible without himself
understanding any of it. This is a curious suggestion -- perhaps you have
something more sensible in mind?
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Aatu Koskensilta
> Conclusive! Obvious! Who could doubt what one learned as a young boy
> in Sunday school?
I don't know. What does one learn about Peano arithmetic as a young boy in
Sunday school?
Aatu Koskensilta
Nothing much. Why?
Aatu Koskensilta
> Come on.
> The Gentzen proof is not obviously understandable.
> N doesn't obviously exist at all.
Gentzen's proof and the existence of the set of natural numbers don't really
have much to do with obviousness of consistency of PA.
Aatu Koskensilta
> On Nov 9, 4:17 am, aatu.koskensi...@xortec.fi wrote:
>> Yes, PA is obviously consistent.
>
> OK, how do we reconcile it with this?
Reconcile in what sense? There is no apparent contradiction between Torkel's
explanation concerning...
... the mistaken idea that "Gödel's theorem states that in any consistent
system which is strong enough to produce simple arithmetic there are
formulas which cannot be proved in the system, but which we can see to be
true." The theorem states no such thing. As has been emphasized, in general
we simply have no idea whether or not the Gödel sentence of a system is
true, even in those cases when it is in fact true. What we know is that the
Gödel sentence is true if and only if the system is consistent, and that
much is provable in the system itself.
and the observation that PA is obviously consistent.
Aatu Koskensilta
> What is "good reason" to you may not be "good reason" to someone
> else. For instance, a hard-core theist would hold that there is not
> "good reason" to discuss the existence of God.
An indifferent atheist might also well find discussing the existence of God
somewhat pointless. Regardless of the question of whether "good reason" is
or is not subjective, it remains a rather trivial platitude that people will
in fact be interested in subjecting this or that to scrutiny, reflection,
doubt, only if presented some incentive to, a "good reason" in a more
mundane sense.
Daryl McCullough
>
>On 2007-11-09, in sci.logic, abo wrote:
>> Conclusive! Obvious! Who could doubt what one learned as a young boy
>> in Sunday school?
>
>I don't know. What does one learn about Peano arithmetic as a young boy in
>Sunday school?
I'm not sure. But an acquaintance of mine explained how the natural
numbers can be represented using lambda calculus. He told me he
learned it in Church.
--
Daryl McCullough
Ithaca, NY
abo
wrote:
> On 2007-11-11, in sci.logic, abo wrote:
>
> > What is "good reason" to you may not be "good reason" to someone
> > else. For instance, a hard-core theist would hold that there is not
> > "good reason" to discuss the existence of God.
>
> An indifferent atheist might also well find discussing the existence of God
> somewhat pointless. Regardless of the question of whether "good reason" is
> or is not subjective, it remains a rather trivial platitude that people will
> in fact be interested in subjecting this or that to scrutiny, reflection,
> doubt, only if presented some incentive to, a "good reason" in a more
> mundane sense.
Somebody, who was in conversation with PS about this subject, asked
him a question, "How do you know?" That would seem to be incentive
enough to provide at least a modicum of scrutiny or reflection.
Newberry
wrote:
> On 2007-11-11, in sci.logic, Newberry wrote:
>
> > On Nov 9, 4:17 am, aatu.koskensi...@xortec.fi wrote:
> >> Yes, PA is obviously consistent.
>
> > OK, how do we reconcile it with this?
>
> Reconcile in what sense? There is no apparent contradiction between Torkel's
> explanation concerning...
>
> ... the mistaken idea that "Gödel's theorem states that in any consistent
> system which is strong enough to produce simple arithmetic there are
> formulas which cannot be proved in the system, but which we can see to be
> true." The theorem states no such thing. As has been emphasized, in general
> we simply have no idea whether or not the Gödel sentence of a system is
> true, even in those cases when it is in fact true. What we know is that the
> Gödel sentence is true if and only if the system is consistent, and that
> much is provable in the system itself.
>
> and the observation that PA is obviously consistent.
There are several issues here.
1) Isn't exhibiting one theory (PA/ZFC) good enough to establish
Lucas's argument?
