Re: Skolem's Paradox and why is math the way it is?
From: KRamsay (kramsay_at_aol.com)
Date: 09/21/04
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Date: 21 Sep 2004 07:02:46 GMT
In article <39d6e584.0409171808.573070f6@posting.google.com>,
troubled6man@yahoo.com (J.E.) writes:
|And for the mathematicians that think this is unneccissary, then how
|do we know everyone is doing the same math if the axioms don't
|describe "real numbers" uniquely?
This is not a job the axioms were ever meant to do.
You're faced with a sort of bootstrapping problem. Some mathematical terms
are of course defined in terms of more elementary ones. But somehow you
need to get the whole system started up. How can we explain it without
getting into an infinite regression?
It's actually a problem with language generally, not just mathematics.
People just tend to notice it in a mathematical context. The way we
bootstrap language is a somewhat messy process in which we don't at all
"define" the initial vocabulary we work with. Instead, we have a process
of informally becoming acquainted with words and their meanings
by observing how they are used.
The concept of a model of a set of axioms isn't a good starting point for
bootstrapping your way up, because it depends upon our already having a
concept of a set of elements (for the domain of the model). If you don't
have such a notion, then it doesn't make any sense to talk about models
of axiom systems. On the other hand, if you do have such a notion, then
not all of the mathematics you are doing is created within one of these
models you are considering; the mathematics you used to discuss the nature
of these models is defined independently of them.
To avoid an infinite regression, you need to have some base theory that
can be understood in its own terms, and not translated into a language
of models of the theory. Perhaps you don't believe the axioms of ZFC,
but believe they are consistent. Then you can prove theorems in ZFC, with
the understanding that you don't believe these results, but only believe
they hold inside models of ZFC. But the theory in which you reason about
the models of ZFC (your "metatheory") has to make sense or else you have
just procrastinated the problem of interpretation for one step.
|How do we know which model is the "intended" interpretation of set
|theory?
A complete answer would require a long discussion of set theory. But take
a simpler special case.
How is it possible for a student to know that when the instructor says
"let f be a function from Z to Z", the intended meaning is what the
student thinks it means? Really, it's essentially the same way as we
come to know that when someone says "apple" they probably mean what we
think they mean.
It's always possible to come up with unintended interpretations that are
formally consistent. If every time I say "apple", you interpret it to mean
"apple if in the northern hemisphere, orange if not", your interpretation
is probably going to stay consistent with what I am saying for a long time.
But it's a far less natural interpretation, which is how we are able to
know that it's not what I mean.
I can try to block such misinterpretations by providing more explanations
of my terms, but ultimately there's no way for a speaker to prevent a
listener from cooking up an artificial interpretation different from the
one intended. Nonstandard models are just another way to cook up such
interpretations. For any given countable model of ZF I can name, the
interpretation of mathematical statements as being about that model is a
much more artificial interpretation than thinking that what I mean by
"apple" in the southern hemisphere is "orange".
Moreover, the interpretation of statements as being about models relies
upon there being a more fundamental level of mathematics in which it's
possible to talk about the models in the first place.
|Why do we care about "uncountably many" real numbers when in reality
|there are only a countable number that we can "prove theorems about"?
You've been writing as if you knew of a well-defined countable set of
real numbers that includes all the ones we can prove theorems about. I
don't think there is such a thing.
Consider for example the set of reals you considered in another posting,
the ones proven to exist in ZFC or maybe it was ZF. This is a countable
set, true. But it's not a set that can be defined in ZFC. The property
of being definable *in those terms* cannot itself be defined in those
terms. There's a subtle catch.
We can extend the language of ZFC so as to make it possible to define that
countable set of reals, but when we do, we are able to define more reals
in that extended language than we could in ZFC itself.
It's not clear to me that there's such a thing as absolute undefinability.
Goedel had a few remarks on the issue. He observes that if we could define
such a thing as absolute undefinability, there aren't any ordinals that are
absolutely undefinable. (If there were any, there would be a least one.
But being the least ordinal satisfying the definition would define it.)
So anything that can be defined in terms of ordinals similarly can't
be categorized as absolutely undefinable. That's a lot!
|Warning-- if the following question offends you, just pretend that it
|is rhetorical and merely answer the OTHER questions. --Is this some
|kind of Platonic conspiracy or vestigal holdover?
