Few questions on forcing, large cardinals

[Ross A. Finlayson]

So isn't forcing simply Dirichlet box / pigeonhole principle? Without
appeal to forcing, simply define arguments in symmetry then as to
transfinite pigeonhole, what's the difference?

How can large cardinals be defined in terms of V (the Universe) when V
isn't a set? How are they cardinals if they aren't of sets?

How can forcing's model be at once model and embedded in the model?
Wouldn't it then be irregular?

Are there any results in recursion/computation that can't simply (or
not so simply) be framed in asymptotics without appeal to transfinite
cardinals?

Are there any results not of transfinite cardinals, solely due
transfinite cardinals?

Thanks,

Ross Finlayson

[Ross A. Finlayson]

On Mar 10, 5:47 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>
wrote:

Looking to http://cantorsattic.info , this is a nice resource for
looking to the day's definitions of what would be "large cardinals",
were they sets or cardinals.

A particular notion of the "large cardinals" is that they are defined
in terms of embeddings into the set-theoretic universe (and
correspondingly in set theory with classes the set and class universe,
except where the distinction or existence is ignored). Then, the
Kunen inconsistency that tops Cantor's attic is that there are no non-
trivial embeddings from the universe V to itself. Yet, identity as
trivial, sees then any pair-wise switch of elements from identity, eg
for a and b that a->b and b->a, being as well an embedding. Then, if
those are all trivial, then so is any function f: S -> S from set S to
itself, which is a rather overbroad definition of "trivial". Would
that remain elementary, in that the function preserves model
isomoprhism, it would where the theorems of the resulting structure so
modeled were the same. Then basically for groups or other features
establishing isomorphisms for all relevant theorems of the structure,
the pair-wise switch among elements identical under isomorphism, would
yield non-trivial elementary embeddings. Basically that is as to
whether, for example, the elements of Z_2, can be unique as elements
of Z_2, from a sub-theory of ZF defining only the binary: that
replacing all the 1's with 0's and 0's with 1's is structurally
indistinguishable, under that all the coded results have the same
import.

f: x e V -> V: x -> x (trivial identity)
f: x e V -> V: x -> ~x (trivial? opposition)

Then, if each structure eg Z_2 various under isomorphism isn't a leaf
or totally uniquely typed, then all mathematical structures are of a
single unified structure, then that structure as consistent as
structure itself could be: would be complete. As there exists
ismorphisms thus elementary embeddings in those structures in vacuo,
there exist all their regular 1-1 compositions defining elementary
embeddings, in their concreteness.

The Universe would be irregular and be its own powerset. This is
known as Cantor's paradox, that there can't be a universe as a regular/
well-founded set because uncountality depends on well-foundedness (and
Regularity and Infinity are the only axioms of ZF restricting
comprehension of quantification). Now, the Universe exists, simple
and structurally from that anything exists. So, the universe is not a
well-founded set, though defined by all its elements, it's naively a
set. Large cardinals aren't set nor cardinals, of ZF.

Basically then from the upper attic of Kunen inconsistency to
Con(ZFC): "Every model of ZFC contains a model of ZFC as an element",
that models of ZFC are not standard nor well-founded, that's just a re-
phrasal of Russell's paradox, that the collection of all the well-
founded sets (which ZFC is) would contain itself: as alluded to there
it does, then there's the simple question: why doesn't that imply
~Con(ZFC)?

I suggest you review the notion of forcing, in logic, and as to
whether structurally, that breaks things. The simplest model: is a
working, structural model. The Universe: is, what it is.

Regards,

Ross Finlayson

[Ross A. Finlayson]

On Mar 10, 5:47 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>
wrote:

http://math.bu.edu/people/aki/14.pdf

"Rather, the focus is on the connections between the combinatorial
properties of the partial order of conditions and structural
properties of the extension."


