Learning Logic and Set Theory
Derek Smith
fill in gaps in my knowledge. I begin reading, and realize this book
mixes together logic and set theory. I then said, well maybe I should
learn formal logic, too. So I begin reading on the web and see
definitions involving statements like a "countable set of variable
symbols" or a "set of function symbols". How do I know what "countable"
or "function" means without axiomatic set theory?
Can anybody recommend a rigorous book for learning mathematical logic
at the beginning graduate level? I'd like to learn foundations and want
to get around any circular reasoning like the above.
Torkel Franzen
> So I begin reading on the web and see
> definitions involving statements like a "countable set of variable
> symbols" or a "set of function symbols". How do I know what "countable"
> or "function" means without axiomatic set theory?
How do you learn anything in mathematics without the benefit of
formalized theories?
MoeBlee
There is a something of a problem when just starting out: For a
rigorous treatment of mathematical logic, we use set theory; and for a
rigorous treatment of set theory, we use a first order predicate
calculus, which is studied under mathematical logic.
I can speak only for how I have personally dealt with this. First, I
learned the basics of using first order predicate calculus while
learning informally why first order predicate calculus is sound. At
this stage, only some very informal and intuitive ideas about sets are
needed, but at the end of this stage, I am very clear about what a
rigorous proof is using first order predicate calculus. Second, now
that I know how to work with first order predicate calculus (and have a
good, yet informal, understanding of why it is sound), I can follow a
treatment of axiomatic set theory (such as Suppes). Third, after having
learned enough set theory, I can return to mathematical logic in a
rigorous context, using set theory, at whatever degree of rigor I
desire, to formulate first order languages and theories and
meta-theorems about them.
Specific books:
To concentrate on proficiency using first order predicate calculus and
to get an informal, intuitive sense of why it is sound, 'Logic:
Techniques of Formal Reasoning' by Kalish, Montague, and Mar may be a
good choice.
Then, for set theory, you have Suppes, which is good. Also, you might
want to compare it with Enderton's 'Elements of Set Theory'.
Then, to return to the subject of mathematical logic, Enderton's 'A
Mathematical Introduction To Logic' is widely used. And this book has
an opening chapter with a summary of most of the set theory that the
book depends upon. After having studied set theory already, this
chapter will be a good summary, or, if you haven't finished basic set
theory, then you can take this chapter "on faith" as a set of
principles that you can use for the book, and fill in the proofs and a
rigorous understanding of the set theory later.
MoeBlee
Dan Christensen
"Derek Smith" <derekl...@gmail.com> wrote in message
news:1133548337.4...@g44g2000cwa.googlegroups.com...
For some quick insights into formal logic and set theory, you might have a
look at my DC Proof program. While it presents a non-standard development of
these topics -- it has been simplified somewhat for ease of use -- you may
find it useful as an introduction to standard texts.
Dan
Download my DC Proof software at http://www.dcproof.com
smnew...@comcast.net
Why not Suppes companion book -Introduction to Logic .I am a big fan
of both .Regards,smn
george
Derek Smith wrote:
> I then said, well maybe I should
> learn formal logic, too. So I begin reading on the web and see
> definitions involving statements like a "countable set of variable
> symbols" or a "set of function symbols". How do I know what "countable"
> or "function" means without axiomatic set theory?
Consider yourself intelligent.
This question is deep.
It is NOT something to be reacted to simplistically
because it is coming from a newbie.
Nevertheless, the simplistic answer is that
"you just have to know". Seriously, every presentation or
explanation of ANYthing MUST take SOME terms as primitive.
To "explain" something to somebody who does not yet understand
it ARGUABLY means to re-express it, translated, in simpler terms
that they ALREADY understand. But they may not (if they are new
enough) yet understand the simpler terms either, in which case the
teacher is re-obligated to re-express in EVEN simpler terms.