2) What did TF intend to say by "in general"? Did he mean
a) the meta, meta-theories in which we establish the consistency of PA
and then ZFC etc. Or did he mean
b) alternative theories e.g. Quine's set theory
The problem in a) is that there seems to be an infinite regress. As
far as b) chances are that we will be able to establish their
consistency just like we established the consistency of PA/ZFC.
Aatu Koskensilta
> Somebody, who was in conversation with PS about this subject, asked
> him a question, "How do you know?" That would seem to be incentive
> enough to provide at least a modicum of scrutiny or reflection.
Surely you know the grounds on which we -- Peter, me, Torkel, and so on --
find PA's consistency obvious by now. On the conception that the naturals
are obtained from 0 by repeatedly applying the "add-one"-operation the
principle of induction
Whenever P is a determinate mathematical property of naturals, if 0 has P,
and whenever n has P, n+1 also has P, all naturals have P
is manifestly true, as is the principle of definition by primitive
recursion, that properties definable by primitive recursion are determinate
and well-defined in the relevant sense. Combining this observation with the
determinateness of properties expressible in the first order language of
arithmetic, that is, those obtainable from the primitive recursive
properties by means of the usual logical operations, leads immediately to
the conclusion that the axioms of PA are all manifestly true, and hence no
contradiction follows from them, by the soundness of the rules of inference
of first order logic.
Now, if someone finds this explanation incomprehensible even after
elaborations, illustrations, gentle persuasion, practice, and so on, I'm
stumped. There's simply nothing I can do but shrug. Of course, people might
be interested in e.g. what can and cannot be proved without appeal to the
totality of the successor function, full induction, etc. for perfectly
sensible reasons -- often we obtain mathematical information beyond than
just that P is true if we know that P is not only true but also provable
from these or those (weak) principles -- but connecting such interests to
rather elusive and incomprehensible doubts is pointless.
Aatu Koskensilta
> 1) Isn't exhibiting one theory (PA/ZFC) good enough to establish
> Lucas's argument?
No.
> 2) What did TF intend to say by "in general"?
He means that in general, if we're presented with an axiomatisable extension
of Q we quite literally have no idea whether or not it is consistent, and
consequently whether or not its Gödel sentence is true or not.
abo
wrote:
"Now, if someone finds this explanation incomprehensible..."
Beware when you need to overstate your case to make a point.
Obviously I don't find your explanation incomprehensible, but I do
find it lacking. It is lacking at the very beginning, in that the
entire point is how or why you think you know that you can always "add
one".
One other thing. Your statement at the end about "connecting such
interests to rather elusive and incomprehensible doubts is pointless"
is a subjective claim hidden as an oracular assertion about which
there can be no dispute. You think it is pointless, no problem with
that. You've been to Sunday School, and you've learned what you've
been told. Good for you! Still, whether the doubts are indeed
pointless or not is an entirely different matter.
Aatu Koskensilta
> Obviously I don't find your explanation incomprehensible, but I do
> find it lacking. It is lacking at the very beginning, in that the
> entire point is how or why you think you know that you can always "add
> one".
That's simply part of our conception of the naturals. I find the idea that
some natural might -- perhaps by accident? -- lack a successor completely
baffling, and can make nothing of it unless it is explained what such a
thing might mean.
> One other thing. Your statement at the end about "connecting such
> interests to rather elusive and incomprehensible doubts is pointless"
> is a subjective claim hidden as an oracular assertion about which
> there can be no dispute.
Anything at all can be disputed.
> You think it is pointless, no problem with that. You've been to Sunday
> School, and you've learned what you've been told. Good for you!
Why do you think I've been to Sunday School?
LauLuna
> On Nov 11, 5:39 am, LauLuna <laureanol...@yahoo.es> wrote:
>
>
>
>
>
> > On Nov 10, 11:38 pm, Newberry <newberr...@gmail.com> wrote:
>
> > > On Nov 10, 12:52 pm, Peter_Smith <ps...@cam.ac.uk> wrote:
>
>
> > > > > b) The human mind does not surpass a machine
>
> > > > > TF is in favor of b.
>
> > > > > If b is the case then I
> > > > > wonder how we can construct a machine that can generate all the truth
> > > > > of PA.