"With forcing so expanded into the interstices of set theory and the
method so extensively amended from the beginning, what is the "it" of
Cohen's forcing and his individual achievement? Cohen discovered a
concrete and widely applicable means of operationally extending a
standard model of set theory to another without altering the ordinals.
The central technical innovation was the definable forcing relation,
through which satisfaction for the extension could be approached in
the ground model. Cohen's achievement was thus to be able to secure
properties of new sets without having all of their members in hand and
more broadly, to separate and then interweave truth and existence."

Quoting Cohen:

"One might say in a humorous way that the attitude toward my proof was
as follows. When it was first presented, some people thought it was
wrong. Then it was thought
to be extremely complicated. Then it was thought to be easy. But of
course it is easy in the sense that there is a clear philosophical
idea. There were technical points, you know, which bothered me, but
basically it was not really an enormously involved combinatorial
problem; it was a philosophical idea."

-- A. Kanimori, "Cohen and Set Theory", pp. 24-25

Symmetry in principle: all and nothing.

Regards,

Ross Finlayson

[fom]

On 3/19/2013 2:18 AM, Ross A. Finlayson wrote:
>
>
> http://math.bu.edu/people/aki/14.pdf
>
> "Rather, the focus is on the connections between the combinatorial
> properties of the partial order of conditions and structural
> properties of the extension."
>
>
> "With forcing so expanded into the interstices of set theory and the
> method so extensively amended from the beginning, what is the "it" of
> Cohen's forcing and his individual achievement? Cohen discovered a
> concrete and widely applicable means of operationally extending a
> standard model of set theory to another without altering the ordinals.
> The central technical innovation was the definable forcing relation,
> through which satisfaction for the extension could be approached in
> the ground model. Cohen's achievement was thus to be able to secure
> properties of new sets without having all of their members in hand and
> more broadly, to separate and then interweave truth and existence."
>

On page 360 he speaks of having to
think of truth in new way.

One result of forcing has been to
consider "truth persistence under
forcing". To the best of my knowledge,
this typifies some of Woodin's work.



"Partiality, Truth, and Persistence" by Langholm

http://books.google.com/books?id=AOGpRxZyfFYC&pg=PA43&lpg=PA43&dq=truth+persistence&source=bl&ots=DQ5YDVLvbR&sig=O42ZBaI_H1pxsc640_83EVE-KDk&hl=en&sa=X&ei=4RdIUZLuK-rY2AXL9IDwAg&ved=0CFUQ6AEwBjgK



Some information about "names" and "descriptions" in
relation to presupposition and the strong Kleene truth
definition used by Langholm

http://plato.stanford.edu/entries/presupposition/#LocConDynTur


And yes, forcing is unobjectionable when you redefine truth.

But, no one told anyone.

[Ross A. Finlayson]

On Mar 19, 1:12 am, fom <fomJ...@nyms.net> wrote:
> On 3/19/2013 2:18 AM, Ross A. Finlayson wrote:
>
>
>
>
>
>
>
>
>
>
>
> >http://math.bu.edu/people/aki/14.pdf
>
> > "Rather, the focus is on the connections between the combinatorial
> > properties of the partial order of conditions and structural
> > properties of the extension."
>
> > "With forcing so expanded into the interstices of set theory and the
> > method so extensively amended from the beginning, what is the "it" of
> > Cohen's forcing and his individual achievement? Cohen discovered a
> > concrete and widely applicable means of operationally extending a
> > standard model of set theory to another without altering the ordinals.
> > The central technical innovation was the definable forcing relation,
> > through which satisfaction for the extension could be approached in
> > the ground model. Cohen's achievement was thus to be able to secure
> > properties of new sets without having all of their members in hand and
> > more broadly, to separate and then interweave truth and existence."
>
> On page 360 he speaks of having to
> think of truth in new way.
>
> One result of forcing has been to
> consider "truth persistence under
> forcing".  To the best of my knowledge,
> this typifies some of Woodin's work.
>
> "Partiality, Truth, and Persistence" by Langholm
>

> http://books.google.com/books?id=AOGpRxZyfFYC&pg=PA43&lpg=PA43&dq=tru...

