Re: Skolem's Paradox and why is math the way it is?
From: J.E. (troubled6man_at_yahoo.com)
Date: 11/21/04
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Date: 20 Nov 2004 17:43:42 -0800
kramsay@aol.com (KRamsay) wrote in message news:<20041119211903.06525.00000114@mb-m24.aol.com>...
> In article <39d6e584.0411171206.c24e54d@posting.google.com>,
> troubled6man@yahoo.com (J.E.) writes:
> [...]
> |I think the best way to teach quantum mechanics is to assume that the
> |wave-function is real (exists), and that the equations describe how it
> |moves, and that's it, in practise that's all you need and every
> |interpretation takes that seriously to the extent that the
> |interpretation takes anything seriously at all.
>
> I mostly agree, but at least one of my physics professors in college
> considered treating the wave-function as real as wrong, wrong, wrong.
> "It applies only to a statistical ensemble, not to just a single
> system." (!) I would feel at least some qualms about leaving students
> with the idea that there's a consensus opinion among physicists about
> the reality of it.
There is a result that predates QM that says "reproducable correlation
implies some degree of systemic common cause". A certain amount of
correlation could be chance, but if again and again and again you see
it, then something has to cause the correlation. Without this
priciple there is no point to science because otherwise no matter how
much you do something someone can say "you were lucky."
OBSERVATIONS of the wave-function apply ONLY to ensemles. But
SOMETHING has to exist to be the common causes, and you can
demonstrate rather conclusively that anything else being the common
cause can be disproven. I'm fine with telling students that there is
no consensus view, but I'd still present the common-cause theorem, and
showing the students how it is inconsist to assume that QM is correct
and that something other than the wave-function is the common cause.
> If the wave-function is real, then are the Everett-Wheeler parallel
> universes real too? I don't have a big problem with the idea that
> they are, but I know a lot of people do, which forces them either to
> abandon thinking that the wave function is real, or to assume that
> it "really" undergoes a reduction process, the nature of which is
> obscure.
THe wave-function exists in configuration space, which is increadibly
large. To have two "Everett-Wheeler universes" interact with each
other is on a par with all the air in your room appearing on one side
of the room a la the statistical mechanics example of the highely
unlikely. I take ufront to claims that there are "parallel universes"
in the sense that they can't interact with each other because if you
put the reverse Hamiltonian in, then it can, but that's just like if
you reverseed every molecule, then you could drive the process of a
glass falling off a table and shattering backwards to turn air and
friction and shards coluding to form a whole glass that has the
momentum to jump up to the table and come to rest. It's possible, but
we aren't going to be able to do it in the lab, but we discuss
thermodynamics of observables and the fundamental micro-laws
separately becuase they are separate, I think Everett-Wheeler is a bad
theory to mix the two fundamentally different concepts as if they are
a single concept. They are two (1) the Quantum wave-function evolves
according to the evolution operator (schrodinger schrodingler-pauli,
dirac, etc. depending on the level of accuracy) and (2) statistically
the physical state of some macrostates (incomplete despriptions of
many many particles) are incredibly difficult to make evolve into a
physical state whose macrostate appears equivalent to the actual
earlier physical state of the system of many particles.
> |> To be completely "unbiased" and transparent with regard to such
> |> possible qualms as doubting that the natural numbers really exist
> |> probably seems like too much of a distraction. So the usual approach
> |> is basically not to worry about it. The platonist and the formalist
> |> will tend to sound the same as they are developing a theory, since
> |> deducing consequences from some assumptions typically sounds just
> |> like you believe the assumptions to be "actually true" in some
> |> domain. And then one can divert discussion of qualms to such venues
> |> as sci.math.
> |
> |This seems rather ahistorical, the problem is that at one point people
> |took the existance of "classes" for granted and it created problems.
>
> Nearly every reference to a "problem" turns out on closer inspection
> to be a reference to Frege. Are you talking about someone other than
> Frege? People seem to have the idea that because Frege wrote a book
> containing a self-contradictory axiom, we should now all be worried
> about the safety of using set theory.
I'll discuss this later in respose to Cantor.
> Most mathematics, incidentally, uses only a relatively uncontroversial
> portion of set theory. People deal with things like sets of real
> numbers all the time, but not so often the parts that depend on
> (say) the axiom of replacement.
Since we haven't proven that ALL the set theory axioms taken together
are consistent, then you'd expect that if a smaller subset works for
physics, that someone would have tried to prove that that smaller set
of axioms was consistent. Is there such a proof?
> |And "the point" of the modern theory is to avoid those problems, so
> |you have to be clear that everybody is doing the same thing so that if
> |someone gets a problem it's clear that the system is to fault, not the
> |person.
>
> Well, Frege succeeded in this as well. He made his own system, and
> remarked that any deduction could be tracked back to the axioms,
> so that if there was a problem, it would be with one of the axioms.
> And then it actually occurred, with the problem lying with axiom 5.
You missed my point, I'm saying that if we eplain axioms badly, then
people could interpret the axioms in two ways (one contradictory, the
other not) and that if other things are defined in terms of the axioms
then the definitions would be different, and then two people could
have the case that one comes to a contradiction and the other doesn't
and they could trace this DISagreement BACK to a disagreement about
the interpretation of the axiom. If the author were dead by that
point, then we wouldn't know what the "correct" theory was, and in
fact we'd have to start the process that the author should have done
in the first place which is to rewrite the axioms so that all the
interpretations are equivalent in what the theory predicts.
> |> | We can assume induction in the langauge and then latter
> |> |show there there is an induction INSIDE the theory as well, so that we
> |> |don't have to use induction outside the theory, but that's very very
> |> |different than proving induction without proving induction. That's
> |> |proving induction in a theory using induction outside the theory.
> |>
> |> Yes, *if* you assume induction for your original concept of "string",
> |> it's an informal assumption that can't come from inside the theory.
> |> Having proven mathematical induction from the axioms, to conclude that
> |> it applies to actual strings is a further step, requiring either
> |> believing the axioms are correct in some sense or something like that.
