dimension theory and the continuum hypothesis

[mitch]

The theorem statement which follows is from
"Dimension Theory" by Hurewicz and Wallman. When
reading it, observe that the first uncountable
infinity is a fixed point relative to transfinite
dimension. Hence, this theorem statement expresses
what one might take to be the truth of the
continuum hypothesis. Whether or not the cardinality
of the continuum is set-theoretically greater than
the first uncountable cardinal, it is collapsed
relative to dimension thoery.

The theorem statement is given as

< begin quote >

"If X has a transfinite dimension alpha,
then alpha is of the first or second
ordinal class"

< end quote >

The language is not necessarily modern here. But
the proof statements are clear,

< begin quote >

"Proof. We have to show that alpha is
less than Omega = omega_1. Suppose the
contrary, and let beta be the smallest
ordinal, Omega <= beta <= alpha, for which
there exists a space B of dimension beta.
[...]"

< end quote >

In either "Principles of Mathematics" or
Introduction to Principles of Mathematics"
Russell makes the observation that counting
dimensions is a different way of counting.
If, in fact, the continuum hypothesis is a
second-order truth that cannot be proven,
one should expect it to be expressed somewhere
in the mathematical literature. Since
dimension theory arises out of topology and
since topology would normally be considered
to be second-order, the theorem statement
above would be a likely candidate for that
occurrence.

mitch

[mitch]

In another thread, the book "Sets for Mathematics"
had been mentioned because of glossary entries that
elaborate on what is meant by "foundation" for the
authors Lawvere and Rosebrugh. As I have come to agree
with the importance of such explanations, I submit
the following paragraph from a research note I wrote
for myself approximately eighteen months ago,

< begin quote >

The material discussed here may be viewed as part of a
structure theory for mathematical logic. In that
sense, it may be called metamathematics. However, the
modern use of that designation is vague, and, the
original use of that designation from Hilbert and
Bernays should be called arithmetical metamathematics
to contrast it from the current exposition. Here,
elements from finite geometry are emphasized. This
view originates from unpublished work interpreting the
negation of the identity relation as coinciding with
topological separations. Finite geometry arises when
extending mathematical separation principles to the
syntactic domain. The methods here may be viewed as
making rigorous the notion of "a prioriness of symbol"
from Hilbert and Bernays. In general, mathematicians
are wary about "symbol shapes" even though that is
precisely what is implicit to logical treatments that
construe mathematical discourse as purely formal and
then reduce proofs to mere rewrite systems. In view
of how topological considerations have motivated this
work, it would be more appropriate to refer to the
current material as topological metamathematics.

< end quote >


Whether or not anyone taking the time to read this
agrees with the latter statements, what I wish to
emphasize is that I acknowledge why certain views are
defended as the "received paradigm". I happen to
believe that "formalism" and "conventionalism" have
been conflated in the literature. But, as far as my
own interests are concerned, if they constitute
"metamathematics" under some strict definition of
"mathematics", then so be it.

While there may be some syntactic notions of forcing
in the literature, the generic forcing theorem in
Jech uses an analogue to Dedekind's completion in
order to form a complete Boolean algebra. The topology
obtained in this way is based upon regular open sets.
And, in his book "Simplified Independence Proofs",
Rosser begins with a statement of topological axioms
to be used. Clearly, while I have not seen the
expression "topological metamathematics" in the
literature, I have not invented it.

For the most part, the material I am posting to
this thread is not taken from the research note
containing the quote above. It does overlap, however,
because much of that note involves characterizing
relations with respect to ortholattices.

mitch

[mitch]

In this post I will try to explain a relation
between counting dimensions and the system of
truth tables which serve as the semantic ground
for propositional logic.

