Date: Dec 9, 2012 3:05 PM Author: ross.finlayson@gmail.com Subject: Re: fom - 01 - preface On Dec 8, 11:21 pm, fom <fomJ...@nyms.net> wrote:

> 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

"I am interested in the foundations of

mathematics and not in how many different

ways a partial picture can be made bigger."

I agree, but for me that means no classes in set theory, that it is

the group noun game, with no winning strategy, that the same paradoxes

that would apply to regular sets would apply to regular classes, or to

regular collections or regular arrays or other regular instances of

"group nouns". What's the class of these ordinals? Not a set, is not

each ordinal in it and via quantification over its elements or

transfinite induction, its construction? Because it would then have

an order type, that is an ordinal (Burali-Forti). There are various

considerations of the "anti-"foundation, for the foundational, for

example of Aczel or into the paraconsistent/dialetheic.

Skolem gives us countable models, here I wonder with regards to

Cohen's forcing that in the development that there's a maximal ordinal

m, or not - do you see that in the development of forcing? Forcing

enables us to resolve some issues with mathematical paradoxes for the

applied, downward Lowenheim-Skolem and Cohen forcing.

http://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem

http://en.wikipedia.org/wiki/Forcing_(mathematics)

With regards to ~(V=L), that the constructible universe V isn't the

universe L, that has a variety of consequences per Feferman, for well-

ordering the reals.

http://math.stanford.edu/~feferman/

https://groups.google.com/groups/search?hl=en&as_q=Feferman+constructible

Many have that the constructible universe is the universe, V = L.

ORD, the class of all ordinals, maps onto any elements of a course of

passage, and transfinite induction up through the constructible,

because each ordinal, finite and transfinite, is in it.

Yes I've seen diagrams of Peano's space-filling curve before, I wonder

what you think of the general position that a point starts at the

origin, a next follows, and that they fill in spiral shells all about

the origin, with no subsequent point ever being closer to the origin

than any precedent, and no point in R^N remaining.

At any rate I'll thank you to continue I find that quite interesting

and rather refreshing. There's much more to your post worthy of

comment.

Regards,

Ross Finlayson