> |
> |As I mentioned before, the PURPOSE of the formal theory was to avoid
> |problems, if you actually are depending on the informal theory (I'm
> |reading Quine now and so far it looks like formulas will be built out
> |of "philosophical statements" and not strings and that statements of
> |set theory will be based on other statements, and that the axioms will
> |likely be about assuming the truth of some statements (which is like
> |interpreting a schemata), I haven't finished it yet, so don't think
> |that's how I'm characterizing Quine, it's just my expectation based on
> |where I am so far and it seems at least not to be circular at this
> |point.
>
> I don't know where your first left parenthesis is supposed to be closed.
I'm thinking that at the "current" end of the paragraph and that the
"rest" of the paragraph was accidentally killed, sorry. I'm not even
sure anymore what induction means, I was reading about the foundation
axiom and I get e-induction, but the "standard" induction depends on a
definition of natural numbers or something, which can't be both
outside the theory and inside the theory at the same time, they have
to be different inductions. If it's neccissary outside the thoery,
then I don't know why it isn't put in as an axiom inside the theory,
since the old one can be brought inside.
> The formal theory has various purposes, many of which could be lumped
> together under "avoiding problems". But there are things that it can't
> do for you. It doesn't, by itself, tell you what the statements of the
> theory are supposed to mean. You can, if you like, punt on that question
> and just proceed to work with the theory formally. But if you are going
> to worry about the questions that depend on the actual meaning of
> statements, it all has to start out somewhere informally.
With IF-logic, there is an informal level, it's about winning
strategies, I can handle that because I know what that means, and that
assigns a "meaning" to every formula as being about a game played in a
model (which is IMO just an agreement with both sides on what the
universe of discourse is and about who wins a game of xey for evey x
and y in the universe of discourse). Nothing more, and nothing less.
When I was reading about the foundation axiom I saw a proof that there
is no set of all games, I thought that was an interesting fact.
> [...]
> |Set theory was billed to me as the type-free be-all theory, and I'm
> |not sure if you are refuting that as a misrepresentation that my
> |teachers made or if yoy are agreeing with them, I can't tell.
>
> I'm not sure what type-free be-all theory means. It's "type free"
> in a sense. It's not like the "type theory" of Martin-Lof or of
> Principia Mathematica.
>
> I don't think there is a be-all theory.
Isn't being a "be all" theory part and parcel of the "standard
interpretation" that everything that could exist for anyone that is
"small" enough to be a set, is a actually a set? The only reason to
insist on this rather than just that enough sets exist to satisfy the
axioms is because one wants to pretend like one can have "an
everything", where the universe of discourse of standard
interpretation set theory is superior to all other universe of
discourses.
Otherwise why is that interpretation consider necissary or even
standard?
> |But IF
> |logic avoids having infinite regresses into higher-order logics so
> |that we CAN sit down and discuss how you make theories, so isn't that
> |worth considering?
>
> What infinite regress into higher-order logic is there for anyone?
Like you say in your previous post, you need SO set theory to define
the strongly inaccessible cardinal, in order to get a faithful model
of set theory, but once you introduce SO set theory, people will want
the "other sets" too, because the whole POINT of introducing that
cardinal was to get "all the sets" that were missing in previous
models. You aren't "succeeding" at getting all the sets.
> | Consistent theories imply strategies. That's a
> |REAL implication. But people complain against IF-logic exactly
> |because it's the right size to do that because it isn't "the right
> |size" for NOT doing that (not the right size for formal deduction).
> |And that's silly because IF-logic has ordinary FOL as a part of it
> |anyway, so anything you do with oFOL you can do with IF-logic, just
> |stop using the / or // symbols.
>
> I don't complain about IF-logic because it's not formal (i.e., has
> no complete set of deductive rules); I only fail to see why once
> one has abandoned formality one should pick IF-logic over languages
> that more directly talk about sets.
I wouldn't say I prefer to use IF-logic to "talk directly about sets",
in fact it's to have that informal level that I think I can share in
common with others, I want a shared theory and I find that I have
difficulty taking the standard interpretation of set theory seriously
whereas I can take the existance of games (finite games even!)
seriously. I don't know of a language that talks about sets. If I
wanted to talk about sets I'd choose some SO axioms, translate their
negations into IF-logic, or them together to get N and then I'd not
that for every theorem "N or T" is a truth in all models, and that is
a statement that I understand, that is falsifiable even if I claim to
have a proof (nice for physics), and that doesn't even require that I
assume N to be false in any model, in fact as long as it isn't true in
a model M, then all the (FO) theorems
T are true in M.
> |> For theories with more realistic goals, I would say self-application
> |> is a bit like being able to lift all the rocks that one can make.
> |> It could be a sign that one is strong. Or it could be a sign that one
> |> has a limited ability to make rocks. IF logic can define its own
> |> truth-predicate, yes. But that's a combination of being strong in some
> |> ways, and being weak in others.
> |
> |In what way do you think it is weak?
>
> You can't say that a graph has three connected components, in it.
Do you have a citation for that result, or better yet can you state
your definition of graph and connected component?
> |> | but that doesn't mean there isn't a
> |> |strong theory that *can* do it's own model theory that has set theory
> |> |(and hence everything based on it) as a component. That's what I'm
> |> |looking for now, and I think the excluded middle is the only thing in
> |> |the way really. There is a subsection of the universe where the
> |> |excluded middle holds, and that's what we call set theory, but it's
> |> |intended models (if it has any) live outside that subsection.
> |>
> |> Why?
> |
> |Every model of set theory lacks a set that should exist as much as the
> |alleged "uncounted real" should exist.
>
> If by "model" you mean a set with an epsilon relation on it, then
> this is correct, but people often mean by "model" either a set *or*
> a proper class with an epsilon relation on it. The cumulative
> hierarchy does not "lack" a set that "should" exist-- it consists
> by definition in all the well-founded pure sets.
