Re: The modern mathematical concept of infinity is indefensible
Aatu Koskensilta
> To allege that "ZFC is inconsistent" EVEN CAN be translated
> into ZFC is simply alleging too much.
What statements in ordinary mathematical English about sets can be
translated into the language of set theory?
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
herbzet
Aatu Koskensilta wrote:
> george writes:
>
> > To allege that "ZFC is inconsistent" EVEN CAN be translated
> > into ZFC is simply alleging too much.
>
> What statements in ordinary mathematical English about sets can be
> translated into the language of set theory?
Is the distinction between "expressible" and "representable"
pertinent here?
--
hz
Aatu Koskensilta
> Aatu Koskensilta wrote:
>
>> What statements in ordinary mathematical English about sets can be
>> translated into the language of set theory?
>
> Is the distinction between "expressible" and "representable"
> pertinent here?
The distinction between "expressible" and "representable" is standard,
but the definitions of these terms is not. This distinction is not in
any apparent way relevant here, whatever definitions one has in
mind.
George seems to be saying that the usual formalisation of "ZFC is
consistent" is somehow incorrect or inadequate, presumably because of
its independence of ZFC. For many statements we have no idea whether
or not their formalisations are provable or refutable in ZFC, yet we
do seem to have no trouble understanding what is at issue when someone
wonders whether such a statement is or is not provable in ZFC. For
such wondering to make sense we must have some criteria for a
formalisation of a statement being correct -- at least in the weak
sense that we can agree that this or that particular formalisation is
a correct translation even if we can't give any general definition of
"correctness" -- independent of provability or refutability in ZFC of
the formalisation.
george
> > To allege that "ZFC is inconsistent" EVEN CAN be translated
> > into ZFC is simply alleging too much.
I *meant* to say FIRST-order ZFC.
On Feb 2, 1:33 pm, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
> What statements in ordinary mathematical English about sets can be
> translated into the language of set theory?
The problem here is not "sets" BUT RATHER *all* sets.
The problem (at least the problem as identified by Godel)
is a vicious circle problem. You can talk about any collection
of sets you like as long as you don't try to talk about a totality
that includes yourSELF.
More to the point, "the language of set theory" is not a locution
that I condone. Set theory can be done in ALMOST ANY language.
When YOU SAY "the language of set theory", YOU MEAN,
"the first-order language whose signature is so simple that it
consists
of 1 and only 1 binary predicate". It obviously DOES NOT MATTER
whether
you spell this predicate "epsilon", "in", OR ANY OTHER particular
name.
As for the unary functors, two of them (Union and Power) are dual,
and are definable, as is the 0-ary functor (the empty set).
THE ACTUAL relevant question IS NOT how you translate to or from
English BUT RATHER how you translate to or from SECOND-order ZFC,
WHICH IS NOT just the same stuff with a 2nd-order language instead of
a first.
But that was a different argument (I was having that one with CM).
george
> George seems to be saying that the usual formalisation of "ZFC is
> consistent" is somehow incorrect or inadequate, presumably because of
> its independence of ZFC.
More specifically, because of its truth in models OF ZFC, because of
the existence of models of ZFC + ~con(ZFC). That is inherently
contradictory
(in light of the model existence theorem) yet also inherently not so,
according to Godel's incompleteness theorems.
First-order models reserve the right to be wrong about EVERY
existential
question, simply because they are only about the things that exist IN
THEIR DOMAIN;
they basically lie about the existence of things that clearly DO
exist, BUT ALSO
CLEARLY GOT EXCLUDED FROM their domain. So I would personally say
that
my qualms are the same kinds of qualms that occur with Skolem's
paradox;
the relevant bijections don't exist in the domain, yet they still
exist.
> For many statements we have no idea whether
> or not their formalisations are provable or refutable in ZFC, yet we
> do seem to have no trouble understanding what is at issue when someone
> wonders whether such a statement is or is not provable in ZFC.
That is not a valid analogy. In the case of con(ZFC) -- or ~con(ZFC),
for that matter --
we indeed DO have NO trouble understanding what is at issue when
someone
wonders whether these statements are or are not provable in ZFC.
We understand PERFECTLY well that they are NOT provable.
We understand PERFECTLY well that DUELING models exist.
However, for the particular case of consistency statements, the MERE
existence of ANY model AT ALL *decides* the question. There is a
disconnect
between what exists in the domain of a particular countable
first-order model and what "really" exists. This problem is the same
as the Skolem paradox, fundamentally.
