fom
Posts:
1,026
Registered:
12/4/12
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Re: fom - 01 - preface
Posted:
Dec 9, 2012 2:21 AM
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On 12/8/2012 10:50 PM, Ross A. Finlayson wrote:
>
> Wouldn't the HOD's order type as model be an ordinal and thus
> irregular, i.e. Burali-Forti?
>
HOD is a class model
I have no interest in, nor give even a
hint of credence to, the meaningfulness
of set models of set theory.
I am interested in the foundations of
mathematics and not in how many different
ways a partial picture can be made bigger.
Quantifiers are particles of transformation
rules in the deductive calculus. Apart
from accepting their usage in the deductive
calculus, one is straying far afield.
Russell distinguished between real and
apparent variables because there is a
difference between interpreting Ax as
an arbitrary choice (the Aristotelian
instruction for proving an ALL statement)
or as an infinite conjunction. The
ontology of the received paradigm for
FOL does not carry the necessity of
presupposition.
In a footnote of his paper describing
the constructible universe, Goedel makes
it clear that the construction presupposes
that every domain element can be named.
The constructible universe is the smallest
transitive class model that contains all
of the ordinals.
From Jech:
A set is ordinal-definable if there is
a formula phi such that
X={u: phi(u, a_1, ..., a_n}
for some ordinal numbers a_1, ..., a_n
===
Given a cumulative hierarchy containing
all ordinals, one can characterize the
class OD of ordinal-definable sets as
OD = U cl{V_beta: beta < alpha}
where cl() is the Goedel closure and
the union is taken over alpha in ORD
After making this characterization,
Jech shows that
There exists a definable well-ordering
of the class OD and a definable mapping
F of ORD onto OD
Next, there is an exercise whose statement
is
If F is a definable function on ORD
then range(F) c= OD
Thus, OD is the largest class for which
one can have a definable one to one
correspondence with ORD
Now, for some given class A,
consider the function
| 1 if xeA and F(z)=1 Az(zex)
F(x) = |
| 0 otherwise
and let B = {x: F(x)=1}
Then B = {xeA | xc=B} is composed of those
elements that are hereditarily in A
That is, B is the largest transitive class
contained in A and may be expressed as
{xeA | TC(x)c=A}
Now,
HOD = {x: TC({x}) c OD}
The next thing to address is the possibility
of forcing. Every forcing model presupposes
that the ground model is partial. That is
fine with set models-- they are partial. But,
forcing on the constructible universe is a
different matter.
To understand what is at issue is to
understand the importance of the phrase
"almost universal". A class is almost
universal when every subclass is included
in an element of the class.
Since every element in the universe has
a power set, every subclass of the universe
is an element.
(Now we have a problem -- the English
that is supposed to explain the formalism
is too vague to explain how to talk about
classes like ORD being part of L but not
being an element of L... the answer is
that "subclass" presupposes the individuation
given by the construction and not the
whimsy of whatever century's pop star logician
(Russell... Frege retracted as did Whitehead
Russell moved on to other things and just
behaved like the jackass he was while his
ideas were nibbled down. Too bad mathematicians
had not been paying attention)).
In any case, one must assume -(V=L) in
order to perform forcing on the constructible
universe.
Now, relative to second-order interpretation,
L=HOD (in Kunen)
> It's of interest to read of a space-filling curve and the general
> position, do you see any curve really fill space? What of a spiral
> real-space-filling curve?
>
a quick internet search will get you many examples
http://www.math.osu.edu/~fiedorowicz.1/math655/Peano.html
> Some have geometry first, others integers first, I'd agree they're
> separate domains of discourse, but of the same domain of discourse.
> Some have points then lines, others points then space, some have zero
> then one, others zero then infinity, and the latters are to an
> appreciable extent more fundamental or primitive.
In his work on the foundations of geometry, Russell
held that the parts of projective space -- that is,
the lines and planes -- are prior to the points.
That same sentiment is in Leibniz and Kant.
And, neither Aristotle nor Leibniz interpreted the
class structure of syllogistic logic as extensional.
For Aristotle, genera are prior to species.
The only sense in which Aristotelian logic places
individuals before genera is in the sense that
the plurality of individuals cannot be vacuous
Numerous authors in the early 20th century
concluded that some sort of fundamental regions
must precede the individuals of set theory. And,
in response to the paradox Lesniewski developed
systems with parts prior to description of class
Whoops!
From Russell:
"When something is asserted or denied about
all possible values or about some undetermined
possible values of a variable, that variable
is called apparent, after Peano.
...
"Whatever may be the instances of
propositions not containing apparent
variables, it is obvious that propositional
functions whose values do not contain
apparent variables are the sources of
propositions containing apparent variables.
...
"Consider a function whose argument is
an individual. This function presupposes
the totality of individuals; but unless
it contains functions as apparent variables
it does not presuppose any totality of
functions. If, however, it does contain
a function as apparent variable, then it
cannot be defined until some totality
of functions has been defined. It follows
that we must first define the totality
of those functions that have individuals
as arguments and contain no functions
as apparent variables. These are the
predicative functions of individuals.
...
"Thus, a predicative function of a variable
argument is any function which can be specified
without introducing new kinds of variables
not necessarily presupposed by the variable
which is the argument.
...
maybe a good example of impredicative is
the "flow of money" in the equation for
monetary inflation. They had some numbers,
they introduced an unmeasurable quantity
to formulate the equation, and now they
attribute inflation to the quantity rather
than the quantity to the inflation.
Until flow of money has an empirical
measure, it is implicitly defined by
the phenomenon it is now interpreted
to explain
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