This is really hard to discuss non-circularly. The words structure,
class, function, set, collection, relation all have definitions "in
the theory" and to use the same words outside of the theory is begging
for confusion. What do you want to take as given? We could have a
third person in the game, and have the third person start talking,
saying "a in M, aea in A, a' in M, a'ea' in A, aea' in E, a'ea in A,
a'' in M, a''ea'' in A, a''ea' in A, a''ea in A, a'ea'' in A, aea'' in
A, a''' in M, a'''ea''' in A, a'''ea in A, a'''ea'' in A, a'''ea'' in
A, aea''' in A, a'ea''' in A, a''ea''' in A, ..." and but where the
third person chooses freely whether to say "xey in A" or "xey in E",
the problem is that the third person might never shuts up for the game
to start, but the A-team and the E-team could agree to make
assumptions and "act as if" the third player said something. The
teams could look at the formula F and as long as they are only
concerned with a winning strategy for ALL models, then the teams can
consider
the fragments of the formula that always have a winning strategy in
any fixed model (a game played after the third player shuts up) and
ask themselves "if A-team wins this fragment O in the yet to be
determined model we eventually get, will one of us be then be able to
always win the whole formula F?", and if so "mark" the formula O-A =>
F-A if A would win and O-A => F-E if E would win. Then if they ask
themselves "if E-team wins this fragment O in the yet to be determined
model we eventually get, will one of us be then be able to always win
the whole formula F?," and if so "mark" the formula O-E => F-A if A
would win and O-E => F-E if E would win. Assuming it was possible to
mark the formula each time, then you consider the four cases:
If F is marked "O-A => F-A" and "O-E => F-A" then F is false.
If F is marked "0-A => F-E" and "O-E => F-E" then F is true.
If otherwise marked, then F if neither true nor false because who wins
clearly depends on what the third person says.
If the teams can't agree that winning one part implies another, the
it's not really considered bad, because this is the original case with
proof-theory, because the F that are true or false correspond to the
sentances "N or T" where T is either a provable theorem (without using
/ or //) or the negation of a such a theorem and N is the IF-FOL
translation of the negations of the second order axioms of which T or
~T is a provable theorem. The only purpose therefore of defining a
model more precisely as a finished whole is to conisder the truth or
falsity of an UNprovable statement. So if makes sense to have a
"type" based theory where first you consider all the provable
statements, and then you could consider things built out of them (let
the provable statements BE the universe of discourse), I myself would
be interested in considering a model where the elements of the model
itself was all the provable statements of the form T="Ex Aa aex <=>
S[a]" where "N or T" is true as defined above for the N correponding
to the set theory axioms, and then saying T1="Ex Aa aex <=> S1[a]" and
T2="Ey Ab bey <=> S2[b]" means we assume that (T1 e T2) in E if
"Ex (Aa (aex <=> S1[a])) & S2[x]" is a provable true theorem as
defined above, and (T1 e T2) in A otherwise. That would be a most
intriging model to me. But I haven't checked to see if it's a model
where the usual theorems are true, but it's a boot-strapping kind of
model where you know what everything actually is, I hope it wasn't too
abstract for you.
As far as I'm considered a model can be anything whatsoever as long as
the players agree what a, b, etc. are in the model and whether "aeb in
A" or "aeb in E" is to hold for each a and b in the model.
I don't understand your claim about the cumulative hierarchy, once you
"finish" the model, someone can take the standard interpretation and
say that some sets are missing, isn't V=L considered "restrictive" by
mathematicians?
> [...]
> |And once you extend the definition of set to have non-excluded
> |middles, the standard proof about the lack of a set with a specific
> |property goes away, the theorem becomes "the graph of the bijection
> |between a set and it's power set does not have an excluded middle,
> |even if it exists". And yes we can make a hierarchy based on
> |equivalnce classes of sets based on graphs with excluded middles,
> |it'll be just like cardinality theory if we do it right.
>
> The proof that a set does not have a bijection with its own power
> set works fine in intuitionist logic, which doesn't incorporate
> the law of excluded middle as a rule of inference. If f:X->P(X)
> is a function, then {x in X : x is not in f(x)} is a set that is
> not in the image of f. If f(y) = {x in X : x is not in f(x)}, then
> y is in f(y) if and only if y is not in f(y). That is inconsistent
> in intuitionist logic.
I've never studied intuitionist logic, and I don't even know what
"inconsistent" is defined to be in intuitionist logic. Assuming "y is
in f(y)" is true leads to ~"f(y) is in f(y)", which is a
contradiction. Assuming ~"y is in f(y)" is true leads to ~~"f(y) is
in f(y)", which is a contradiction. So we are lead to conlude that "y
is in f(y)" is neither true nor false. This is a problem if you have
an excluded middle, but otherwise, what's the big deal. I suspect
from talking to you that intuitionist logic DOES have an excluded
middle, but that it has a limited power to discuss that fact and a
limited ability to infer from that fact.
> I think you're avoiding this conclusion by considering it possible
> for some statement to be true if and only if it is not true. I
> understand how IF-logic permits such a thing to occur, but it's
> not convincing to me that this makes good sense.
And I don't understand what IT MEANS to not have an exluded middle and
disallow that. We are coming from different worlds and I don't know
the basis for your ideas, while you know that my meanings are derived
from the semantics of games. So it's a bit unfair for me to be
explaining things to you. And there is going to be huge problems
because we define implication differently, I use a stronger version
than use, it is more expressive and says more, but therefore is has
fewer rules of inference. So my biconditional says A <-> ~A means
that
(~A or ~A) & (A or A) which is logically equivalent to ~A & A which
means that A cannot be true or false, full stop.
> Once we have a logic with three truth-values, true, false and
> indeterminate, I don't see how it can be invalid for me to start
> talking about which of the three bins a sentence falls into. The
> claim that a certain sentence simply fails to be true, i.e., is
> either false or indeterminate, appears to make logical sense. It's
> just not a claim that can be expressed in IF-logic.