Eventually, the iterated simplification HAS to stop. Eventually we
have
to get down to terms that are simple enough that the student agrees
that
he DOES just GET them.
"Countable" and "function" are two of these terms.
Seriously, neither set theory nor first-order logic as we know
them is competent to define them, not without presuming them
as part of the machinery, anyway.
Aatu Koskensilta
> "Countable" and "function" are two of these terms.
> Seriously, neither set theory nor first-order logic as we know
> them is competent to define them
How does saying that a set is countable iff there is bijection between
it and the set of a natural numbers fail to define countability?
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Derek Smith
understanding that first-order logic along with the axioms of set
theory provide sound and even intuitive definitions for terms like
"countable" and "function". But how do you define first-order logic?
Well the definitions I've seen assume that we have things called "sets"
and "functions". What's the point of the definitions provided by set
theory if we have to assume these objects exist just to build the
definition?
This has been racking my brain and slowing me down for the past two
days, so I'm just going to ignore it and and continue on with my
studies. But is this the current state of foundations or is there a way
to circumvent this problem?
Barb Knox
"Derek Smith" <derekl...@gmail.com> wrote:
It's inevitable. By Tarski's theorem (a formal version of the Liar
paradox), no sufficiently strong formal system can consistently define
what "truth" means in that system. So to fully describe ANY formal
system you need to use another (usually stronger) system. "It's turtles
all the way down".
(Note that some knowledgeable posters to this NG disagree with that
view. I'll leave it to them to state their side of it.)
--
---------------------------
| BBB b \ Barbara at LivingHistory stop co stop uk
| B B aa rrr b |
| BBB a a r bbb | Quidquid latine dictum sit,
| B B a a r b b | altum viditur.
| BBB aa a r bbb |
-----------------------------
Torkel Franzen
> It's inevitable. By Tarski's theorem (a formal version of the Liar
> paradox), no sufficiently strong formal system can consistently define
> what "truth" means in that system. So to fully describe ANY formal
> system you need to use another (usually stronger) system.
There is no truth predicate associated with the concept of a
formal system.
Keith Ramsay
Derek Smith wrote:
|The problem isn't with the definition of countable. It is my
|understanding that first-order logic along with the axioms of set
|theory provide sound and even intuitive definitions for terms like
|"countable" and "function".
It all depends on what you mean by "provide definitions".
The definitions of "countable" and "function" in set theory
terms only tell you what they mean if you already understand
the terms used to define them. In particular, you have to
know what domain the variables range over, and what the relation
denoted by "epsilon" means. If you don't know what "epsilon"
means, the definitions won't tell you what "countable" or
"function" mean.
The axioms of set theory play no role in providing these
definitions.
|But how do you define first-order logic?
|Well the definitions I've seen assume that we have things called
"sets"
|and "functions". What's the point of the definitions provided by set
|theory if we have to assume these objects exist just to build the
|definition?
You have to understand the terms without the use of a formal
definition of first-order logic. They're assuming you have an
informally developed understanding of what sets are.
|This has been racking my brain and slowing me down for the past two
|days, so I'm just going to ignore it and and continue on with my
|studies. But is this the current state of foundations or is there a
way
|to circumvent this problem?
To the extent that I understand what's bothering you, it seems
to me that you're expecting the formal development to accomplish
something that it just isn't able to do, because it would be
circular.
The formalism has no power of its own to reach out of the page and
grab you. If it means anything to you, it's because you already have
some idea of what the terms mean. Now, one common process is
to start with some informal foundation, then go back and formalize
it. This is essentially what you are seeing done. But the informal
understanding is needed to boot-strap the process.
Keith Ramsay
george
Torkel Franzen wrote:
> There is no truth predicate associated with the concept of a
> formal system.
Association is subjective and psychological.
People are free to associate anything they want to with whatever
formal system you present them. They are also "unfree" in the sense
that once you present them a formal system, both they and you
are constrained to agree that it has the components that it in fact
has.