It begins with the fact that the sixteen truth
tables satisfy the axioms of a finite affine
geometry. One can see this if one studies negations
and de Morgan conjugations as involutions on the
set of truth tables. Note, however, that there is
actually a commutative diagram involving three
different involutions. I refer to the third as
conversion because it exchanges the conditional
for the reverse conditional (the converse). So, if
one performs this analysis, one will obtain three
involutions -- 'conversion', 'conjugation', and
'negation' -- organized as a commutative diagram.

Because each affine geometry is associated with
a projective geometry, I worked out a labeling
for a 21-point projective geometry so that the
line at infinity would have the terms 'NOT', 'NO'
'ALL', 'SOME', and 'OTHER'. While logicians have
developed many notions of "quantifier", the success
of compositional logics in the late nineteenth
century really depended upon being able to represent
the classical square of opposition. The significance
of Russell's paradox is that representability of
the square of opposition fails when the qualifying
predicate "is a set" is used in the representations.
Russell's type hierarchy (specific to his usage of
the term 'type') solves the problem because every
application of "is a set" occurs with respect to a
universe of discourse with a greater "order".

The terms above are thought of with respect to
the square of opposition. However, they refer only
to the syntactic representations,

SOME -> Ex
OTHER -> Ex~
ALL -> ~Ex~
NO -> ~Ex

whose relationship to one another depends on a unary
negation ('NOT'). In general, unary negation has been
portrayed as a connective. But it is eliminable
because of the complete connectives NOR and NAND. It
really has no essential role in propositional logic,
although it is essential to extensions to propositional
logic which extend the system analogously to what has
been done above.

Both deontic logic and modal logic may be viewed
in this way.

When examining the labeling used for the projective
geometry, what became apparent had been the fact that
a finite affine geometry on sixteen points has twenty
lines. It turns out that the labeling can be done
in such a way that fifteen terms for truth tables can
be mapped into line names. Under this labeling, all
five of the terms on the line at infinity map into
line names.

Now, there exists a 20-element ortholattice with
a 16-element Boolean block. If one exchanges 'NOT'
for 'NTRU' (false) in the bottom of the 16-element
Boolean block, then the quantifier terms label the
remaining four elements of the lattice. And, since
those elements are located in an 8-element Boolean
block, it is the order theory which situates them
as complements to one another.

This particular lattice also has a representation
as an orthologic. Such representations are summarized
in what is called an orthogonality diagram. An
orthogonality diagram for the lattice discussed here
is given by



OTHER
*
/ |
/ |
/ |
/ |
/ | NOR NIF
SOME *------*-------------*
|\ /|
| \ / |
| \ / |
| X |
| / \ |
| / \ |
|/ \|
*-------------*
NIMP AND


The labels on the vertices correspond to the atoms
of the ortholattice. The two Boolean subblocks share
one atom. So, the triangle denotes an 8-element
Boolean lattice with three atoms, while the complete
graph on the quadrilateral denotes a 16-element
Boolean lattice. Because they share an atom, they
share the corresponding co-atom and the top and bottom
of the lattice. This is what is called an 'atomic
amalgam' in the theorh of ortholattices.

Now, observe that the triangle is a complete
graph and the (degenerate) projection of the
2-simplex into the plane. As denoted above, the
quadrilateral is a complete graph. With four
vertices and six edges, it is the projection of
the 3-simplex (tetrahedron) into the plane. In
fact, the infinity of regular polygons in the
plane corresponds with the sequence of complete
graphs associated with those polygons. Moreover,
each such complete graph may be thought of as
corresponding with an n-simplex of appropriate
dimension projected into the plane.

Consequently, one can situate the orthogonality
diagram above into a sequence of orthogonality
diagrams beginning with



* *
/ | / |
/ | / |
/ | / |
/ | / |
/ | / |
*------*----- * *------*-------------*
| / |\ /|
| / | \ / |
| / | \ / |
| / | X |
| / | / \ |
* | / \ |
|/ \|
*-------------*


The first diagram consists of two triangles
sharing a vertex. Both are complete graphs.
So, the diagram consists of two complete graphs
sharing a vertex.