Huh? Truth is about a winning stratgey in all models, same with
falsity. Being neither can mean totally different things. It could
mean that it has a winning strategy for one team in one model one for
the other team in another model, or it could mean that there is a
model where the sentance has no winning strategy for either side. Why
does it make sense to lump these cases together? What we care about
is truth, which means winning in all models. For instance if N is the
negation of an axiom and T is a theorem of the axioms, then "N or T"
is a validity, true, true in all models, in all worlds. And there are
TWO ways to "negate" that CONCEPT, to say "false in all worlds" or to
say "not the case that it is true in all worlds". IF-logic does the
FIRST, because the point is that you write SENTANCES and then you
assert that their truth means something about THE DOMAIN OF DISCOURSE,
this is what we do in physics, we state "the world is such that T is
true", it's what philosophers do. To discuss anything else invovles
actually quantizying over possible worlds, which I didn't think people
were still serious about doing.
> This is really what I want Hintikka to tell me: why am I mistaken
> when I think I've made an assertion such as "the domain of discourse
> is finite" that is true *exactly* when a certain sentence of IF-logic
> fails to be true. In what way is this an incoherent sentence? It
> seems as though he simply ham-strings his theory to make it impossible
> to say certain things in it.
It's an funny difference of opinions because you think he ham-stringed
his theory, and it seems to me that you want him to ham-string his
theory when he hasn't. Validities are what we care about, things true
in all worlds, when you look at a sentance S and say "I want to
describe a universe where this is true", then if the sentance is oFOL
then you can write the validity "~S or S" down consider the first part
to be a statement of the world of discourse and a latter part a
theorem of the universe. If that's all you want to do then you don't
need IF-logic, you could pick some first order axioms and write N=~A1
or ~A2 or ... or ~An and then be assured that for every provable
theorem T, the statement "N or T" is a validity, and so you could if
you thought the axiom system A1 & ... & An described the universes of
discourse you had in mind consider that to be a description. The
point of IF-FOL is to have a stronger version of N where is is not the
case that "N" has have a winning strategy for the A-team in ANY model,
but where the validities "N or T" can still describe all the universes
where every T has a winning strategy for the E-team for every validity
"N or T" and in THAT sense "N" is a description of the worlds under
consideration. It's a much much stronger statement about
conditionality, the whole point of IF-logic is to have stronger more
expressive statements.
"(~A or B)" is stronger than "A=>B"
"~A" is stronger than "it is not the case that A is true", so we can
say things that you can't otherwise say in FOL.
It's is a WEAKER claim about the universe of discourse, it merely says
"consider the worlds where the theorems are true" not "consider the
worlds where the (second order) axioms are true". You want nonsesnse?
It IS nonsense to say "consider a FO universe of discourse where the
SO axioms are true" if you just want the FO axioms and oFOL theorems,
then IF-FOL is useless. If you want to translate SO axioms into FO
language without assuming a universe of discourse that INCLUDES SO
entities. Then instead of considering the universe where all the
axioms are true you should instead consider the universe of discourse
where all the theorems are true, do you really not get it? The IF
logic claim is more powerful because when you actually translate SO
axioms into IF-FOL you get statements that do NOT have contradictory
negations in second-order logic, that is because they simply do NOT
have winning strategies for either side, so INSTEAD of considering
universes where you can VERIFY the truth of the axioms, you consider
the ones where it is impossible to VERIFY the negation of the axioms.
It is simply astonding that the ordinary proofs of theorems carry over
to IF-FOL validities because it is a much much stronger claim to say
that the theorems are all true in models where the axioms are not
false, then to MERELY say they are true when the axioms are true,
because the INTENDED axioms are NOT true in any model. I'm trying to
explain why what you are asking for is nonsense, but I don't know if
you can see it.
> Wittgenstein gave an example of an incoherent description: "when it's
> five o'clock on the Sun". He imagines someone asking, "what does that
> mean?" and the answer is, "Just like what it means to be five o'clock
> here-- but on the Sun". I can imagine that some of the things I think
> and talk about are confused in sort of this way: it only seems to me
> that I'm considering a well-defined proposition. The only way I can
> see how it would make sense to consider a system like IF-logic to be
> ultimate is if I were confused in such a manner about the things that
> I say that appear not to be expressible in IF-logic.
The claims you want to make (contradictory negations of IF-FOL
formulas) are actually either IF-FOL formulas (in the special case
where the original formula was actually logically equivalent to an
oFOL formula) or the negations are ACTUALLY SO statements and they
require a CHANGE of the universe of discourse to INCLUDE SO entities.
If you want to stay FO, then I'd be hard pressed to come up with some
stronger and more expressive logic than IF-FOL, but maybe there is
one. I think Hintikka had an extended IF-logic.
> [...]
> |I can grant that no deception was intended if you think it was just a
> |technical inaccuracy, I'm a bit scarred (as in maimed, not as in
> |afraid) in that I still do not know the order in which to resolve
> |things, but I'm still hoping this book of Quine's I'm reading will get
> |everything in the right order.
>
> It seems a bit unlikely that there was an intention to deceive, although
> those of us reading you on sci.math have no first-hand knowledge of it.
>
> I think it will be mighty difficult for you to "order" your development
> unless you admit the necessity of starting informally, for at least a
> brief time, or you decide to go "formal" all the way and not care about
> such things as whether the theory has a model.
Which theory are you talking about? Quine is discussing statements,
and Hintikka has games, either of those seems fine to treat
informally, they are both just abstractions of VERBS, things I can do
personally. And so it is disprovable and subject to experimental
observation ultimately.
> Lorenzen likened the process to the process of building a ship at sea.
> With one ship, you can build another one inside of it. But if you are
> just tossed out into the ocean, with some building materials, you have
> no choice but to learn to swim, and then maybe build your system while
> swimming.
>
> Figure out what original undefined terms you are willing to accept.
> Then decide what axioms about them you are willing to accept. Then
> go from there.
I'm trying to do that, but I no one is defining set theory, I could
"learn to swim" and then I could "build something", but how do I know
what I personally build "is a boat"?
> [...]
> |I think this is
> |about describing separation badly, I don't think it's about logic or
> |language at all, so is "EF ~ (Ax Ey Az (zey) <-> (zex and 0=F[z]))" a
> |good descrpition of what is intended by separation or not? I still
> |don't know.
>
> What is intended by separation being false, do you mean? If I'm not
> misunderstanding your notation, this is what I remember as being the
> second-order separation axiom (negated).