Barb Knox is probably competent to allege "an associated truth-
predicate" for ANY formal system you might care to present her.
I would further insist that the system has NOT been coherently
presented
UNTIL the hearer can understand such a predicate. The fact that the
predicate
may not be total or may not be fully "definable" is IRrelevant. The
whole
point is that it is precisely THAT predicate that is not definable (as
opposed
to the myriad others that are). Therefore, the notion of that (truth)
predicate
is coherent. It is also inevitable.
george
Aatu Koskensilta wrote:
> george wrote:
> > "Countable" and "function" are two of these terms.
> > Seriously, neither set theory nor first-order logic as we know
> > them is competent to define them
>
> How does saying that a set is countable iff there is bijection between
> it and the set of a natural numbers fail to define countability?
I DIDN'T SAY that THIS failed to define countability!
I said that FIRST-ORDER LOGIC failed to define countability!
IF you translate this definition into first-order logic THEN it fails
to define countability BY the Lowenheim-Skolem theorem.
Torkel Franzen
> IF you translate this definition into first-order logic THEN it fails
> to define countability BY the Lowenheim-Skolem theorem.
What odd notion of "define" are you using here?
Aatu Koskensilta
> Aatu Koskensilta wrote:
>
>>george wrote:
>>
>>>"Countable" and "function" are two of these terms.
>>>Seriously, neither set theory nor first-order logic as we know
>>>them is competent to define them
>>
>>How does saying that a set is countable iff there is bijection between
>>it and the set of a natural numbers fail to define countability?
>
> I DIDN'T SAY that THIS failed to define countability!
It's puzzling then that you said that set theory is not competent to
define countability.
george
Aatu Koskensilta wrote:
> >>How does saying that a set is countable iff there is bijection between
> >>it and the set of a natural numbers fail to define countability?
and reiterated,
> It's puzzling then that you said that set theory is not competent to
> define countability.
This is NOT puzzling AT ALL.
Set theory is not competent to define the set of natural numbers,
(at least not at first order), so obviously a definition in terms of
them
is no help. The set required to exist by the axiom of infinity can be
bigger
than the actual natural numbers. You can define them in higher-order
logic
but going to higher-order logic completely moots the need for set
theory
anyway.
george
> "george" <gre...@cs.unc.edu> writes:
>
> > IF you translate this definition into first-order logic THEN it fails
> > to define countability BY the Lowenheim-Skolem theorem.
Torkel Franzen wrote:
> What odd notion of "define" are you using here?
NO *odd* notion, ASSHOLE.
It would help a LOT if you could upgrade from YOUR disability
about making unsupportable value judgments instead of just
dealing with the technical issue.
Besides, you are an expert.
You know damn well the relevant senses in which first-
order logic fails to define the things that it fails to define.
You are not entitled to ask this question.
It isn't necessary to use uncouth language in order
to make onself insufferable.
Torkel Franzen
> You are not entitled to ask this question.
OK, so I take it it's not possible to say in what sense the standard
definition fails to define countability.
Keith Ramsay
george wrote:
> Aatu Koskensilta wrote:
> > >>How does saying that a set is countable iff there is bijection between
> > >>it and the set of a natural numbers fail to define countability?
>
> and reiterated,
>
> > It's puzzling then that you said that set theory is not competent to
> > define countability.
>
> This is NOT puzzling AT ALL.
I'm sure it isn't to you.
> Set theory is not competent to define the set of natural numbers,
> (at least not at first order), so obviously a definition in terms of
> them
> is no help. The set required to exist by the axiom of infinity can be
> bigger
> than the actual natural numbers. You can define them in higher-order
> logic
> but going to higher-order logic completely moots the need for set
> theory
> anyway.
Your point seems to be simply that other models exist of axioms
commonly used in the theory, and that the term relativized to that
other model can be different from the usual referent of the term.
But I don't think this accords with what people usually mean when
they speak of being able to define a term.