Note, then, that the second diagram consists
of two complete graphs sharing a vertex. This
is precisely the form that the following element
and all of its successors will have. The
triangle will be represented uniformly in each
diagram while the other complete graphs will
proceed in relation to pentagons, hexagons,
septagons, octagons, and so forth (nonagon?).

For anyone reading this post who is familiar with
embedding theorems in combinatorial topology, recall
that every abstract simplex of dimension n can be
embedded into a real space of dimension n+2. Since
this presentation involves projections of geometric
simplexes, one cannot make a direct comparison to
that result. However, the arithmetic is correct.
The number of vertices from the projected simplex
is increased by two in the orthogonality diagrams
described here.

As this post is now quite lengthy, let me point
out that there is an "off-by-two" error in the
correspondence with how one would naively interpret
this structure. The diagrams above would denote
'1' and '2', whence the logical structure is
associated with "the first plural natural number".
Since semantics must assume a plurality of truth
values it is only natural that 'logic' needs to be
understood with respect to some 'fixed point'
associated with plurality. In particular, that
number needs to be two because any multiplicity
of truth values can be reduced to 'true' and
'not true'. Indeed, the notion of rules of
detachment in proofs dictates this reduction.

I will explain the labeling for those orthogonality
diagrams in another posting.

mitch

[mitch]

Any mathematician who uses the fraction,

1/4

thinks of what is denoted singularly. But, that
fraction is a distinguished term among an infinity of
fractions. It is distinguished by being that
fraction expressed in lowest terms.

Similarly, when working with periodic functions,
every mathematician expresses the result in relation
to the principal branch of the angle measure. There
are, in fact, an infinity of expressions which would
be numerically correct.

In the link,

https://www.math.wisc.edu/~miller/old/m873-05/setplane.pdf

what is listed as Theorem 1 is a result of Davies
discussing a partition of the plane. The
distinguishing feature of these classes is that each
possible distance is represented only once between the
points in a class.

Normally, a metric is understood as a function
into the non-negative real numbers. In fact, however,
this only makes sense if the real number is understood
in relation to its difference with respect to zero.
So, it is mapping a pair of terms from a space to a
line segment with endpoints, except when the terms
denote the same elements.

Davies result shows that the continuum hypothesis
provides for a similar system of line segments. Since
it is based upon a countable partition, the partition
containing the origin of |R^2 would be a good candidate
to serve as the distinguished partition just as the
fraction,

1/4

is the distinguished element among those fractions
taken to be equal to it.

Because there are no line segments of zero length,
this understanding of the partition can only be
arrived at by considering "apartness" or "diversity"
relations.

Davies paper does not appear to be available
online without a subscription. The abstract
can be found at the link,

https://www.researchgate.net/publication/231947429_Partitioning_the_plane_into_denumerably_many_sets_without_repeated_distances

In contrast with the theorem statement in the earlier
review paper, the abstract describes the problem as
suggested to Davies by Erdos. In that statement, it
is a matter of finding partitions in which any four
points determine six different distances.

This configuration would be a complete graph on
four points. Before you judge what is suggested
here too harshly, look at the orthogonality diagram
for 'logic' in the post discussing logic, plurality,
and counting dimensions.

I cannot prove the continuum hypothesis. My
reasons for believing it to be a truth of mathematics
arise from how the continuum is understood through
its metric structure.

mitch

[Ross A. Finlayson]

About the provability of CH, you might find some ways
to address the forcing argument as to its independence
from ZFC in terms of what statement there is that is
extra-ZFC in the method of forcing (for trans-finite
Dirichlet box).

Some might have that these cardinals dense in themselves
(for Not CH) have initial ordinals so that if it's
consistent that they exist, then they do, because it's
consistent that they exist.

As of other developments in set theory you can find
some varying views on the interpretation of what
the forcing method is, here.