Yes, negated, that's the form we want because we "or" the negations of
the axioms together to get N so that for a closed oFOL formula T we
can assert "N or T" is true to assert that something is a theorem.
What are the other second order axioms? Do we need a second order
foundation? A second order replacement (substitution, collection)?
> [...]
> |> Don't confuse rigor with formality. Only a formalist needs to have
> |> "formula" defined *inside* of a formal theory separately from outside.
> |
> |I'm unfamiliar with your definitions of "realist", "formalist" and so
> |on, it just wasn't covered in my education. Some people on this
> |Usenet group have told me to take a set theory class, I have, they
> |didn't cover those terms. Are they covered inmost classes and was I
> |just unlucky?
>
> I don't know of many places offering courses that would cover it.
> Philosophy of mathematics is not a big field, and I wouldn't be at
> all surprised if you went somewhere where there simply weren't any
> courses in it. You could try something like the Benacerraf and Putnam
> book.
Is that book consistent with the definitions you were using?
> |I'm fine with IF-logic saying that some string represent well-defined
> |games and that some games have winning strategies for one side, and
> |some for the other, and some don't. I'm find with someone making a
> |claim that the set theoretical universe is such as to make the string
> |(~A1)or(~A2)or...(~An)or(T) true (when fleshed out with the right
> |axioms for A1 through An and the theorem for T,
>
> Actually, the chunk V_a of the set-theoretical universe I was referring
> is supposed to make these negated axioms ~Am fail to be true. Of course
> one can't say simply "fail to be true" in IF-logic. The string
> (~A1)or(~A2)or...(~An)or(T) is supposed to hold true in _all_ domains
> of discourse, since for all of the ones other than the intended chunk
> of the set-theoretical universe, one of the (~Am) holds. And for that
> intended one, (T) does.
Close, (~A1)or(~A2)or...(~An)or(T) is a validity (true in all models)
if T is true in all models where (~A1)or(~A2)or...(~An) is NOT true,
even if (~A1)or(~A2)or...(~An) is neither true nor false in that
model. In fact there is no proof that a model exists where
(~A1)or(~A2)or...(~An) is actually false. So IF-FOL conditional (and
negation) are STRONGER than implication or contradictory negation.
The statement (~A1)or(~A2)or...(~An)or(T) is a validity is stronger
than the oFOL statement A=>T for a FO A (and a SO A cannot even be
stated in oFOL).
> | and if they say that
> |it's true for all theorems, then that just leads to the question "what
> |are the theorems", but that isn't confusing because we are forever
> |talking about actual games and these questions are about what elements
> |can be selected for substitution in the games and which atomic
> |sentances AeB are going to be true, and which are going to be false.
> |It's forever a discussion about the rules of the game, no infinite
> |regress into types.
>
> It seems, then, that we have found a dictionary between whatever I
> might have to say mathematically that concerns this large submodel
> of the cumulative hierarchy (the sets having rank less than the
> smallest inaccessible cardinal) can be translated into a language
> that you can understand!
I'm still not sure that there is such a submodel that fits the
standard interpretation. If I look at the "set theory validities",
i.e. the valid formulas "N or T" where T is an oFOL closed formula and
N is the IF-FOL translation of the SO negations of the SO set theory
axioms SO alternated together, then it can be true in all models, but
that doesn't mean that there is a model where N is actually false. It
just means that "N or T" is a validity.
> |This is FINE for physics because in physics we ALSO play verification
> |and falsification games in the laboratory, so I can make them match
> |up, and we make the SAME kinds of satements about the universe.
>
> But you don't play uncomputable strategies against the universe,
> so far as I know. IF-logic only works the way that it does because
> one is implicitly assuming the possibility of using uncomputable
> strategies.
Where did computable come from? The universe itself plays the part of
the initial falsifier, the existance of the winning strategy means you
can defeat the universe at the game, every time. If you can proof the
validity, then you can show that you could win the game "N or T" in
any model. So I'm not making the assumption that the proofs construct
all the validities, so why should I? And you are totally forgeting
that the game is usually a premise as well as a conclusion, so to say
that something is continuous, you might be more precise and say that
the game "Ax Ay (~(0<y) or Ed Ar ~(|r|<d) or |f(x+r) - f(x)| < |y|))"
is true for f, and the meaning of the sentance is that there is a
winning strategy for the game, you could THEN use that strategy to
prove other things about the function, like that "Aa Ab (a<b &
f(a)<f(b)) => (Az ~(a<z<b) or (Ec a<c<b & f(c)=z))", you don't have to
CONSIDER what the word "computable" means to do any of this.
Validities is based partly on the fact that an oFOL statement T always
HAS a strategy for one side, so then you can assume a strategy for one
side and try to prove that a strategy exists so that the E-team wins
the game "N or T" and then you assume a strategy exists for the other
side and try to prove that a strategy exists for "N or T" for the
E-team. The word computable (whatever it means) never comes up, the
fact that T is oFOL and has a strategy for either side is useful.
Maybe you are bothered that there are two subjects, logic, the study
of formulas, and set theory the study of sentances T such for a fixed
N (unkown to me) the formulas "N or T" are validities. Set theory
needs logic, it really does, I didn't think this was a problem.
> How else can it be true that either for every x, ~P(x), or there
> exists an x such that P(x)? The only way to win that game is to
> be able to search the whole domain of discourse to check whether
> any of the elements satisfies P!
I don't know what P(x) is, but saying "Ax ~P(x)" is true MEANS that if
you searched the universe that ~P(x) is true for every x, "Ex P(x)" is
true means that some choice of x makes P(x) true, whether "[Ax ~P(x)]
or [Ey P(y)]" is a validity or not is just a matter of whether P(z) is
true or false for every z, if it is, that's enough, then you know it's
a validity. The point of a validity is that since it is true for any
play of the game in any model, it is subject to disproof, that's the
best you can hope for. Proof in science is impossible, so having a
verb that serves as capable of disproof is the best you can hope for.
You can't perform every experiment in every place at every time, but
since the claims are universal, you can attempt disproof to your
heart's content.