> For such wondering to make sense we must have some criteria for a
> formalisation of a statement being correct --
Oh, BULLSHIT. You DON'T need any GENERALIZED criterion.
All natural langauge statements are NOT created equal. SOME natural
language statements happen to have implications for model-existence.
THOSE are attackable in ways that others would not be.
> at least in the weak sense that we can agree that this
> or that particular formalisation is
> a correct translation
This is completely ass-backwards. You may never find out what a
CORRECT
translation is but it can be EASY to show that a translation is NOT
correct.
Science, experiment, falsifiablity, ad nauseam.
herbzet
Aatu Koskensilta wrote:
> herbzet writes:
> > Aatu Koskensilta wrote:
> >
> >> What statements in ordinary mathematical English about sets can be
> >> translated into the language of set theory?
> >
> > Is the distinction between "expressible" and "representable"
> > pertinent here?
>
> The distinction between "expressible" and "representable" is standard,
> but the definitions of these terms is not. This distinction is not in
> any apparent way relevant here, whatever definitions one has in
> mind.
>
> George seems to be saying that the usual formalisation of "ZFC is
> consistent" is somehow incorrect or inadequate, presumably because of
> its independence of ZFC. For many statements we have no idea whether
> or not their formalisations are provable or refutable in ZFC, yet we
> do seem to have no trouble understanding what is at issue when someone
> wonders whether such a statement is or is not provable in ZFC.
Yes, and no -- your statement allows of different interpretations.
George's reply takes it literally, but I think what you meant is
not "whether such a statement is or is not provable in ZFC" but
"whether such a statement is correct" -- meaning, true in the
intended model of ZFC.
> For
> such wondering to make sense we must have some criteria for a
> formalisation of a statement being correct -- at least in the weak
> sense that we can agree that this or that particular formalisation is
> a correct translation even if we can't give any general definition of
> "correctness" -- independent of provability or refutability in ZFC of
> the formalisation.
Right -- provability or refutablility in ZFC is an insufficient general
criterion for correctness. Any consistent extension of ZFC will have
a plurality of models -- which one are we talking about?
When we talk about what's true or false in the universe of sets, what
universe are we talking about? A finitely given formulation cannot
single out that universe (not at first-order anyway).
Yet your point, if I understand it correctly, is that we definitely
have some universe in mind when we talk about mathematical propositions
being true or false -- or also, in taking a mathematical proposition to
have a definite meaning.
I hope I'm not over-simplifying your position.
Still struggling with the idea of an essentially incomplete mathematics
-- which is to say, a mathematics with an essentially indeterminable
model (at first-order, anyway).
--
hz
Aatu Koskensilta
> George's reply takes it literally, but I think what you meant is not
> "whether such a statement is or is not provable in ZFC" but "whether
> such a statement is correct" -- meaning, true in the intended model
> of ZFC.
Perhaps I have been too subtle for my own good again. We do wonder
about many statements whether or not they're provable in ZFC. This is
not the same question as whether or not the statement is true. But, if
such wondering makes sense, we must have some criteria by which to
judge whether a formalisation of the statement in question in the
language of set theory is correct, in the sense of expressing the
mathematical content of the statement. This is because the notion of
being formally provable, unprovable, refutable in ZFC applies to
formal sentences in the language of set theory, not to statements in
ordinary mathematical English. We might ask, for example, whether
Goldbach's conjecture is provable in ZFC. Here we presuppose that
there is some formalisation of the conjecture in the language of set
theory, the formal provability of which is the object of our
curiosity. But if the correctness of formalisations of such statements
-- and recall that Goldbach's conjecture has the same logical form as
"ZFC is consistent" -- depends on their provability or refutability in
ZFC it seems we should allow for the possibility that the
formalisation does not "correctly" express the conjecture at all,
should we find it is independent of ZFC. In fact, George's doctrine,
taken literally, implies that no mathematical statement is ever
independent of ZFC -- though of course various formal sentences are --
since no such statement can be "correctly" formulated.
Of course, in reality no one has any doubts about the correctness of
such formalisations -- the formalisation is routine matter,
essentially amounting to just spelling out the statement of the
conjecture in terms of basic set theoretic notions -- and the same
applies just as well to the statement of consistency of ZFC.