It's not a conspiracy, because those who regard Platonism in mathematics
as valid are simply supporting what they see as truthful. On the other
hand, those who see it as a vestige are merely hoping that the truth,
as they see it, will win out.
I say that because I don't think the relationship between "Platonism"
or "realism" and their competitors has changed much in a long time. It's
been discussed (usually without much success) for a really long time,
and I just don't see a trend either in favor or against Platonism. The
only way that I can see a person concluding that the discussion was going
to turn decisively one way or the other, is if they believe they know
which side is right, and believe somehow that the truth will win out.
In article <39d6e584.0409180746.204fda1d@posting.google.com>,
troubled6man@yahoo.com (J.E.) writes:
|> Second-order logic (the recursively axiomatizable version of it,
|> anyway) is still deductively incomplete, so it doesn't make much
|> difference.
|
|Have you studied IF-friendly logic, and why do we use set theory
|instead of just pure logic?
As far as I can tell, "IF-friendly" logic is a topic considered only
by Hintikka. By the way that it handles dependencies between variables,
it permits one in effect to assert the existence of functions having
certain properties. As far as most of the important differences between
first-order logic and second-order logic are concerned, that puts it
closer to second-order logic.
The reasons why set theory is used are more practical than theoretical.
If you want to reexamine the 19th century development of real analysis,
you can feel free to consider whether they could have accomplished the
same things a different way. If they had known of a better way to do it,
they would have taken it.
One could use second-order logic as a starting point, and it wouldn't
necessarily make much difference. We could write down second-order axioms
that uniquely specify a "large chunk" of the universe that the language
of ZF is meant to talk about. By a "large chunk", I mean that it includes
the set theoretical representations of all the objects that mathematicians
other than set theorists consider: all the reals, all the functions from
the reals to the reals, and so on.
| Is there are reason that we want sets
|with more numbers than we can prove things about. It seems silly to
|use a bigger, more complicated model than necissary. As a physicist I
|don't want more numbers in my model than required.
Bigger and more complicated do not go hand-in-hand. The set of reals
provably existing in ZFC is a relatively complicated set. The step in the
construction of the real number system that causes the big "expansion"
in "cardinality" is that of considering an arbitrary function from the
natural numbers to the rational numbers. The notion of an arbitrary
function from one set to another is much simpler than almost any notion
of a function that can be defined by restricted means.
[...]
|No, we do NOT because every theorem can be interpreted in the
|countable model, where there are NOT uncountably many singletons (in
|the class of all sets). Why isn't the countable model the "intended"
|model, and why can't we fix the axioms so that this illusion of more
|members goes away?
That should be "a" countable model. There are uncountably many of them!
The only way that it makes sense to call a model a "countable model" is
if you believe in the existence of a one-to-one correspondence between
the natural numbers and the elements of the model. That correspondence
is external to the model.
It's this bootstrapping issue again. If you don't believe in any
mathematical objects to begin with, you don't ever get to the point where
you believe that there are such objects as "models of ZFC", so there's
no question here. If you do believe in certain mathematical objects, then
by the time you get around to defining the notion of a countable model of
set theory, the model is then an object that exists in a context of other
objects, which belie its claim to including "all sets". From the model
itself, we can construct additional sets that aren't in the model.
Consider: if ZFC is consistent, then there exists a model of ZFC in which
the false axiom "ZFC is not consistent" holds. Inside of itself, there are
no models of ZFC. But we know that if such a model exists, then it's not
the class of all sets, because the model is itself one of the sets that's
not a member of the model.
|> Well, even if you don't assume Platonism, it's still a theorem that
|> there are uncountably many real numbers. So surely that's a good
|> reason to "care" about uncountably many real numbers?
|
|I disagree, there is a theorem, that says for any given model, and any
|injection in that model, there exists another number that is in the
|reals that has no preimage.
The theorem doesn't say anything about models. Don't revise the language
by inserting your own interpretation of it into it.
In order to avoid an infinite regression, there has to be on some level
a language which is understood in its own terms, and not reinterpreted as
being only about models in some metatheory.
|It only proves ONE more number,
It's important to remember that that's all that's needed. There seems to
be a strong temptation to think that uncountability should mean something
more than what the definition says. To resist the temptation, just keep
reminding yourself, it means *nothing* other than the fact that there is
a one-to-one function from the integers into the set, but no one-to-one and
onto function (i.e. one-to-one correspondence) between the set and the
integers.