>
> > > > Why should any machine be able to generate all the truths of (the
> > > > language of) PA? After all, we can't do that either.
>
> > > How can we construct a machine that can generate all the truth of PA
> > > that we can?
>
> > Good question.
>
> > In 'Inexhaustibility' TF poses the following question:
>
> > It seems that whenever human logico-mathematical reason (HLMR) sees as
> > evident a set of axioms, it also sees as evident the proposition that
> > those axioms are consistent (which is a kind of reflection principle).
> > But, if HLMR is consistent and sufficiently rich, that proposition
> > does not always follow from those axioms (by Gödel's second theorem).
> > So, if there is an initial and sufficiently rich set of logico-
> > mathematical truths that must be included in HLMR and HMLR is closed
> > under that kind of reflection principle, there is no algorithm
> > representing human logico-mathematical reason.
>
> > As I interpret TF, he denies the conclusion by alleging
>
> > 1. It might happen that there is no such thing as a definite HLMR.
>
> > 2. Even if HLMR exists, human finiteness precludes the possibility
> > that it is closed under that reflection principle: humans will
> > hesitate as things grow increasingly involved.
>
> > TF's position (very akin indeed to Hofstadter's) seems questionable to
> > me because it fails to recognize the existence of an ideal legality in
> > human reason, that is different from what humans can actually perform,
> > and that he, TF, is implicitly invoking while reasoning.
>
> This is the part I did not understand. And this is how I interpreted
> it:
>
> a) We are absolutely certain about the truths of PA, even those PA
> cannot prove
> c) Therefore we can write a computer program emulating ZFC that can
> generate the truths of PA
> d) We are absolutely certain about the truths of ZFC, even those ZFC
> cannot prove
> e) There is a theory X in which we can prove the consistency of ZFC
> f) Therefore we can write a computer program emulating X that can
> generate the truths of ZFC
> g) We are not certain about the truths of X
>
> Maybe I read him wrong?
>
>
>
> > Nevertheless, I think TF's arguments show clearly why Lucas's and
> > Penrose's arguments fail. They both are assuming implicitly that:
>
> > A. HLMR is a definite object
> > B. HLMR is closed under some reflection principle(s).
>
> > Clearly, A and B does not follow from Gödel's theorem.
>
> > Regards- Hide quoted text -
>
> > - Show quoted text -- Hide quoted text -
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -
I interpret that your a), b), c),... are just an example (for actually
they are not true). If so, what you say is similar to what I think TF
said. TF suggests that we would eventually get to such a convoluted
theory that we would hesitate in applying the reflection principles in
order to get new truths.
We do believe:
a') The axioms of PA
b') a')+ Con(a')
c') b') + Con(b')
etc.
TF says that at some too high level things would get so difficult that
we would stagger or would be simply unable to go on.
What I argue is that though this is in fact so, the argument fails to
distinguish the logico-mathematical legality enclosed in human reason
from what humans can effectively accomplish., i.e. what is logically
possible for humans from what is physically possible for them.
The necessity of reducing logical impossibility to physical
impossibility is one of the weak points of AI.
Regards
abo
wrote:
> On 2007-11-12, in sci.logic, abo wrote:
>
> > Obviously I don't find your explanation incomprehensible, but I do
> > find it lacking. It is lacking at the very beginning, in that the
> > entire point is how or why you think you know that you can always "add
> > one".
>
> That's simply part of our conception of the naturals.
This is just the ontological fallacy (from a property of a thing you
can infer the existence of a thing). You say that you have a
conception whereby a natural number always has a successor,. Fine;
the only things that will be natural numbers are those things that
have a successor. But that doesn't imply that there exist any natural
numbers. Similarly, one could say that part our conception of God is
that He is an absolutely perfect being. But that doesn't imply that
there is any being who is absolutely perfect.
I'd add that it seems to me worthwhile to distinguish our conception
of what a natural number is, and our conception of what the natural-
number sequence is. I think it is incorrect to say that part of our
conception of what a natural number is is that it have a successor.
We do have a conception of naturals, and we'd agree that 2 is a
natural number; yet it just can't be, from the fact that 2 is a
natural number, that 10^10^10^10 exists and is a natural number, which
would in fact follow were every natural number always to have a
successor (which is a natural). I'd agree with you that our
conception of the natural-number sequence is that every natural in the
sequence has a successor (in the sequence); but then the question just
becomes whether there is any such sequence.