> |"The
> |universe is such that when I do this experiment I get this kind of
> |results," and one can make DIFFERENT models to help CHOOSE new things
> |to TEST (in both math and physics). And based on the results, you
> |might decide to change the rules of the game (new axioms), or just
> |make new definitions to make existing questions easier (for people) to
> |ask, verify, or falsify.
>
> This story holds equally well regardless of what axioms we choose
> to use, however. In fact, if all you care about is generating
> testable predictions, you should refrain from worrying about such
> things as whether the mathematical axioms have a model of the kind
> people think they do. Just deduce consequences! Seriously.
You have to make a choice about WHAT to deduce consequences FROM, I
still don't have a basis to start with. Circular definitions of
separation aren't good, and I don't understand what the correct theory
is supposed to be, how do I know that what I'm deducing consequences
from is the same thing (or equivalent) to what everyone else is?
> [...]
> |The physics classes I've taken I can teach myself, why is math so into
> |hiding things? I couldn't honestly teach set theory today, even a
> |basic one, because I haven't seen a logical presentation. My physics
> |teachers would answer questions when the students got together and
> |demanded resolution (like when we asked to know how you know when to
> |treat a stick as single object versus each molecule like a separate
> |object, versus each atom as an object versus electrons and nucleuses
> |versus electons and quarks and gluons), and they did so in
> |non-circular ways.
>
> Almost nothing you've asked about has been about the "logical
> development of the subject". (Do you have any questions of the
> form, "Is X a theorem of Y?" or "Is X an axiom of Y?") There
I've asked many times what the axioms are. What are the intended
axioms of set theory? Is separation one of them or does some SO
replacement and regularity take care of it, or is "limitation of size"
considered better? I apologize for being unclear on this, I'd love to
know the correct axioms, the ordinary FO axioms seem to have bad
models, so better ones are fine with me, I'm not sure that the new
ones I'd get are better but how can I know until I see them first?
> has been scarcely any strictly mathematical (as opposed to
> philosophy of math) question in this discussion. You ask things
> like "How do theorists know that the SEQUENCE to generate h
> [Planck's constant] exists in ZF?" that doesn't have any clear
> meaning.
Now you can't hold me responsible for whatever other people bring to
the discussion. I would wonder if ZF is the right theory in which to
create a model that PREDICTS the value of h [Planck's constant], but
as far as I'm considered the value we measure in the lab is a rational
number. The point is that people assert that "everything" is in set
theory, but they go OUTSIDE set theory to assert that, and I don't
know WHERE that brazen confidence comes from. ANYONE can just ASSUME
that all sets are in their theory. Here's an axiom:
"Axiom of assertion: For all x", then you assert the "standard
interpretation" that "all sets exist", which "cleary includes every
subset of every set". If I make it this extreme then everyone knows
it's silly, everyone admits that there is more than one axiom, and the
burden of proof is on the believers in the standard interpretation to
say the "every set" exists in the intended model, not to say "well we
have every set, see the Axiom of Assertion, so just take the class of
all sets and the normal epsilon class-relation and that's a model with
every set", that's is SO clearly based on a wild meaning of the axiom
of assertion that isn't really in the axiom at all.
> I don't think physics classes and mathematics classes are different
> the way you think. Over and over, in physics we were taught
> theories that we knew were not quite correct, but which we were
> supposed to value as useful approximations to the truth, helpful
> in seeking better approximations later. Our main job was to work
> with them, in spite of whatever imprecisions were involved.
In physics if they had a wrong theory, they'd state at the very very
beginning that it was wrong, that's very different than saying that
theorems are true for a whole semester.
> There were a number of places where we had to fudge. Infinities
> appear in certain places that one just has to accept as being
> not quite right. The energy in the electric field of a charged
> point-particle is infinite, and the electric field diverges to
> infinity at the particle. What is the electromagnetic force on
> a charged particle? They gave us a formula, except of course it
> can't be correct; the product of the charge and the electric
> field vector at it is undefined, because the electric field goes
> to infinity at the particle itself. So we hand-wave a little bit
> about the particle probably not being *exactly* a point particle,
WHAT! Those can all be fixed by being more careful. NO ONE cares
about "the force on a charged particle," I took electrodynamics after
electrodynamics and I kept asking "why don't we compute the trajectory
field of the charged particle instead of E? And the answer is that
it's harder and no one cares. But it's up to the people who care
about something to make it work, the purpose of most physics classes
is not to teach you how to find analytic solutions, it's to develope
physical intuition so that you can recognize correct results and fix
problems caused by doing numerical approximations sloppily and to get
approximate answers so that you don't have to get the true answer for
every situation.
> and say a few words about test particles whose charge is arbitrarily
> small and so on. The notation "A << B" is tossed around with gay
> abandon, meaning usually that we're supposed to pretend that
> (A/B)^2 might as well be 0. The business about the force law is,
> incidentally, not so trivial to correct.
The force-law is exceedingly complex because accelerating particles
radiate, so you can't just use ma=q(sigma_other-particles E), but
every course I took, I was told "no one cares" about the motion of
charges particles, I think it's partly because most people are only
looking for approximate answers anyway. These are academic anyway,
because most instructors will be honest and say that real applications
solve the equations with computers with approximations anyway, the
derivations of analytic solutions are just to get practise taking
extreme versions of the laws and recognizing how different physical
parts dominate the physics in different regemes, it's about physical
intuition, the PDEs are the PDEs and you solve them numerically in
practise, it's not like we HIDE Maxwell's equation from students.
> What exactly is the Dirac delta function? Well, it's not really
> a function, and telling you what it precisely is, is beyond the
> scope of this course. :-) This willingness to work with an ill-defined
> concept is not just accepted; actual pride is taken by physicists
> in their willingness to dispense with precise definitions and
Distribution theory, if we want to have a pissing contest about who
has better teachers, then I can concede that many many bad physics
teachers exist, in fact that's what I'd like to change, that's my goal
to improve physics teaching. But the physics teachers *I* had were
honest about what they knew and didn't know and they would only say
they were sure when they were, and they would differentiate between
what the books we had said and what the instructor meant, when
something is a derivation versus a "consistenty check", when something
was an approximation and to what degree it was a valid approximation.