You can, if you like, iterate the construction. If we have a sequence
a0,a1,... of reals, then we can get another sequence b0,b1,b2,... of reals
by repetition. Then, however, since the a's and b's taken together are
still countable, we can keep going and get another sequence.
This process can be continued transfinitely, to get a set of reals that
corresponds to each countable ordinal. That's aleph-1 reals. (The axiom
of choice was used.)
| and since
|the set of all reals in ONE model is actually countable (from the
|outside) then it seems that the "uncountably many reals" are ghosts in
|the wind. Seems like fancy talk to say that the set of bijections
|between the reals is incomplete because our axioms made it so. Why
|not fix the axioms?
No, the lack of a bijection is a fact. All that reworking the axioms
could do is disguise it or express it in different terms.
In article <39d6e584.0409180736.151b86c7@posting.google.com>,
troubled6man@yahoo.com (J.E.) writes:
[...]
|Well, for instance, people say there are uncountably many reals, and
|I've seen the cantor diagonal thoerem, but that just says that set
|theory doesn't contain abijection from the reals to the naturals
|within itself.
Don't confuse a theory with the domain it talks about.
When we consider all bijections, it's like considering all orange
objects. It makes no more sense to ask whether we might be accidentally
considering only a subset of the bijections (when we say we're talking
about all of them) than it would to say, "But maybe when you talk about
'all orange objects' you are accidentally leaving some orange objects
out."
| It actually only produces one other number, so no
|matter how many theorems you prove, and no matter how many times one
|uses the diagonal arguement, one can only DEMONSTRATE countably many
|real (i.e. prove they exist) and in fact there is a countable set that
|one can interpret all the axioms of set theory on and IT contains a
|set of that can be interpreted as "the reals" and that set is
|countable. Why all the blather about uncountability then?
Remember again the definition of uncountable.
[...]
|So if another physicist walked up to me and said that Occam's
|razor says the other numbers don't exist, how can I argue against him?
I don't know exactly how Occam explained his razor, but it seems clear
that it's better to prefer *simple* theories than it is to prefer ones
that keep to a minimum the things that they have in them (if it's at
the expense of being less simple).
All of these "trimmed down" sets of reals are more complicated than
the full set of reals.
It's possible to write down axioms equivalent to the axioms of ZF that
are reasonably short (in total length). So there's little hope of using
Occam's razor to cut mathematics down to something simpler; such an
argument would hinge on a fairly delicate weighing of theories that
differ in complexity only a little bit. It's nothing like getting rid
of an independent constant in a physical theory.
In article <39d6e584.0409182125.2d1a5ab1@posting.google.com>,
troubled6man@yahoo.com (J.E.) writes:
[...]
|I have
|real concerns (as a scientist) about how to make representations of
|mathematical objects, and I don't see how maintaining the fiction that
|real numbers that we can't describe somehow "exist" helps anything.
|It is platonic mathematicians NOT logicians that claim these numbers
|exist.
The notion of "absolutely impossible to describe" has problems with it,
as I suggested above.
|As for forgetting
|logica, after reading "The Principles of Mathematics Revisited" by
|Jaakko Hintikka, I'm leaning the other way and considering abandoning
|math and using logic instead, I don't know why math is supposed to be
|better.
It contains mathematical logic.
| It's not like anyone can prove that ZF is consistent.
Perhaps you've heard of Goedel's second incompleteness theorem, which
precludes there being a proof in ZF of the consistency of ZF (unless
ZF is actually inconsistent). There's no real point in trying to work
out a loophole.
| Has
|anyone proven the countable model of ZF to be consistent? My first
|and primary concern is understanding the physical universe and my care
|for math or logic is as a foundation for that enterprise.
Logicians don't differ all that much from any other mathematicians in
how they consider these issues.
In article <39d6e584.0409182108.956a219@posting.google.com>,
troubled6man@yahoo.com (J.E.) writes:
|The Berry Paradox, is not well-defined in my book, since "describable
|in less than forty english words" is not well formed. There is no
|such number, and there is no proof that such a number exists.
Exactly. Just as there's no countable set containing all numbers
definable in any way whatsoever.
Keith Ramsay
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