> I find the idea that
> some natural might -- perhaps by accident? -- lack a successor completely
> baffling, and can make nothing of it unless it is explained what such a
> thing might mean.
Here's one way to picture it: after some very big point, the naturals
fade away, ever so gradually.
>
> > One other thing. Your statement at the end about "connecting such
> > interests to rather elusive and incomprehensible doubts is pointless"
> > is a subjective claim hidden as an oracular assertion about which
> > there can be no dispute.
>
> Anything at all can be disputed.
I agree with you. But surely you realize - because you surely you
intend it - that your style tends at times to be oracular, where you
assert something as if dispute is impossible. I usually find it more
appropriate, for instance, to say, "I find it obvious that..." instead
of "It's obvious that...".
>
> > You think it is pointless, no problem with that. You've been to Sunday
> > School, and you've learned what you've been told. Good for you!
>
> Why do you think I've been to Sunday School?
>
Because you have?
Newberry
wrote:
Right. So now the question is how do we reconcile the absolute
certainty that PA is consistent with Goedel's theorem, which says that
the consistency of PA is unprovable. It seems that you just proved it.
You can prove it in ZFC? First of all I do not know if the ZFC proof
is the same one as the manifest truth proof. Secondly, is ZFC
consistent?
Herman Jurjus
> On 2007-11-12, in sci.logic, abo wrote:
>> Somebody, who was in conversation with PS about this subject, asked
>> him a question, "How do you know?" That would seem to be incentive
>> enough to provide at least a modicum of scrutiny or reflection.
>
> Surely you know the grounds on which we -- Peter, me, Torkel, and so on --
> find PA's consistency obvious by now.
> Now, if someone finds this explanation incomprehensible
Who says they do? Perhaps they just find it 'too easy' as an answer?
> I'm stumped.
Glad that you admit that you don't understand the issue.
> There's simply nothing I can do but shrug.
Glad again that you admit it yourself.
But the feeling might be more mutual than you think, you know.
[Everything with a grain of salt, as usual.]
--
Cheers,
Herman Jurjus
Aatu Koskensilta
> Right. So now the question is how do we reconcile the absolute
> certainty that PA is consistent with Goedel's theorem, which says that
> the consistency of PA is unprovable. It seems that you just proved it.
Gödel's theorem does not imply that the consistency of PA is unprovable in
any absolute sense, only that there is no formal derivation of "PA is
consistent" in PA.
> You can prove it in ZFC? First of all I do not know if the ZFC proof
> is the same one as the manifest truth proof.
In ZFC one would probably just show that the finite von Neumann ordinals are
a model of PA.
> Secondly, is ZFC consistent?
Sure.
Aatu Koskensilta
> Who says they do? Perhaps they just find it 'too easy' as an answer?
Given that the consistency of PA is an obvious triviality it should not be
surprising the answer is easy.
> Glad that you admit that you don't understand the issue.
I do indeed find it utterly baffling people should worry about the
consistency of PA, all the while accepting, apparently without any qualms,
much more abstract mathematical statements unprovable in PA. There are
exceptions, of course, such as Edward Nelson, whose objections and doubts
are understandable and interesting, even if totally unrelated to the way the
naturals are usually conceived, and the way we usually reason in
mathematics.
LauLuna
>
> - Mostrar texto de la cita -
If I don't misunderstand your query on infinite regress along the
hierarchy of theories, you are posing an 'ultimately philosophical'
question: where does our confidence in PA ultimately stems from?
Well, it originates from our confidence in reason, in rational
evidence. That is what Lotze called 'Selbstvertrauen der Vernunft',
i.e. reason's confidence in reason.
We believe some propositions because we are able to derive them from
evident truths. We believe evident truths because we rely on reason.
We rely on reason for no reason?
Regards
Aatu Koskensilta
> We rely on reason for no reason?
Relying on reason, and accepting evident truths, is very reasonable.
abo
wrote:
There are
> exceptions, of course, such as Edward Nelson, whose objections and doubts
> are understandable and interesting, even if totally unrelated to the way the
> naturals are usually conceived, and the way we usually reason in
> mathematics.