>
> Some information about "names" and "descriptions" in
> relation to presupposition and the strong Kleene truth
> definition used by Langholm
>
> http://plato.stanford.edu/entries/presupposition/#LocConDynTur
>
> And yes, forcing is unobjectionable when you redefine truth.
>
> But, no one told anyone.

If you might elucidate on that, it may help to establish the context a
bit more firmly to the gallery.

A model of ZF might be ill-founded, or even where it's not, it
basically includes ZF and all its sets which is the Russell set. (As
it is, upon inspection.) Cohen's forcing (of an ordinal structure)
would have that then the resulting items are modeled as ordinals.

Then, of large cardinals, it is somewhat a misnomer, cardinal, when
they're not of regular/well-founded objects of the set theory, yet
forced to be.

As an extension of Skolemization, Cohen's forcing is upward (past the
entire model of regular set theory), yet, there is also downward in L-
S. Why doesn't that correspond to the irregular and "Kunen
inconsistency" (Cantor paradox) forced into the regular? This may be
considered with regards to Levy collapse.

For infinity in the numbers, we start counting, and it doesn't end,
from that there is infinity, in the numbers.

Are there, any results solely due transfinite cardinals, not of
transfinite cardinals? Via forcing, there are results of transfinite
cardinals, not due transfinite cardinals, but only of, transfinite
cardinals.

Regards,

Ross Finlayson

[JT]

On 19 mar, 16:17, "Ross A. Finlayson" <ross.finlay...@gmail.com>
wrote:

No the infinity is not ***in the numbers*** there is infinity but it
ain't numerical.

[fom]

On 3/19/2013 10:17 AM, Ross A. Finlayson wrote:
>>
>> And yes, forcing is unobjectionable when you redefine truth.
>>
>> But, no one told anyone.
>
> If you might elucidate on that, it may help to establish the context a
> bit more firmly to the gallery.
>

It is not a mathematical issue.

Forcing changes what it means for something to
be true in mathematics if the outcome is to
define truth in terms of "truth persistence
under forcing".

Tarski-based semantics is replaced by the kind of
thing that is discussed in the book by Langholm.

To change the classical problem is not the same
as solving the classical problem.

For example, suppose it to be true that partial
systems are not diagonalizable.

I make this guess simply because a presumption
of the diagonal argument is a presumption that
"all" of the objects have been given a locus
in the list.

Non-diagonalizability is a "truth" of partial
systems.

It is not a truth of total systems.

So, what other "truths" are different?

[fom]

On 3/19/2013 10:37 AM, JT wrote:
>
> No the infinity is not ***in the numbers*** there is infinity but it
> ain't numerical.
>

There are many number systems that are not
related to natural numbers in the way you
interpret the word: Complex numbers, Quaternion numbers,
Octonian numbers, Icosian numbers, Miniquaternion numbers,
Hurwitz numbers, etc.

What defines a number system abstractly is
whether or not someone has constructed an
arithmetic for the system of objects.

Infinity in the absolute sense is, of course,
plural and not subject to arithmetical treatment
as an object.

But, Cantor did develop an arithmetic that can
be put in correspondence with the usual natural
number arithmetic and extends to what are called
transfinite numbers.

Because of deliberations on the opinions of people
who feel as you do, I try to be careful. As much
as possible, I use the term 'transfinite number'
because that distinguishes from the sense of
infinity that is necessarily plural.

[Ross A. Finlayson]

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, for there to be numbers counting upward, to infinity,
there are numbers counting backward, from infinity. In a very
fundamental sense, then infinity is _in_ the numbers.

As well, number theorists, who work on theorems about the structure of
natural integers, sometimes define a point at infinity, on the line of
numbers. The line of numbers goes on forever and contains only
numbers, the point at infinity is on it. Sometimes, for example, the
theorems are about there being a prime number at infinity, others, a
composite and even the composite of all the numbers.

In real analysis, there are as well considerations of there being the
point or points at infinity and negative infinity. This is even with
staying in what otherwise is a Euclidean geometry with the parallel
postulate, where other geometries find there isn't even the parallel
postulate that parallel lines don't meet (at some point at
infinity).