For what purposes you might find of ZF+CH (that they
are distinct domains, the orders of magnitude as
represented by the initial ordinals of cardinals)
you might build that into some model (that they
are distinct domains) where still in ZFC it is
consistent that they are transitive domains,
as it were, that nonstandard processes or supertasks
variously cross domains as in accord with some notion
of the transfer principle, or model expansion/collapse.

In finite combinatorics you might find that
enumerating the subsets of a set is O(2^n).
Once those are generated for the indices
in space terms O(n 2^n) ~= O(2^n) of the sum
of the lengths of all the combinations, for
O(1) indices' space and access, then those
can be read out in parallel in O(n).

Then, some notion like simulated annealing
or "quantum processes" (or about their
implementation as analog processors or
computers) might see that the model of
orders of magnitude could have various
implementations as that the overall (or
outer) models' expansion or collapse
would maintain the various distinctness
of domains of the inner model, as it were.

[khongdo...@gmail.com]

On Sunday, 9 October 2016 01:07:00 UTC-6, mitch wrote:

> I cannot prove the continuum hypothesis. My
> reasons for believing it to be a truth of mathematics
> arise from how the continuum is understood through
> its metric structure.

Seems like mathematicians and logicians these days would look
at being impossible to prove in meta level as some sort of a
intellectual/knowledge disease. Why is that, I wonder?

[khongdo...@gmail.com]

When can we once for all get rid of the Hilbert-era mistake which is
basically that _anything_ could be proven - it's just a matter of time?

[Ross A. Finlayson]

That's "independence" not "impossible to prove".

Either CH or Not CH are provable (as they were
each shown independent of ZF) given some various
stipulation as to expanding or restricting
comprehension as modern logic hasn't yet
arrived at, for narrative instruction.

Your "language model" (which is just a partial
model of a space of sentences then for moving
it around in truth tableau or from column to
column when things change) isn't a language
model about which inference proper itself can
be made so as to establish these facts (as it
were) of model theory about the model (and
the objects of the model).

That's neither good nor bad, it just is,
(some language model template), but, it
is only what it is, which is the role of
careful and constant definition, _then_
for the establishment of convention for
the relaxation of declaration as established.

Some would find its misappellation indefensible,
for example formalists (which includes most
practices of the communication of logic,
and particularly the establishment of determinism).

[khongdo...@gmail.com]

On Sunday, 9 October 2016 10:28:33 UTC-6, Ross A. Finlayson wrote:
> On Sunday, October 9, 2016 at 8:54:30 AM UTC-7, khongdo...@gmail.com wrote:
> > On Sunday, 9 October 2016 01:07:00 UTC-6, mitch wrote:
> >
> > > I cannot prove the continuum hypothesis. My
> > > reasons for believing it to be a truth of mathematics
> > > arise from how the continuum is understood through
> > > its metric structure.
> >
> > Seems like mathematicians and logicians these days would look
> > at being impossible to prove in meta level as some sort of a
> > intellectual/knowledge disease. Why is that, I wonder?
>
> That's "independence" not "impossible to prove".
>
> Either CH or Not CH are provable (as they were
> each shown independent of ZF) given some various
> stipulation as to expanding or restricting
> comprehension as modern logic hasn't yet
> arrived at, for narrative instruction.

Apparently you don't understand my "prove in meta level".

_Assuming only_ ZF be consistent and have a model M, would you think that
it's a distinct possibility that it's impossible to demonstrate - prove in
meta level - (G)CH is true in M?

[Peter Percival]

khongdo...@gmail.com wrote:

> When can we once for all get rid of the Hilbert-era mistake which is
> basically that _anything_ could be proven - it's just a matter of time?

What makes you think that the Hilbert-era mistake, as you call it, was
not recognized as a mistake years ago?