What are Nelson's objections and doubts which you understand? As near
as I can tell, he complains about the Platonic existence of the
naturals and then takes out his doubts on induction, all the while
still assuming the existence of said naturals.
Newberry
wrote:
> On 2007-11-13, in sci.logic, Newberry wrote:
>
> > Right. So now the question is how do we reconcile the absolute
> > certainty that PA is consistent with Goedel's theorem, which says that
> > the consistency of PA is unprovable. It seems that you just proved it.
>
> Gödel's theorem does not imply that the consistency of PA is unprovable in
> any absolute sense, only that there is no formal derivation of "PA is
> consistent" in PA.
Thanks for the lecture but that does not help us to get around the
problem.
>
> > You can prove it in ZFC? First of all I do not know if the ZFC proof
> > is the same one as the manifest truth proof.
>
> In ZFC one would probably just show that the finite von Neumann ordinals are
> a model of PA.
Is it the same as the manifest truth proof?
>
> > Secondly, is ZFC consistent?
>
> Sure.
How do we prove ZFC consistency?
>
> --
> Aatu Koskensilta (aatu.koskensi...@xortec.fi)
Newberry
I was mainly asking if I interpreted Franzen correctly. We are sure
that PA is consistent and we prove it in ZFC. We are sure that ZFC is
consistent and we prove it in some metatheory. But we are not sure if
this meatatheory is consistent. Is this what he is saying?
you are posing an 'ultimately philosophical'
> question: where does our confidence in PA ultimately stems from?
>
> Well, it originates from our confidence in reason, in rational
> evidence. That is what Lotze called 'Selbstvertrauen der Vernunft',
> i.e. reason's confidence in reason.
>
> We believe some propositions because we are able to derive them from
> evident truths. We believe evident truths because we rely on reason.
>
> We rely on reason for no reason?
>
Newberry
> Penrose's arguments fail. They both are assuming implicitly that:
>
> A. HLMR is a definite object
> B. HLMR is closed under some reflection principle(s).
>
> Clearly, A and B does not follow from Gödel's theorem.
>
The issue is that we can conclude with certainty that G is true. (PA
is consistent because the axioms are manifestly true.) Thus far no one
explained how we could construct a machine that would do the same.
MoeBlee
> I was mainly asking if I interpreted Franzen correctly. We are sure
> that PA is consistent and we prove it in ZFC. We are sure that ZFC is
> consistent and we prove it in some metatheory. But we are not sure if
> this meatatheory is consistent. Is this what he is saying?
I wouldn't take that as an accurate summary of his view. Rather, he
has a main point in his discussion of skepticism. If you go back to
read it, it's really difficult to miss what that point is.
MoeBlee
LordBeotian
"Daryl McCullough" <stevend...@yahoo.com> ha scritto
> Well, here's an attempt at describing an informal metatheory that
> captures a lot of human metatheoretic reasoning:
>
> 1. Every axiom of ZFC is true.
>
> 2. For every statement Phi in the language of ZFC,
> Phi <-> Phi is true.
>
> 3. If T is any theory in the language of ZFC, and every
> axiom of T is true, then every theorem of T is true.
>
> This informal theory can prove Con(ZFC) and
> Con(ZFC + Con(ZFC)), etc. And it's all perfectly
> mechanical; you can write a program to work out
> all the consequences of rules 1-3.
What does it mean "etc." here?
LordBeotian
"Newberry" <newbe...@gmail.com> ha scritto
>So let's confine ourselves to PA for now. We can prove that it is
>consistent, that is we have proven G. How did we manage to do that
>without running into a contradiction? We did not simply add Con(T) as
>another axiom, we proved it. I suppose we proved it in some metatheory
>M. How do we know that M is consistent?
To know that theory (PA or M) is consistent we don't necessaily need a
"proof".
kleptoma...@hotmail.com
When one says one has proved a theorem "for sure", it means one has
proved it from axioms that one is "sure" are true. Consistency does
not enter the picture.
Newberry
> "Newberry" <newberr...@gmail.com> ha scritto
Right. So it means that Lucas and Penrose are rigtht, and Franzen is
wrong?