For proofs of induction and for bounded cases, the remarkable thing is
that results are derived with only needing finite and unbounded
elements. But, there are true results of these objects, that may well
best be explained, with infinity, in the numbers.

Regards,

Ross Finlayson

[fom]

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?

http://en.wikipedia.org/wiki/Cantor-Bernstein-Schroeder_theorem


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.

[Ross A. Finlayson]

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.

Regards,

Ross Finlayson

[fom]

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.

[Ross A. Finlayson]

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?
2) are there any results due transfinite cardinals, not of transfinite
cardinals?
3) is not an irregular model of ZF non-well-founded?
4) does not a model of ZF contain itself?
5) is ZF not a model of itself?

Regards,

Ross Finlayson

[fom]

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

HOD=HOD^HOD

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.

[Ross A. Finlayson]

1) Forcing might be better thought of as that there's an ordinal
greater than all ordinals.
2) That may as well be stated as that the Borel hierarchy is in terms
of ranks of countable ordinals.
3) An irregular model is not well-founded.
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".
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.

Regards,

Ross Finlayson

[fom]

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,

M(z)={z|psi(z)}

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

{z|M(z,k)}

{<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

<M,E>

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

[phi(x)]^<M,E>

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.

Pass.

[Ross A. Finlayson]

An analysis pointing out perceived deficiencies in forcing:

http://groups.google.com/group/sci.math/msg/c855b91976dde22e

Basically forcing "scales" the universe then as to where transfinite
induction (over transfinite ordinals) is through all of them. The
difference between this and plain transfinite induction is as to the
difference between induction and transfinite induction. Then it is as
to transfinite Dirichlet box.

Borel's hierarchy in terms of finite languages and computability is as
to countable ordinals and even more simply polynomials in omega.

A model of ZF _is_ ill-founded. Whether in "naive" set theory or not,
with its concomitant paradoxes of Russell, Cantor, and Burali-Forti as
are well known, it's in ZF: or not, and as a model of ZF, includes
all theorems of ZF, and then some, else ZF could model itself. (Which
it doesn't, directly.)

So,

a) model theory is in the meta, and in naive set theory

b) forcing introduces elements that would exist in ZF that have
properties of elements that wouldn't exist in ZF

c) large cardinals presuppose a universe (and aren't regular sets nor
cardinals)

d) models of regular theories are irregular (as are large cardinals)

e) transfinite ordinals and polynomials in w support transfinite
Dirichlet box

f) there are non-trivial elementary embeddings V -> V, else in pure
sets no elements of structure with models (under isomorphism) are
primary, and all are concrete/constructible

g) there's a non-trivial elementary embedding V -> V, v -> V\v

h) the paradoxes of naive set theory with regards to HOD transitive
closure as regularity aren't resolved in their demurral

Regards,

Ross Finlayson

[Ross A. Finlayson]

On Mar 24, 10:20 am, "Ross A. Finlayson" <ross.finlay...@gmail.com>
wrote:

So: what theory are large cardinals in, and, is it consistent, in
that theory, to force them into ZF? Because, if they're independent
of ZF, the axioms establishing their definition, large cardinals,
isn't there a theory, with them?

One might aver to NFU (New Foundations with Universes, vis-a-vis, New
Foundations with Ur-Elements) and then having large "sets", compared
to the universe (or domain of discourse, as it were) of ZF(C), but
then they're classes, and there's only one proper class. And, in a
pure set theory: it's a set.

Regards,

Ross Finlayson

[fom]

For my part Ross, I do not want to misdirect
you. It has been a very long time since I
had been studying set theory closely enough
to talk about large cardinals.

My primary investigations had been with respect
to how the sign of equality is treated within the
theory. The model theory of forcing models is
directly relevant to that understanding. The
specifics of large cardinals and their axioms
are not.

I hope I have been of some help to you.

[Ross A. Finlayson]

Dichotomy is a basic reasoning. Everything or rather any thing is
defined by what it isn't. The "context", of a thing or assembly, is
basically the universe setminus the thing: v ~ V\v.