Do you consider real analysis competent to define "continuous
function"? I don't see any essential difference between this
example and yours.
Keith Ramsay
Daryl McCullough
>>> IF you translate this definition into first-order logic
>>> THEN it fails to define countability BY the Lowenheim-Skolem
>>> theorem.
Is there a formal definition of what it means to "define" something?
To me, it seems that for a formula to be said to define something,
you have to rely on an *interpretation* of the language that formula
is written in. So, in particular, a formula Phi(x) is said to
define "x is countable" in case
Phi(x) is true in the interpretation
<-> x is a countable set
The Lowenheim-Skolem theorem just implies that there are
interpretations of the language of set theory for which
Phi(x) fails to express (absolute) countability. But that
isn't a failure of *set* theory's power to define things.
I think what the Lowenheim-Skolem theorem shows is that
we can't eliminate the need for an interpretation. The
interpretation can't itself be specified unambiguously
using just first-order logic.
--
Daryl McCullough
Ithaca, NY
george
Daryl McCullough wrote:
> Is there a formal definition of what it means to "define" something?
!@#$%!
There does not NEED to be a FORMAL definition!
And coming from YOU of all people, this question is, I repeat,
maddening! IF there were a formal definition, YOU would know it!
YOU are an expert! YOU therefore may NOT ask THIS question!
For once TF is a step ahead of you; knowing dang well that formality
was not relevant, he instead asked me what "odd notion" of definability
I was using. This at least establishes that notions plural are both
extant
and relevant. However, "first-order definability" is HARDLY "odd"
among them.
> To me, it seems that for a formula to be said to define something,
Oh, please. This is not a matter of opinion. The general linguistic
behavior of the community does factually settle the issue.
> you have to rely on an *interpretation* of the language
And this is just idiotic bullshit, Darryl. The WHOLE POINT is that
first-order theorems are INDEPENDENT of the interpretation.
> So, in particular, a formula Phi(x) is said to
> define "x is countable" in case
>
> Phi(x) is true in the interpretation
> <-> x is a countable set
If Phi can be true in some interpretations and false in others,
then the question must arise about whether x can be countable
in some interpretations and not countable in others. This is
ridiculous. The question was whether the FORMULA ITSELF,
whether phi, by itself, defined countability. This by definition
cannot be an interpretation-dependent question, ESPECIALLY
not in the first-order context where ALL the important stuff
(i.e. all the provable stuff) is interpretation-INdependent ANYhow.
This line was not worth pursuing.
Daryl McCullough
>Daryl McCullough wrote:
>> Is there a formal definition of what it means to "define" something?
>
>!@#$%!
Merry Christmas to you, too, George.
Daryl McCullough
>There does not NEED to be a FORMAL definition!
>And coming from YOU of all people, this question is, I repeat,
>maddening! IF there were a formal definition, YOU would know it!
>YOU are an expert!
I am not! I'm a physics major. I only have a Masters Degree, at that.
I'm not an expert in anything. Don't accuse me of being an expert.
>> To me, it seems that for a formula to be said to define something,
>
>Oh, please. This is not a matter of opinion.
If there is no formal definition of what it means to
define something, then yes, it is a matter of opinion.
>The general linguistic behavior of the community does
>factually settle the issue.
According to the community, "countable" is perfectly
definable in the language of set theory.
>> you have to rely on an *interpretation* of the language
>
>And this is just idiotic bullshit, Darryl. The WHOLE POINT is that
>first-order theorems are INDEPENDENT of the interpretation.
What relevance is that? Whether a formula defines something
or not has nothing to do with theoremhood. Perhaps you are
thinking that a formula Phi(x) defines a set S if
x in S <-> Phi(x) is provable
but by that criterion, only recursive sets are definable. That
is *not* the usual notion of "definable". The usual notion, as
I've said, is via a standard *interpretation* of a language.