Daryl McCullough
>> To know that theory (PA or M) is consistent we don't necessarily need a
>> "proof".
>
>Right. So it means that Lucas and Penrose are right, and Franzen is
>wrong?
That doesn't follow at all. The question that Penrose asked was:
Is it possible for there to be a computer program P(x) such that
P(x) halts and returns true
<->
x is the Godel number of a statement that human
mathematicians can become convinced is an absolutely
unassailable truth
It is irrelevant whether the set of "unassailable truths"
are provable or not.
Daryl McCullough
Sorry, I thought it was obvious. We can define a sequence of theories
T_n as follows:
T_0 = ZFC
T_{n+1} = that theory whose axioms consist of all the axioms of T(n)
plus the additional axiom Con(T(n))
Then the informal theory described can prove Con(T_n) for every n.
Newberry
wrote:
His point was that the human mind surpasses any machine. If the human
mind can comprehend a truth that cannot be formally proven then the
human mind surpasses any computer.
Example: "if a set of axioms is manifestly true then the theory is
consistent" is just as compelling as e.g. "~A, A v B |- B." It is not
clear how any machine can prove that a theory based on manifestly true
axioms is consistent. At least in this thread we failed to explain how
it could be conclusively proven.
Bill Taylor
> >> in Sunday school?
>
> >I don't know. What does one learn about Peano arithmetic as a young boy in
> >Sunday school?
>
> I'm not sure. But an acquaintance of mine explained how the natural
> numbers can be represented using lambda calculus. He told me he
> learned it in Church.
And don't forget that Kleeneness is next to Godelness!
-------------------------------------------------------
Bill Taylor W.Ta...@math.canterbury.ac.nz
-------------------------------------------------------
And God said
Let there be numbers
And there *were* numbers.
Odd and even created he them,
He said to them be fruitful and multiply
And he commanded them to keep the laws of induction.
-------------------------------------------------------
LordBeotian
"Newberry" <newbe...@gmail.com> ha scritto
>> >So let's confine ourselves to PA for now. We can prove that it is
>> >consistent, that is we have proven G. How did we manage to do that
>> >without running into a contradiction? We did not simply add Con(T) as
>> >another axiom, we proved it. I suppose we proved it in some metatheory
>> >M. How do we know that M is consistent?
>>
>> To know that theory (PA or M) is consistent we don't necessaily need a
>> "proof".
>
> Right. So it means that Lucas and Penrose are rigtht, and Franzen is
> wrong?
How would you draw this conclusion?
LordBeotian
"Newberry" <newbe...@gmail.com> ha scritto
> His point was that the human mind surpasses any machine. If the human
> mind can comprehend a truth that cannot be formally proven then the
> human mind surpasses any computer.
Every truth can be formally proven. Just take the statement of the truth as
an axiom.
> Example: "if a set of axioms is manifestly true then the theory is
> consistent" is just as compelling as e.g. "~A, A v B |- B." It is not
> clear how any machine can prove that a theory based on manifestly true
> axioms is consistent. At least in this thread we failed to explain how
> it could be conclusively proven.
I think ZFC can prove that any theory which has a model is consistent, and
also can prove that any theory whose axioms are true in a model is
consistent.
LauLuna
wrote:
> Relying on reason, and accepting evident truths, is very reasonable.
I completely agree.
But, is that a good reason to rely on reason?
Regards
george
wrote:
> Given that the quoted passage is perfectly clear
No, it isn't.
Gee, that was easy.
Even talking about the clarity of *a* passage in the context
of an overall take on a global issue is silly to begin with.
The meaning and the clarity of the passage have obvious
prior dependencies on the coherence of the context. The quoted
piece involved 2 different statements on p.105 and a longer prior
one from p.55. That triple does NOT fall under the definition of
"the quoted passage". "Passages", instead. And once there are
3 of them then they get to have 3 different degrees of clarity.
Not to mention relevance.
The mere fact that the passages purport to talk about truth at all
is unfortunate. That is necessary if you are trying to debunk other
people's misconceptions but there is an underlying hubris here,
an underlying claim to gnosis, that is far more objectionable for
being SIMPLY IRRELEVANT than it is for being conceited.