Western tradition has then Kant, Hegel, and Heidegger as the
philosophical leaders of the concepts of "Nothing", and "Being", or
existence, then for Heidegger also "Time". The Universal is often in
passage ascribed to the divinity, but it's as simple to start with
"Nothing" and "All" as to start with zero, then one.

Then in terms of identity or tautology, equality, even the very
concept of equality is seen to vary: for all that n object evinces as
identity, and all the things that come together as it as tautology.

Then that is rather airy but the basic consideration of equality
consequent all structure has a variety of reasonings to so define it,
of basically sameness and convergence.

Forcing is a reflection that via properties of the universe, there is
as much tendency from greatest to smallest as least to most, here of
the definition of ordinals from zero, in the universe, here of
ordinals and as such: ordinal. Then, where technical philosophy
founds logic founds mathematics, forcing in logic is that there is a
universe with its self-similar and self-containing properties (a la
Kant's Ding-an-Sich or thing-in-itself) of here set theory: a
universe. Forcing is an axiomatization, of the maximal, for a theory
that the maximal would make inconsistent.

Then in the paraconsistent one can find simple enough that the
_direct_ results of ZF are still so, but that _eventual_ or _extended_
results are not, when _direct_ results of as intuitive a notion as
that there is an "All", and consequences that there is, leave the
happy fiction that ZF could be consistent when its own model, and so
modeling it, would invalidate it. Yet, specific to that notion is
that as well is justified at least as reasonable (given reason)
grounds that universe-like structures and deduction of their existence
and the import of all's existence has: not just an interminable
cascade of apologizing to the universe with ever greater and never
great non-set cardinals in set theory, but here theory, and set
theory, accommodating the dialetheic.

All and Nothing, Nothing and All: same.

So: forcing is an axiomatization: of a universe-like set. And, it's
a contrivance: for theory and its universe.

A theory: is the theory of everything.

Regards,

Ross Finlayson

[Ross A. Finlayson]

On Mar 16, 3:44 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>

wrote:
> On Mar 10, 5:47 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>
> wrote:
>
>
>
>
>
>
>
>
>
> > So isn't forcing simply Dirichlet box / pigeonhole principle?  Without
> > appeal to forcing, simply define arguments in symmetry then as to
> > transfinite pigeonhole, what's the difference?
>
> > How can large cardinals be defined in terms of V (the Universe) when V
> > isn't a set?  How are they cardinals if they aren't of sets?
>
> > How can forcing's model be at once model and embedded in the model?
> > Wouldn't it then be irregular?
>
> > Are there any results in recursion/computation that can't simply (or
> > not so simply) be framed in asymptotics without appeal to transfinite
> > cardinals?
>
> > Are there any results not of transfinite cardinals, solely due
> > transfinite cardinals?
>

> Looking tohttp://cantorsattic.info, this is a nice resource for

Then, the notion that the Kunen inconsistency establishes that there
are no non-trivial embeddings of elements, has these notions of
counterexamples: that there are finite or infinite isomorphic
structures, and that opposition isn't trivial.

Opposition is to map each element v of the universe V to the element V
\v. As it's a universal set, it contains as elements V\v, basically
the complement of elements {v}. Then, what about when v is itself an
irregular collection, eg of ordinals? The idea here is to figure out
where V\v = v. That gets into notions like: the Russell set, with
and without its sputnik of quantification, that R_realized \r =
R_specification, but that: r = R_specification, that R_realization \
r = R_specification = r, for R the Russell set and r itself. These
self-complementing sets (in the universe) are then a similar notion as
to the collection of groups with a natural operation of composition
aside auto-annihilation: Grp as a group. Basically in the universal,
there is that disjoint themselves, some objects are still themselves,
and that, in union, some non-empty objects union themselves are
empty. These are structural features that follow from deduction.

