fom
Posts:
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Registered:
12/4/12


Re: fom  01  preface
Posted:
Dec 9, 2012 2:21 AM


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. BuraliForti? >
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 ordinaldefinable 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 ordinaldefinable 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 wellordering 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 secondorder interpretation, L=HOD (in Kunen)
> It's of interest to read of a spacefilling curve and the general > position, do you see any curve really fill space? What of a spiral > realspacefilling 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