> it seems you're suggesting
> Franzén managed to write something eminently comprehensible without himself
> understanding any of it.
"True" in natural language is complicated.
The mere existence of "this sentence is NOT true" proves
THAT. Choosing to talk about some of this stuff as though it were
straightforward is part of the disease, not of the cure.
> This is a curious suggestion -- perhaps you have
> something more sensible in mind?
You're being entirely too charitable.
Just the usual cantankerous nihilism.
george
wrote:
> Gentzen's proof and the existence of the set of natural numbers don't really
> have much to do with obviousness of consistency of PA.
Are you saying that PA continues to remain obviously consistent
even in the context where it is insisted that N is a proper classm and
just plain can't be a set?
Given that the model existence theorem is a theorem of SET theory,
that is going to be a little complicated. There is a context in which
we
can PROVE that a theory is consistent if and only if it has a model.
If N doesn't exist then it isn't a model.
So is it always obvious how to prove that some other model exists?
Or are you insisting that some context-for-debating-consistency-in-
which-
models-in-general-are-not-relevant has *higher*, *prior* claim or
status
than THE USUAL context, in which models of consistent theories MUST
exist?
Daryl McCullough
>
>On Nov 13, 9:14 pm, stevendaryl3...@yahoo.com (Daryl McCullough)
>wrote:
>> Newberry says...
>>
>>
>>
>> >On Nov 13, 1:49 pm, "LordBeotian" <pokips...@yahoo.it> wrote:
>> >> To know that theory (PA or M) is consistent we don't necessarily need a
>> >> "proof".
>>
>> >Right. So it means that Lucas and Penrose are right, and Franzen is
>> >wrong?
>>
>> That doesn't follow at all. The question that Penrose asked was:
>> Is it possible for there to be a computer program P(x) such that
>>
>> P(x) halts and returns true
>> <->
>> x is the Godel number of a statement that human
>> mathematicians can become convinced is an absolutely
>> unassailable truth
>>
>> It is irrelevant whether the set of "unassailable truths"
>> are provable or not.
>
>His point was that the human mind surpasses any machine. If the human
>mind can comprehend a truth that cannot be formally proven then the
>human mind surpasses any computer.
No, that doesn't follow at all. You're applying a double standard.
You're only requiring that the human be able to "comprehend" a truth,
while you're requiring that the computer be able to *prove* it. To
make it a fair comparison, you either require both to prove the
statement, or require neither to prove the statement.
>Example: "if a set of axioms is manifestly true then the theory is
>consistent" is just as compelling as e.g. "~A, A v B |- B." It is not
>clear how any machine can prove that a theory based on manifestly true
>axioms is consistent.
Why does it matter whether the machine can prove it,
if the human can't prove it, either? As I said, for
a computer program to be as powerful as a human in
recognizing truth, all that's necessary is for the
program to be able to *recognize* true statements.
It's not necessary that the program be able to
*prove* them.
Having said that, there is actually no problem in proving
that a true theory must be consistent: Truth is preserved
by logical deduction. A contradiction cannot be true.
Therefore, it is impossible to deduce a contradiction
from any true theory.
Newberry
How can a computer recognize that PA is consistent?
>
> Having said that, there is actually no problem in proving
> that a true theory must be consistent: Truth is preserved
> by logical deduction. A contradiction cannot be true.
> Therefore, it is impossible to deduce a contradiction
> from any true theory.
>
> --
> Daryl McCullough
> Ithaca, NY- Hide quoted text -
Daryl McCullough
>> Why does it matter whether the machine can prove it,
>> if the human can't prove it, either? As I said, for
>> a computer program to be as powerful as a human in
>> recognizing truth, all that's necessary is for the
>> program to be able to *recognize* true statements.
>> It's not necessary that the program be able to
>> *prove* them.
>
>How can a computer recognize that PA is consistent?
By producing an output "true" when given the
input question "Do you believe that PA is consistent?".
Are you asking how one would go about *programming*
a computer program that would emulate a human's
mathematical reasoning? If so, I have no idea.
