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Topic: Few questions on forcing, large cardinals
Replies: 17   Last Post: Mar 30, 2013 1:21 PM

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Registered: 12/4/12
Re: Few questions on forcing, large cardinals
Posted: Mar 24, 2013 2:42 AM
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On 3/23/2013 10:23 PM, Ross A. Finlayson wrote:
> On Mar 23, 5:01 pm, fom <> wrote:
>> On 3/23/2013 6:46 PM, Ross A. Finlayson wrote:

>>> On Mar 23, 3:34 pm, fom <> wrote:
>>>> On 3/23/2013 5:09 PM, Ross A. Finlayson wrote:
>>>>> On Mar 23, 2:44 pm, fom <> wrote:
>>>>>> On 3/23/2013 4:34 PM, Ross A. Finlayson wrote:
>>>>>>> In a sense, infinity _is_ the numbers. Start from even more
>>>>>>> fundamental objects than natural numbers as elements. Like the
>>>>>>> numbers, they are as different as they can be and as same as they can
>>>>>>> be, where they are each different in not being any other and each same
>>>>>>> in being defined by that difference. There's no stop to that, it's
>>>>>>> gone on, forever. Then, in a way like when you look into the void, it
>>>>>>> looks into you

>>>>>> Is this your way of saying that if you look
>>>>>> into the void, you and the void become one?

>>>>>> Just kiddding....
>>>>>> I think Cantor would appreciate your sentiment that
>>>>>> the numbers of Cantor's paradise are more fundamental
>>>>>> than those of Kronecker's torment.

>>>>> I wouldn't say that infinity, even in the numbers, is either of those
>>>>> things. In ZF, Infinity is _axiomatized_ to be an inductive set, and
>>>>> a well-founded/regular one, that's not a given. Calling that the
>>>>> universe, Russell's comment is that it would contain itself.

>>>>> There's a case for induction, as it were, that each case exists. Then
>>>>> it is to be of deduction, not fiat by axiomatization, from simple
>>>>> principles of constancy and variety, the continuum.

>>>>> In a theory with sets as primary objects, a set theory and a pure set
>>>>> theory, numbers would be very rich objects indeed, as not just
>>>>> individual elements by their elements, but all relations of numbers.
>>>>> Set theory (well-founded, as it were, regular or that objects are
>>>>> transitively closed) is at once over-simplification, to talk about
>>>>> anything besides sets, and over-complexification, to talk about itself
>>>>> when any universal statement is in the meta.

>>>>> There are no numbers in a pure set theory. To call the natural
>>>>> integers a set, it contains only numbers, for the Platonists: elements
>>>>> of the structure, of numbers, as: none exist in a void.

>>>> It is odd. In some sense, modern mathematics actually
>>>> treats its objects as urelements relative to set theory.
>>>> Looking at Hilbert, he makes statements whereby his formalism
>>>> is intended to supersede the class-based constructions of
>>>> Dedekind.

>>>> Your frank statement that a set is not a number reflects
>>>> that sentiment.

>>> Particular finite sets are called ordinals, set-theoretic operations
>>> on them are defined that give the same results as Presburger/Peano
>>> arithmetic of the natural integers. The negative integers aren't
>>> simply the complement as in finite-word-width machine arithmetic, but
>>> again simple enough operations on sets (with the only ur-element being
>>> the empty set) give a "model" of the integers. Rationals are defined
>>> simply enough as equivalence classes over any pairs of integers,
>>> besides zeros, the reals then see the Least Upper Bound as axiom.
>>> These are all to match number-theoretic features, and largely suffice
>>> for integers and rational numbers, but not so obviously do sets
>>> suffice to represent thusly elements (and all of) the continuum of
>>> real numbers.

>>> Then, though, to call the empty set the number zero: wouldn't that be
>>> the number zero wherever there's an empty set? Building upwards to
>>> have particular sets for each of of the finite integers: then to
>>> build the numbers as sets, is to build all the relations of the
>>> numbers as sets, not just as to a set-theoretic model of only that set
>>> of numbers' operations: but of all instances, besides the schema.
>>> Where the ur-element is any thing, it so implies all other things,
>>> and is so implied. The collection and aggregates of sets or
>>> categorization or refinement of types or partition or bounding of
>>> division, are all of the same corpus.

>>> Here back to the questions as above:
>>> 1) is not forcing simply transfinite Dirichlet box?
>> I am not sure what you mean by this.
>> However, forcing might be better thought of as comparable
>> to Euclid's proof that there is no greatest prime.

>>> 2) are there any results due transfinite cardinals, not of transfinite
>>> cardinals?

>> The Borel hierarchy is defined in terms of the first
>> uncountable ordinal. Hence, results in descriptive
>> set theory that depend on that definition may count.
>> I do not have enough knowledge of that branch of
>> study to comment further.

>>> 3) is not an irregular model of ZF non-well-founded?
>> What is your definition of irregular?

>>> 4) does not a model of ZF contain itself?
>> There are relativizations of models. So, one question
>> in set theory is whether
>> where HOD are the hereditarily ordinal-defined
>> sets and HOD^HOD is HOD relativized within itself.
>> In this sense, models may have representations
>> within themselves. But, once again, expertise
>> is lacking here.

>>> 5) is ZF not a model of itself?
>> ZF is an axiomatization. The question is not
>> well-construed.

> 1) Forcing might be better thought of as that there's an ordinal
> greater than all ordinals.

That is not quite what you should take from my statement.

What determines that a forcing model is "bigger" than its
ground model is that for the set of "names" in the forcing
language, there is one name for every object in the ground
model and one name for which there is no such object.

The ordinals are "special" as the "spine" of the model. They
can be collapsed onto lower ordinals as given in the ground
model. But, it is probably not correct to view the manipulations
in forcing as adding ordinals at the top of the hierarchy.

> 2) That may as well be stated as that the Borel hierarchy is in terms
> of ranks of countable ordinals.

Probably a better statement.

> 3) An irregular model is not well-founded.

I passed on this one as I recall.

> 4) There's a relativization of ZF down to the countable and even to
> omega. Then that a model of HOD, hereditarily ordinally-definable,
> isn't itself HOD is again: Russell's "paradox".

I am not certain that the nature and existence
of countable models should be considered as having
the same sense as relativization.

Simply put, relativization involves reinterpretation
of quantified formulas in the sense of

[phi(x)]^M for some class M

Ex(phi(x)) becomes Ex(xeM /\ phi(x))

Ax(phi(x)) becomes Ax(xeM -> phi(x))

Since classes are associated with the grammatical forms
of naive set theory,


One can also speak of a different sense of relativization.
Let k be a parameter. Relative to the parameter k, let two
classes be given by


{<p,q>|peM, qeM, /\ E(p,q,k)}

In this case, if E satisfies the axioms when interpreted
as the membership relation over M, then


is a model of set theory. Relativization in this case
is denoted with


for a given formula.

Ex(phi(x)) becomes Ex(xeM /\ phi(x)^<M,E>)

Ax(phi(x)) becomes Ax(xeM -> phi(x)^<M,E>)

The additional complexity of the formulas indicates that
the membership relation is reinterpreted by a definite
class specification.

> 5) ZF as theory is all its theorems. That as all the sets that don't
> contain themselves, again via Russell, does. I'll agree it's a direct
> question as to the content of ZF, simply construed.


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