Then for isomorphic structures, that is as simple as any system with a
language with two elements, that any two elements satisfy the
properties of being distinct elements with regards to the structure,
that any are interchangeable. Unless the stucture is actually all the
possible structure that could be, which it would thus be, there is a
non-trivial (non-identity) simple transposition of those elements.
Models are concrete, or not, and if not, there are non-trivial
elementary embeddings in the universe of them, under isomorphism, and
if so, then universe as concrete model is irregular: and thus is the
domain.

That doesn't much speak to whether it's hypocritical to have large
cardinals, bereft properties of cardinals and sets, forced into the
space of regular cardinals (well-founded cardinals as it were). There
either are or aren't "inconsistent multiplicities": defined by their
elements, that the universe exists or the "domain principle" is true
has merit as a truism. To apply a class distinction, for example to
ordinals as defined by their elements and containing lesser ordinals,
sees that there is courtesy their construction: no difference, to
have quantification over elements, or not, and that it's inconsistent
to have both classes and sets without having both defined by their
elements and their structure.

Regards,

Ross Finlayson

[Ross A. Finlayson]

Kunen -> Grp as a group.

Here I am reading off that via elemental embedding,
the element and its complement in the naive universe,
its context, here this simple universal relation of
composition and auto-annihilation is making the set-
theoretic universe also a group.

Kunen is with Burali-Forti and Russell and Cantor
on the set size limits and paradoxes of size and domain.

The Kunen inconsistency here is "no complete universe,
cardinals are incomplete under universals, here though
with that cardinals in ZF are regular". (Including large
cardinals.)

Then, here, universal and group actions, are over
sets as over cardinals.

It seems a neat establishment, this in the
notation, here references in group actions,
in set theory.

I'd certainly admit that's clearly easy to read off.

[Rupert]

On Monday, March 11, 2013 at 1:47:51 AM UTC+1, Ross A. Finlayson wrote:
> So isn't forcing simply Dirichlet box / pigeonhole principle?

No.

> Without
> appeal to forcing, simply define arguments in symmetry then as to
> transfinite pigeonhole, what's the difference?

This bears no resemblance to forcing.

> How can large cardinals be defined in terms of V (the Universe) when V
> isn't a set? How are they cardinals if they aren't of sets?

Large cardinals are sets; you can find the standard definitions in any textbook.

> How can forcing's model be at once model and embedded in the model?
> Wouldn't it then be irregular?

Which textbook about forcing are you referring to?

> Are there any results in recursion/computation that can't simply (or
> not so simply) be framed in asymptotics without appeal to transfinite
> cardinals?

I don't know of any results in recursion theory at all that require appeal to transfinite cardinals.

> Are there any results not of transfinite cardinals, solely due
> transfinite cardinals?

This isn't grammatical and appears to make no sense.

[Mostowski Collapse]

Large cardinals are not really defined. Defined
would mean you have a definition (in ZFC):

my_lrg_card(X) :<=> ...

And then you could prove:

Existence:
exists X my_lrg_card(X)
Uniqueness:
forall X,Y (my_lrg_card(X) /\ my_lrg_card(Y) => X=Y)

But often you cannot prove Existence in ZFC.
So you simply need to assume it.

Some such assumptions, called Large Cardinal Axioms,
have the nice property, that you can

use them to show something about ZFC. Like ZFC
has a model, and thus is consistent.

https://en.wikipedia.org/wiki/Large_cardinal

[Mostowski Collapse]

An early discovered large cardinal was an inaccessible
cardinal. Thats basically a cardinal k such that:

forall a (a < k => 2^a < k)

So its member are closed over power set, and thus
k would give a set universe itself. If you iterate
it, you get univeses inside unverses etc.., like

universe axiom of Grothendieck. But alas we have:

"As is the case for the existence of any inaccessible
cardinal, the inaccessible cardinal axiom is unprovable
from the axioms of ZFC."
https://en.wikipedia.org/wiki/Inaccessible_cardinal#Existence_of_a_proper_class_of_inaccessibles

[Rupert]

On Wednesday, January 22, 2020 at 1:09:48 PM UTC+1, Mostowski Collapse wrote:
> An early discovered large cardinal was an inaccessible
> cardinal. Thats basically a cardinal k such that:
>
> forall a (a < k => 2^a < k)

That is a strong limit cardinal. An inaccessible cardinal in addition has to be regular and uncountable.