The issue is not whether we *currently* know
how to make an artificial intelligent computer
program. The issue is whether Godel's theorem
implies that it is impossible. It doesn't
imply any such thing.
There is what I think is a pretty air-tight argument
that no single human can do any mathematical reasoning
that is noncomputable: A real human has a finite
memory capacity, and so there are only finitely
many different statements of mathematics that we
can ever hold in our heads at one time. So the
collection of all statements that any *actual*
human would ever claim to be "unassailably true"
is a finite set. Every finite set of formulas
is computable.
Now, you could argue about what an idealized
human could do, where we idealize the human
to have an infinite memory capacity. Could
such an idealized human do something that
no Turing machine could do? Well, it depends
on the details of how the "ideal human" is
idealized.
Newberry
Bingo!! You got it! So we have the human mind surpasses any machine
and no single human can do any mathematical reasoning that is
noncomputable. A contradiction! That is what I was trying to say all
along.
abo
wrote:
>
> There is what I think is a pretty air-tight argument
> that no single human can do any mathematical reasoning
> that is noncomputable: A real human has a finite
> memory capacity, and so there are only finitely
> many different statements of mathematics that we
> can ever hold in our heads at one time. So the
> collection of all statements that any *actual*
> human would ever claim to be "unassailably true"
> is a finite set. Every finite set of formulas
> is computable.
>
A finite machine can't compute the palindrome or multiplication
function, so it would seem by the same token neither any actual human
nor any actual computer can "do" multiplication. This strikes me as
perhaps not what people have in mind.
Daryl McCullough
>> There is what I think is a pretty air-tight argument
>> that no single human can do any mathematical reasoning
>> that is noncomputable: A real human has a finite
>> memory capacity, and so there are only finitely
>> many different statements of mathematics that we
>> can ever hold in our heads at one time. So the
>> collection of all statements that any *actual*
>> human would ever claim to be "unassailably true"
>> is a finite set. Every finite set of formulas
>> is computable.
>
>Bingo!! You got it! So we have the human mind surpasses any machine
>and no single human can do any mathematical reasoning that is
>noncomputable. A contradiction! That is what I was trying to say all
>along.
No, we *don't* have that the human mind surpasses any machine.
There is no reason to believe that's true.
Daryl McCullough
>A finite machine can't compute the palindrome or multiplication
>function, so it would seem by the same token neither any actual human
>nor any actual computer can "do" multiplication.
I think that's all perfectly true. We can't multiple two
trillion-digit numbers. We can't compute the palindrome
of a trillion-letter word.
Daryl McCullough
Whoops! What I meant was that we can't decide whether a
trillion-letter word is a palindrome.
Newberry
wrote:
> Newberry says...
>
>
>
> >On Nov 14, 4:56 pm, stevendaryl3...@yahoo.com (Daryl McCullough)
> >wrote:
> >> Why does it matter whether the machine can prove it,
> >> if the human can't prove it, either? As I said, for
> >> a computer program to be as powerful as a human in
> >> recognizing truth, all that's necessary is for the
> >> program to be able to *recognize* true statements.
> >> It's not necessary that the program be able to
> >> *prove* them.
>
> >How can a computer recognize that PA is consistent?
>
> By producing an output "true" when given the
> input question "Do you believe that PA is consistent?".
>
> Are you asking how one would go about *programming*
> a computer program that would emulate a human's
> mathematical reasoning? If so, I have no idea.
Easy. Just add this axiom: (Ex)P(x, #("F") --> F
Newberry
wrote:
But we do. We know that G is true. Proof:
The axioms of PA are manifestly true
PA is consistent
"PA is consistent" is equivalent to G
G QED
This proof cannot be formalized. We can prove the consistency of PA in
ZFC. We believe ZFC to be true. We can prove the consistency of ZFC
only in a stronger theory about which we are not sure that it is true.
We observe two things
1) This proof is not the same as the manifest truth proof
2) Since we have not proved the consistenvy of ZFC we do not know if
the proof of the consistency of PA is not a falsehood. So our
certainty that PA is true does not come from this ZFC based proof.
If the manifest truth proof were formalizable and the axioms/rules
were added to PA we would have a contradiction.
Hence it is not formalizable i.e. the human mind surpasses any
machine.