I had very good physics teachers, and the fact that it could have been
much better is just a case that physics teaching has room for
improvement. But there is a cultural difference too, so I'll discuss
IF-FOL validities to give an example.
Some validites like "N or T" have "proofs" but others could just
"appear" to be true top someone informally, but since you can't
manually check over all strategies over all models, someone might "act
as if it were a validity", and the fact that "disproof" is possible
leads some safe feelings for physicists, since they ALWAYS operate in
a world where disproof is possible no matter what. So it's not
uncomfortable to explore the "what if" and make hypothesis about the
consequences of the "what if", if the consequences of the "what if"
lead to things that you earlier didn't think to check in the lab and
the lab backs it up, that's good. If later the math falls apart, then
you just look for new math, not matter how often the math falls apart,
the laboratory observations are still there (that's why I *dispise*
how some labs actually THROW AWAY data based on a computer saying that
it didn't matter, it's really bad if someone later disagrees with your
theory or your math because those experiments are difficult and
expensive and you'd have to do them again, nothing should be thrown
away).
> just work intuitively with stuff. One E&M instructor told us that
> "a mathematician would say" that f(x)=e^x does not have a Fourier
> transform, but that he would say it does have a Fourier transform,
> but we just don't understand what kind of thing it is! We solved
> the problem of a plane wave incident on a circular obstacle by
> "integrating" a function that behaves like e^{i * (x^2+y^2)}
> as x and y go to infinity. (Where it oscillates rapidly, pretend
> like it cancels out....)
Um, plane waves don't exist in the lab, and I doubt they ever will, so
that's all just to test a limiting case anyway isn't it, so what's the
big deal?
> Experimental sciences all depend on the arrow of time in a way that
> is not very often explained. I mention this little gotcha partly
> because to me it has a family resemblance the alleged circularity
> involved in using a formal system to prove results about itself.
> One is forced, ultimately, to make assumptions about the initial
> state of the universe, just as in mathematics one is forced, ultimately,
> to assume something about how it applies to the world.
A theory can be used two ways, to assume something about the physical
state of the universe and test the predictions of the theory, or to
assume the theory is correct and use the observations to deduce things
about the earlier (or inaccessible to observation part of the) state
of the universe, success comes from BOTH holding together well, and of
course there is a possibility for more than one model to fit all the
data. No one in physics can say "SR is a true model of the universe"
with any authority, or any model, there can always be another model,
and we don't that it's consistent, or even that it matches experiment,
but we're more clear about the assumptions and the consequences than
my math classes were about set theory.
> Little bits of slop everywhere. Please note that I'm not complaining
> about the courses I took or the instructors I had. I think they
> did a fine job overall, and they were right to expect us to put
> up with some of the messy bits of the subject, basically not to
> get too hung up on the parts that have to be fudged. We were
> expected to keep going in spite of it.
If they were honest about what was legitimate and what was for
reassurance or other touchy-feely reasons, that's fine. Confusing one
with the other on their part isn't smart though.
> The difference in this respect is the in mathematics, we come ever
> so much closer to getting it exactly correct, and ironically get
> to suffer as a result. People come to expect a much higher degree
> of exactitude. If you were not traumatized by the inability of your
> physics courses to explain *precisely* how we are supposed to get
> away with all this stuff, but were traumatized by your mathematics
> doing so-- the only explanation I can find is that you approached
> the two with very different expectations.
I read my physics textbooks years before taking my physics classes,
and with about 3 years more than the "math prerequisites" so it was
usually the case that I could tell what was actually intended from
reading the book, and if a book annoyed me, then I'd supplement it
with another book. There are lot's of good physics books that start
at simple parts and go far, but most math books seem to not want to
start at the beginning, but instead with set theory, which makes the
latter subjects easier if you assume set theory, but there just
weren't good set theory books that I could find. You also have to
remember that reading physics textbooks years early and taking math
classes with the Moore method are about as different as night and day.
> Now, the one thing you've suggested is that at least one of your
> mathematics professors failed to acknowledge the imprecision in
> his remark about the axiom of comprehension. I would have to hear
> just exactly how that exchange went between the two of you before
> I would assume that this is so. If he did pretend to be absolutely
> precise when he was not, however, I'm sorry that he did.
I could simply erase the incident from my mind if I just knew what to
replace it with, I still don't. I want some authority that a
particular theory IS "set theory", making up my own axioms doesn't
tell me that what "I'm doing" is the set theory everyone else is
doing.
> I wouldn't say that it's much worse than the physics professor who
> once was diagonalizing a matrix in class, and did it wrong. I pointed
> out his mistake, and he muttered that I "couldn't" have figured it out
> like that, erased his work, and then did it over more carefully.
I was called (by my friends) into physics classes to sit on on their
classes to call their professors on their mistakes, the only time I
failed was when a professor "started where they left off yestersay
(before I sat in)" in a class I hadn't taken, so it took about 45
minutes for me to recognize that "that isn't a wave". People are
sometimes sloppy, you shouldn't just trust instructors but should you
should attempt to understand everything they say (if the subject is
important to you) and all the reasoning and justification, just
trusting someone to be right is bad news. My frustration about math
is about AXIOMS. If I can't find uncircular books to back up my
teachers (and wasn't allowed when taking the course) and the books
aren't clear and the teachers weren't clear about the definitions then
I don't even know what the subject IS, so to compare that to a physics
class being sloppy about a momentum-eigenstate in a (rigged) Hilbert
space or a plane wave solution or something isn't fair at all, it's
more like if your physics class said that SR was based on the
assumption that "the speed of light is 'true' is all *mumble* frames".
That's preposterous, and just because you can "speak informally" to
disguise sloppiness doesn't make it better, it should be fixable. I
should be able to find out what the axioms are, full stop, this isn't
supposed to be a power-trip or a head-game. In physics I can find
books and experiments to make sloppiness clear, usually bringing some
more math to the table fixes things in physics, and "more math" might
fix the problems in math too, but I can't find books to fix it. I
could make up my own axioms, but that wouldn't be "set" theory, it'd
be "something" theory. To know what "set theory" is is like knowing
what "SR" is, I have to have the definitions or the consequences one
or the other.