[Mostowski Collapse]

For universe axiom of Grothendieck, you would
also allow countable, and then not all inaccessible
cardinals lack proof of existence.

For example V_omega provable exists, and is even
a model of ZFC \ AOI.

"Die einzige in der Zermelo-Fraenkel-Mengenlehre
ZFC bekannte unerreichbare Kardinalzahl ist ℵ0. Die
Existenz weiterer unerreichbarer Kardinalzahlen
kann im Rahmen dieser Theorie nicht bewiesen werden
(die Widerspruchsfreiheit derselben einmal angenommen),
sondern muss durch ein neues Axiom postuliert werden."
https://de.wikipedia.org/wiki/Grothendieck-Universum#Unerreichbare_Kardinalzahlen

[Mostowski Collapse]

But frankly I have not a working memory right now
about weak/strong and regular/singular.

It seems you can use weakly inaccessible k and
Gödels constructive universe to already get:

L_k |= ZFC

"Weak limit cardinals become strong limit
cardinals in L because the generalized continuum
hypothesis holds in L."
https://en.wikipedia.org/wiki/Constructible_universe#L_and_large_cardinals

[Peter Percival]

Mostowski Collapse wrote:
> An early discovered large cardinal was an inaccessible
> cardinal. Thats basically a cardinal k such that:
>
> forall a (a < k => 2^a < k)

Both k = 0 and k = omega satisfy that!

[Ross A. Finlayson]

Large cardinals aren't cardinals, nor (regular) sets.

I encourage anyone to read Cohen's formalism of forcing
as he applied it to the undecideability of CH, and GCH.
It should be familiar in argument after Skolem and collapse
that he compactifies the space, and basically establishes
trans-finite Dirichlet, and most people don't quite much
know about its neat, simple, clear, and direct formalism
instead having established the trivia about its import to
questions of decideability of CH.

Large cardinals: often ordered by "discovery".

Of course Kunen inconsistency is way on top of all that.

[Ross A. Finlayson]

Heh, no it is grammatical (-ly correct), that real analysis
for example is courtesy countable additivity, and it can be
built besides without trans-finite cardinals _per se_,
it would be interesting to establish something courtesy
trans-finite cardinals not simple due other features of
the objects in their spaces as it were.

I'm referring to Cohen's "Independence of the Continuum
Hypothesis", I and II, and about what you can read off
my opinions that stand here about it. (That are his.)


I don't care to fault you, but, if you can't read,
it's not my fault. And, one would do well to validate
the grammatical correctness of my usual long-winded
approach, not that it's ever wrong: just too strong.

Or, "RTFM". TLDR: reading comprehension (and accuracy,
and speed, for digestion and retention) is an important life skill.

TLDR: Johnny can't read. (And didn't consult the sources.)


Trans-finite induction is a very usual notion in the establishment
of induction over ordinals generally. (And, all of them.)

[Mostowski Collapse]

Nope, you are talking nonsense.

Well if you add a large cardinal axiom,
like for example:

exists X my_lrg_card(X)

Then there will be at least one ordinary
set X that is my_lrg_card.

This will happen in all models that
satisfy ZFC and the large cardinal

axiom. Has nothing to do with forcing.

[Mostowski Collapse]

In set theory cardinals are usually taken
as ordinals, which in turn are taken a sets.
A cardinal is just the least ordinal with

a certain size, and therefore a set. To
have a large cardinal already exist as a
class before you lift the class into a set

via an axiom, requires another for of
definition statement for the large cardinal.
A definition statement such as:

my_lrg_card(X) :<=> ....

has X as a whole as a set argument, and
does not describe the large cardinal
as a class. To have the large cardinal

described as a class you would need:

X = { x | my_lrg_card_member(x) }

You could then operate with the large
cardinal as a class. But this gives you no
power above ZFC, since ZFC has already

these classes as syntactic sugar. Only
if you postulate a large cardinal axiom
you change the picture.