>> So, in particular, a formula Phi(x) is said to
>> define "x is countable" in case
>>
>> Phi(x) is true in the interpretation
>> <-> x is a countable set
>
>If Phi can be true in some interpretations and false in others,
>then the question must arise about whether x can be countable
>in some interpretations and not countable in others.
You pick a *standard* interpretation, and then x is countable
iff Phi(x) is true in *that* interpretation. The equivalence
Phi(x) is true
<->
x is countable
only holds for the standard interpretation. For other interpretations,
the equivalence may not hold. "Countable" may not even be definable
for some interpretations.
>This is ridiculous.
>The question was whether the FORMULA ITSELF,
>whether phi, by itself, defined countability.
Without an interpretation, a formula doesn't define anything.
It is a formula, together with an interpretation that defines
a collection.
>This line was not worth pursuing.
As far as I understand (which doesn't mean a lot, since I'm
not an expert), what I've been telling you is the usual
notion of what it means for a formula to define something.
george
Daryl McCullough wrote:
> >> To me, it seems that for a formula to be said to define something,
I interrupted,
> >Oh, please. This is not a matter of opinion.
> If there is no formal definition of what it means to
> define something, then yes, it is a matter of opinion.
I insisted,
> >The general linguistic behavior of the community does
> >factually settle the issue.
And got THIS bullshit as a response:
> According to the community, "countable" is perfectly
> definable in the language of set theory.
That is just utter bullshit.
"The community" does not use terms like "the language of "
this, that, or the other, in any consistent way anyhow.
The standard classical paradigm of first-order logic has a
PRIOR definition of a FIRST-ORDER LANGUAGE.
The question of whether something is or isn't a first-order language
is LOGICALLY PRIOR to the question of whether it is or isn't
"the language of set theory" or "the langauge of arithmetic" or
the language of whatever. Finally, "the community" has been taught
the Lowenheim-Skolem theorem, so the community knows damn well
that countability is not first-order definable, not in the language of
set
theory OR IN ANY first-order language.
> >> you have to rely on an *interpretation* of the language
> >
> >And this is just idiotic bullshit, Darryl. The WHOLE POINT is that
> >first-order theorems are INDEPENDENT of the interpretation.
>
> What relevance is that? Whether a formula defines something
> or not has nothing to do with theoremhood.
Again, I repeat, utter bullshit.
There is no DIRECT connection with theoremhood, but IN LIGHT OF
the completeness theorem, IN LIGHT OF the fact that (syntactic)
theoremhood
does not just HAPPEN to CORRELATE with (semantic) logical consequence,
with AGREEMENT ACROSS interpretations, whether a candidate definition
does or does not get elected has a LOT to do with theoremhood, IF the
definitions
are being phrased IN A FIRST-ORDER LANGUAGE and that paradigm is being
presumed as relevant.
> Perhaps you are thinking that a formula Phi(x) defines a set S if
Oh, shut up.
In order for that to be coherent, one would need a prior definition of
"set" --
GOOD LUCK!
ANd I told you in any case that "a formal definition of 'definition'"
was NOT relevant
here. What is relevant instead are some basic minimal standards that
anything
purporting to be a definition would need to meet; definability would
need to be
scientific rather than religious, i.e., to be falsifiable, NOT
verifiable.
And, moreover, NOT, necessarily, DEFINABLE.
> but by that criterion, only recursive sets are definable.
I repeat, shut up.
Sets simply have nothing to do with this.
That is a completely other can of worms.
> That is *not* the usual notion of "definable".
Dipshit, I repeat: if there were a standard, YOU WOULD KNOW IT.
You're an expert. Your disclaimers at the top to the contrary just
indefensible hypocrisy. I have a way of bringing out the worst in some
normally good people.
> The usual notion, as I've said,
Well, YOU'VE said YOUR opinion, but that IS NOT sufficient to make
your opinion "usual".
> is via a standard *interpretation* of a language.