> |But the math proffessors just say "why are you so
> |interested in foundations, I thought you liked physics?", I'm just
> |simply tired of being discriminated with, I'm certain that they talk
> |not in circles amongst themselves,
>
> I've never had a lot of conversations with set theorists, but I can
> assure you, conversations between professors of mathematics are no
> more formal, rigorous, and ultra-precise than their conversations
> with students. They are much less liable to be tripped up by
> inadvertent imprecisions of the kind you've been complaining about,
> and consequently they take less care to avoid them amongst themselves.
> The kind of stuff you've been agonizing about is just not worried
> about! When writing papers, of course, there's a third standard that
> is in some ways more formal, but also allows for details that the
> (professional) reader doesn't need to be left out.
There should be some source SOMEwhere that is clear about the
definitions of the field, shouldn't there?
> |and I think it's down right rude to
> |hide the actual logical developement from people just for not being
> |"in the club", this isn't middle school this is science.
>
> Perhaps you think that at some later date, the students who mean
> to go on to become professional set theorists get taken aside and
> have the "real" development of the subject taught to them. I don't
> think so.
Then how do they know, I've seen lot's of books at a naive informal
level and many advanced level books that assume you already know
things not covered in the informal books, where are the course
materials for the intermediate level courses, where are the books of
intermediate level?
> | I consider
> |math to be science.
>
> Then try treating it like one.
>
> The key objections to be made to a physical theory are that its
> predictions are observed to be incorrect, it's been superceded by
> a more comprehensive theory, and that it's needlessly complex.
> The corresponding objections can be made of a mathematical theory
> too, in principle, but I see you making an awful lot of objections
> that are of a completely different kind.
In IF-FOL I can imagine an observational refutation of a proposed
validity (by demostrating game play inconsistent with that
description), but I don't see how you can "observe" other things to
invalidate a theory, the theorems are treated formally if you object
to the English, and then someone will continue using the English, so
for most people it's just a waste of time to "observe" anything in
math.
> Allow it to be just as sloppy and messy as physics is and nearly
> all of those remaining troubles are gone! If you are still of the
> impression that you can't tell "what ZF is", try naming some statement
> which you either have trouble knowing how to formalize, or don't know
> whether it's an axiom, or a proof which you don't know whether it's
> valid, and we can surely clear it up for you.
Is Ax ~Ef (Ea aex & {0,{0,a}}ef) & (An Ab (bex => ({n,{n,b}}ef => (Ec
cex & {nU{n},{nU{n},c}}ef & ceb)))) the foundation axiom? Is it
considered part of "set theory"? I have more questions like this, but
for all I know I'm already kill-filed by everyone.
> [...]
> |There is the translation from English to set theory and back
> |constantly, there is no way for me to tell that I'm doing it the same
> |way. The point is that someone can say "X is a theorem: Y, QED" but
> |if you don't know WHAT it is a thoerem of, then I can't turn around to
> |the guy next to me and say "X is a theorem" because if I'm asked
> |"theorem of what" then I can't answer, so the "theoremhood" of the
> |statement isn't really proven (I can't carry it around with me or
> |apply it outside of the set theory class), it was only stated as a
> |theorem of SOMETHING. Only after we know the axioms can we know that
> |X is IN FACT a theorem of THAT axiom system.
>
> We don't depend upon axiom systems in the way that you imagine.
> When someone claims that a result is a theorem, they mean that it
> has been proven, period. End of sentence. Not, "proven in..." but
> "proven".
What? You don't seriously mean "proven without assuming any standards
of proof or axioms", do you? You could always say "(N or T) is a
validity", instead of saying "T is a theorem", and that's be more
clear, but then I'd probably be happy because then someone would STATE
what N is, which is the big thing I don't know. If somehow you handed
be an algorithm to compute all the Ts, then I could postulate an N,
but if I don't have all the Ts and I don't know N then how can I make
my own theorems, or recognize a theorem T when I see one?
> It is true that having proven a result, your proof can be examined
> to see what kinds of axioms would suffice for it. If you ask what
> it's a theorem in, you are liable to get the answer "ZFC", not
> because the proof has been examined in this way, but just because
> it's so rare for any other assumptions to be used. If one does use
> an assumption not supported by ZFC, it's expected that one will
> mention it somewhere in the course of the proof.
But what IS the ZFC axiom system? Does it have separarion?
Collection? Replacement? FO? SO? Foundation?
> [...]
> |> I had in mind the common situation where one has a first-order theory.
> |> In that case, we have the Goedel completeness theorem that says the
> |> logical consequences of a set of axioms are the same as the consequences
> |> that can be deduced using standard first-order logic.
> |
> |Are you sure about how you stated that? I'm assuming "standard
> |first-order logic" is ordinary first order logic (so not IF-logic or
> |SOL), but with IF-logic you can make first order statements that
> |aren't statements of ordinary first order logic, so you claim seems,
> |... a bit sensational. If it's true, then that's great, but I want to
> |know if that's what you meant.
>
> Hintikka misleads the unwary reader by causing him (i.e. you) to
> think that "first order statement" has come to mean something more
> than a statement in "ordinary" first-order logic. It has done so
> only in Hintikka's own terminology. He claims his system should
> count as first-order logic, but this is not what anybody else means
> by it.
I have no idea what anyone else means, Hintikka has formulas, they
have quantification of individuals of the same type (individuals of
the domain of discourse) that's FO. SO involves formulas that
quantify over a second type, some SO formulas are logically equivlanet
to oFOL formulas, others to IF-FOL formulas, others to neither. But
IF-FOL doesn't assume the existance of another type.
> Certainly the Goedel completeness theorem is only for first-order
> logic, not IF logic.
>
> Keith Ramsay
I would still be unclear what you mean, you'd have to define "logical
consequence" and "deduce with oFOL", and I'm unclear about the point
too. IF-FOL has stronger expressions than implication. FOL is weak
because you have to know that the axioms are TRUE in a model before
knowing that the theoresm MUST be true, but with IF-FOL you only need
that the axioms not be false to assert that the theorem MUST be true.
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