> I hope I'm not over-simplifying your position.
My position regarding indeterminacy in mathematics is not at issue at
this junction. My simple observation is just that in talking about
this or that being formally provable, refutable, etc. in ZFC we
presuppose there's an accurate translation of this or that in the
language of set theory. For if there isn't, the observation that the
statement is not provable in ZFC is as uninteresting as the
observation that the continuum hypothesis or the existence of
inaccessible cardinals is not provable in PA due to the fact that
these statements can't be expressed in the language of arithmetic in
the first place.
george
> Right -- provability or refutablility in ZFC is an insufficient general
> criterion for correctness.
The assumption of the existence of "correctness" is ridiculous hubris.
> Any consistent extension of ZFC will have
> a plurality of models -- which one are we talking about?
You don't have to "extend" to get this "plurality".
G1 says/means/proves THAT EVERY axiom-set satisfying its
hypotheses has dueling models of EVERY SENTENCE that is
not a theorem or the denial of one (and there MUST ALWAYS be
such sentences). This is not specific to ZFC; indeed, it's general
to everything more complicated than Robinson arithmetic.
As to which one we're talking about, THAT IS PUTTING THE CART BEFORE
THE HORSE. THAT is a direction-of-fit ERROR. We don't NEED to be
talking about ANY particular model! WE COULD INSTEAD be talking about
the consequences of the axioms! We could instead be CARING about the
theory AS OPPOSED to about the model! The model is going to decide
questions that the theory doesn't. If you CARE which way somesuch
question is decided THEN THE OBVIOUS remedy is SIMPLY TO ADD
your PREFERRED (NOT "correct"!) decision about it AS AN AXIOM!
george
In the case of Peano Arithmetic, we have always been able to
say, "the one that you get when you lift the first-order schema to
second-
order". Chris Menzel says that that justs lifts the ambiguity to your
choice of a semantics, but fuck him. There is every bit as obviously
one known standard semantics for 2nd-order logic AS THERE IS
FOR 0th AND 1st. So the point is, using THAT semantics, you get
ONE model. Something analogous happens in ZFC as well (the
only axiom-schema we have there is Replacement, if you do it right --
pairing and separation become theorems), but in that context, it seems
MORE like BEGGING the question, since the nature of sets was
what the original question was trying to be about.
My point was simply that rather than talking about formalizing
natural language about some allegedly understood universe of sets,
we should talk instead about what happens with the 2nd-order version
of the same axiom-set. If that also produces categoricity (for ZFC
as it did for PA) then the ambiguity ends. But the "correctness"
debate
REMAINS because nobody's axioms are any more or less correct than
anybody else's, for questions you can't prove an answer to.
herbzet
george wrote:
> herbzet wrote:
> > Right -- provability or refutablility in ZFC is an insufficient general
> > criterion for correctness.
>
> The assumption of the existence of "correctness" is ridiculous hubris.
>
> > Any consistent extension of ZFC will have
> > a plurality of models -- which one are we talking about?
>
> You don't have to "extend" to get this "plurality".
What I said is correct. What you said is correct.
> G1 says/means/proves THAT EVERY axiom-set satisfying its
> hypotheses has dueling models of EVERY SENTENCE that is
> not a theorem or the denial of one (and there MUST ALWAYS be
> such sentences). This is not specific to ZFC; indeed, it's general
> to everything more complicated than Robinson arithmetic.
>
> As to which one we're talking about, THAT IS PUTTING THE CART BEFORE
> THE HORSE. THAT is a direction-of-fit ERROR. We don't NEED to be
> talking about ANY particular model! WE COULD INSTEAD be talking about
> the consequences of the axioms! We could instead be CARING about the
> theory AS OPPOSED to about the model! The model is going to decide
> questions that the theory doesn't. If you CARE which way somesuch
> question is decided THEN THE OBVIOUS remedy is SIMPLY TO ADD
> your PREFERRED (NOT "correct"!) decision about it AS AN AXIOM!
Good.
Btw, Presburger arithmetic is complete, but has a model of every
infinite cardinality, since it (obviously) has a model of infinite
cardinality. But they're not "dueling" models: what's true in one
model is true in all models.
So even in a complete theory at first order, we don't necessarily
know specifically "what" we're talking about -- that is, a single
model (up to isomorphism) is not necessarily compelled by the axioms
of even a complete theory at first-order.
--
hz