[Mostowski Collapse]

Do you know a large cardinal, which has a

my_lrg_card_member

predication?

[Mostowski Collapse]

Anyway the error you made reminds me of
Mückenhausen, he is also ignorant of any
part and whole distinction,

and does not understand what the objects
of a set theoretic domain are, and how FOL
can talk about these objects.

[Ross A. Finlayson]

Large cardinals aren't cardinals, ordinals, nor sets,
in ZF(C), nor do they have initial ordinals.

Forcing (relativizing the domain, to the enumerable)
and "large cardinals", are different things, indeed.

Now you should read "Independence of the Continuum
Hypothesis" and follow its formal development if you'd
rather be interested in what all's the switch-er-roo.

[Ross A. Finlayson]

Over the past few years, the "Mostowski Collapse" poster
really has demonstrated some improvements in its understanding,
what from a more informed point might have seemed very ignorant.

[Peter Percival]

Ross A. Finlayson wrote:

> Large cardinals aren't cardinals, ordinals, nor sets,

Clearly they are all of those things.

> in ZF(C)

Typically they
(i) are to believed to be consistent with ZF,
(ii) imply ZF's consistency, and thus
(iii) can't be proved, in ZF, to be consistent with ZF if ZF is consistent.

> , nor do they have initial ordinals.

What does that mean?

[Peter Percival]

Ross A. Finlayson wrote:
> On Thursday, January 23, 2020 at 2:21:05 AM UTC-8, Mostowski Collapse wrote:

[...]

>
> Over the past few years, the "Mostowski Collapse" poster
> really has demonstrated some improvements in its understanding,
> what from a more informed point might have seemed very ignorant.

You, on the other hand, have only ever written incoherent garbage.

[Ross A. Finlayson]

Heh, dupe.

Hasty overgeneralizations often result in the wrong.

I accept what insult that is as unfortunately,
both to and from you.

I.e., it's not so much unlucky, unfortunate, just, ..., sad.
I'm inured to that but, can't really make the horse drink.
(I'd even be a bit surprised if it did.)

Anyways large cardinals these days is a bit the backwater
after "dynamical measure theory" and real results in real
analysis. There was a lot of hope that it would inspire
a modern (contemporary) compelling foundation, but different
notions of "continua" instead seem to be the interest more
generally (ultrafilters and the metrizable, line continuity,
properties of Cantor space and distributions of numbers,
in their space, ...).

In, ..., "the logic".

[Mostowski Collapse]

I guess Rossi AI Robot has discover
another kind of dark numbers,

since his large cardinals, are neither
cardinals, ordinals, nor sets.

Thats quite amazing!

LMAO!

[Mostowski Collapse]

Whats the Rossi AI Robot criteria
for dark number? WM has a number must

be addressable. Whats it in Rossi AI
Robot? A cardinal must rest off a

space filling curve?

LMAO!

[Mostowski Collapse]

In the following I write |- _ as a short
hand for ZFC |- _. The Russel Set is not a
large cardinal, since we already know that
its non-existence as a set can be proved.

If Ru is the predication that u is a R
ussel set, then we can prove:

|- ~Ru

Which is the same as:

|- forall u (~Ru)

Which is the same as

/* we can state this for the Russel set */

|- ~exists u Ru

If such a proof is available for a predication,
then we know that in all models of ZFC, an object
satisfying the predication is not possible. For
a large cardinal property this is not the case.

We cannot prove their non-existence. Because a
similar proof for a Large Cardinal property LC,
would give read:

/* usually large cardinals don't have this statement */

|- ~exists u LCu

The typical phrasing is that exists u LCu is consitent
with ZFC, this precisely means:

/* what the phrasing consistent with ZFC means */

exists u LCu |/- f

But now using right implication introduction we get:

/* the phrasing is equivalent that existence cannot be denied */

|/- ~exists u LCu

[Ross Finlayson]

Cohen's forcing is because of illative ordinals and pair-wise sets.

That it's sort of "Dirichlet for Poincare" helps explain things.