This is so idiotically ass-backwards as to be insufferable.
In the first place, once there is more than one interpretation,
HOW EXACTLY is anyone supposed to INDICATE WHICH is "standard"?
If this indicability existed prior to the creation of the
thing-that-has-myriad-
interpretations (that thing is usually some sort of formal structure,
subsuming
some particular first-order language), then WHY WAS THERE EVER ANY NEED
for the creation of the formal approximation? My point plainly and
simply is
that THERE SIMPLY NEVER WAS any such need; it was ALWAYS the case that
IF the alleged "standard" could be specified via some other linguistic
means,
THEN THE WHOLE ENTERPRISE was conductable in THAT language.
The formal approximation is just irrelevant. One retreats to the
formal approximation
because theoretical results have been proven about it, because it
offers a modicum
of tractability as a result of that. But once that happens, the LIMITS
of that tractability
pose DIFFICULT PHILOSOPHICAL problems for ALL STRONGER treatments, VERY
MUCH INCLUDING the one via which the original standard was specified.
> >> So, in particular, a formula Phi(x) is said to
> >> define "x is countable" in case
Shut up.
YOU DON'T KNOW.
> >> Phi(x) is true in the interpretation
> >> <-> x is a countable set
This is incoherent. If "Phi(x) is true" has to be qualified
by "in the interpretation" then "x is a countable set" MUST ALSO
be similarly qualified. The fact that you didn't do this makes your
formulation UNGRAMMATICAL.
OBVIOUSLY, if Phi(x) can be true in some interpretations and false
in others, then you are going to collapse into the absurdity that
x is countable AND it isn't.
> >If Phi can be true in some interpretations and false in others,
> >then the question must arise about whether x can be countable
> >in some interpretations and not countable in others.
>
> You pick a *standard* interpretation, and then x is countable
> iff Phi(x) is true in *that* interpretation.
That is infinite regress, DUMBASS.
That MERELY removes the question of "definability"
FROM whether "countable" is "definable in the language"
TO whether "the standard interpretation" is definable in the
language. OBVIOUSLY, NO interpretation CAN be definable
IN the object language! Defining interpretations REQUIRES a meta-
language! If you are defining something in terms of the standard
interpretation then you are defining it in the language IN WHICH THAT
interpretation is defined/indicated AND NOT in the language you CLAIM
to be defining it in.
That point is moot in any case.
To the extent that any particular standard model occurs for any
first-order theory with multiple models (which, by Godel 1, is all of
them,
all of them with any decent amount of strength, anyway), everybody
advocating that standard suffers under the nearly unberable BURDEN of
proof
that THAT model DESERVES to be THE standard. There are A FEW first-
order theories where you can meet that burden, and PA is one of them.
But PA basically meets the burden by having only 1 model up to
isomorphism
AT SECOND order, and by further having only 1 first-order model that
agrees
with that one. FIRST-ORDER THEORIES IN GENERAL ARE NOT so lucky.
And in any case, even in this PARTICULAR case, to say "we've solved the
problem
of definability by appealing to the standard model" is just bullshit.
We were talking
about definability IN A FIRST-ORDER LANGUAGE and the "solution" is by
appeal
TO A SECOND-order construct.
> The equivalence
>
> Phi(x) is true
> <->
> x is countable
>
> only holds for the standard interpretation.
That was MY point, DUMBASS.
> For other interpretations,
> the equivalence may not hold.
DUH.
Ergo, ipso facto, phi DOES NOT define "countable".
> "Countable" may not even be definable
> for some interpretations.
"Countable" is not definable FOR ANY interpretation,
DUMBASS, because it is not a MATTER of interpretation!
It is a PRIOR concept TO interpretation! BEFORE you could know
WHICH of these various models was "standard", before you could
even DEFINE/CHOOSE the standard among all the models of PA,
you would HAVE to ALREADY *know* what COUNTABLE means!
> >This is ridiculous.
>
> >The question was whether the FORMULA ITSELF,
> >whether phi, by itself, defined countability.
>
> Without an interpretation, a formula doesn't define anything.
That's completely idiotic, Darryl.
Obviously, in the case of any and everything that IS provable,
at first order, the interpretation SIMPLY DOES NOT MATTER.
That's WHY we BOTHER proving things.
> As far as I understand (which doesn't mean a lot, since I'm
> not an expert), what I've been telling you is the usual
> notion of what it means for a formula to define something.
Your understanding is wrong.
Your acquaintance with what is "usual" is not as broad
as you think it is, either. Moreover, even if you were to prove that
69% of successful researchers believe that it is reasonable to talk
about interpretation-dependent definitions at least 69% of the time
(which would imply that your line is tolerated almost half the time,
which would, under normal circumstances be an adequate defense),
it would still be the case that a moment's reflection would reveal
that,
here as elsewhere, a practice that has become common/convenient,
among those who know, is, logically, just sloppy.
David C. Ullrich
Just curious about something:
You know that a lot of the people you're talking to are
very well aware of the L-S theorem. In spite of this
many of them, seemingly people who know what they're
talking about, are saying that "countable" is in fact
definable in set theory.
What I'm curious about is this: Given the above, do
you think it's possible that your definition of
"definable in set theory" is different from the
standard definition?
************************
David C. Ullrich
Daryl McCullough
>Daryl McCullough wrote:
>> According to the community, "countable" is perfectly
>> definable in the language of set theory.
>
>That is just utter bullshit.
Well, you are wrong. "Countable" is perfectly definable
in the language of set theory, Skolem-Lowenheim or not.
You are not being reasonable. If you want to propose
your own definition of "definability" go ahead, but
don't appeal to the "community" when you are stating
something that is at odds with the way everyone else
uses the term.
Daryl McCullough
>Oh, shut up.
Fine.
Tom
> By Tarski's theorem (a formal version of the Liar
> paradox), no sufficiently strong formal system can consistently define
> what "truth" means in that system. So to fully describe ANY formal
> system you need to use another (usually stronger) system. "It's turtles
> all the way down".
>
> (Note that some knowledgeable posters to this NG disagree with that
> view. I'll leave it to them to state their side of it.)
Dear Ms Knox,
If I may firstly utter a word of apology for intruding, I would also
say that even Tarski himself (e.g. in his much praised Introduction),
and entirely for pedagogical concerns, deliberately exercises analogous
compromises (e.g. through making no distinction between sets and
classes). IMHO, it might be entirely plausible to tell a newbie that
e.g. sets are classes, or that perhaps proof is truth, as long as it is
_crystal clear to both parties_ that such compromises are involved.
(After all, it took Goedel to determine the epistemological status of
PM.)
Ms Knox, I have tremendous respect for your person, and your
competence. The attention to detail you exhibit is the main
characteristic of personality I strive to develop. Nevertheless, IMHO,
if teaching involves no compromise at all, many a newbie may never make
it to the core (which Hofstadter himself describes as the matching of
patterns).
Thank you very much indeed and I am sorry.
Happy New Year to All.
Kindest regards,
Tom
Daryl McCullough
>Daryl McCullough wrote:
>> >> To me, it seems that for a formula to be said to define something,
>
>I interrupted,
>> >Oh, please. This is not a matter of opinion.
Okay, according to another thread ("Metamathematically True or False?"),
Smullyan makes the following distinction:
A formula Phi(x) expresses a set S of natural numbers if
Phi(n) is true <-> n is an element of S
A formula Phi(x) represents a set S of natural numbers if
Phi(n) is provable <-> n is an element of S
I assume that both of these can be generalized to sets other
than sets of naturals. Either one could plausibly be used as
a definition of the concept that Phi is a definition of S, but
the concept of "expressing" a set generalizes to non-r.e. sets,
while the concept of "representing" does not.