--
Do, as a concession to my poor wits, Lord Darlington, just explain
to me what you really mean.
I think I had better not, Duchess. Nowadays to be intelligible is
to be found out. -- Oscar Wilde, Lady Windermere's Fan

[Peter Percival]

khongdo...@gmail.com wrote:
> On Sunday, 9 October 2016 10:28:33 UTC-6, Ross A. Finlayson wrote:
>> On Sunday, October 9, 2016 at 8:54:30 AM UTC-7, khongdo...@gmail.com wrote:
>>> On Sunday, 9 October 2016 01:07:00 UTC-6, mitch wrote:
>>>
>>>> I cannot prove the continuum hypothesis. My
>>>> reasons for believing it to be a truth of mathematics
>>>> arise from how the continuum is understood through
>>>> its metric structure.
>>>
>>> Seems like mathematicians and logicians these days would look
>>> at being impossible to prove in meta level as some sort of a
>>> intellectual/knowledge disease. Why is that, I wonder?
>>
>> That's "independence" not "impossible to prove".
>>
>> Either CH or Not CH are provable (as they were
>> each shown independent of ZF) given some various
>> stipulation as to expanding or restricting
>> comprehension as modern logic hasn't yet
>> arrived at, for narrative instruction.
>
> Apparently you don't understand my "prove in meta level".
>
> _Assuming only_ ZF be consistent and have a model M, would you think that
> it's a distinct possibility that it's impossible to demonstrate - prove in
> meta level - (G)CH is true in M?

It is known that there are models in which CH is true and models in
which not-CH is true.

[Ross A. Finlayson]

Here we're talking about the
Continuum Hypothesis, that
Aleph_alpha+1 = 2^Aleph_alpha.

Goldbach's conjectures concern
themselves with the distribution
and addends of primes (vis-a-vis,
their integer composition, where,
they're primes). Goldbach's conjectures,
absent proof, hold up well because
they can be checked and for some
effectively large upper bound
courtesy mechanical computing,
it is known that Goldbach's
conjectures hold up to various
known bounds and then that any
counterexamples would follow
certain various forms as follows
various relations in number theory.

So far, this applies to most statements
in number theory as have the status of
being a well-researched conjecture.

Now, what's relevant to whether a given
Goldback conjecture could be undecideable
(independent of number theory) here is
basically about the notion that there
would be some undecideable fact about
the numbers basically about a function
that belies all finite inputs, then
whether or not it has the extrapolated
output or the deduced output.

Basically this is a question about
the integers in terms of, for example,
whether the point at infinity is a
prime, or a composite (as an example).
This is a question about the integers
to establish quite fundamentally some
root question that imbues the integers,
which are mostly defined in terms of
their unboundedness, as what may apply
about their compactness.

So, it _is_ in that sense possible that
a conjecture of Goldbach might be independent
of usual clear direct inductive naive
definitions of the integers, then for
where, if that is so, then one might
consider the stipulation as about the
integers (and basically the closure of
their operations) to so establish more
clearly or definitely a model of the
integers as would decide this "infinity-
point-independent" fact about the integers.

It's not established (deterministically,
definitively, and directly) that is so, just,
not that it is not, those are conjectures.

Then the question would properly be about
whether there are "infinity-point-independent"
facts about the integers (and some have that
there are), then what those are.

CH is established (courtesy Goedel and Cohen)
as independent of ZF.

A given GC is not established as
independent of "number theory".

"Infinity-point-independent"
statements about integers are,
as definition (and example),
independent of "number theory",
as are various known concerns
of compactification of the integers
(being "infinity-point-independent").

[Peter Percival]

Ross A. Finlayson wrote:

> Now, what's relevant to whether a given
> Goldback conjecture could be undecideable
> (independent of number theory) here is

[...]

> So, it _is_ in that sense possible that
> a conjecture of Goldbach might be independent
> of usual clear direct inductive naive
> definitions of the integers, then for

[...]

If Goldbach's conjecture is undecidable then it is true in the standard
model.

[Ross A. Finlayson]

A (or "the") standard model, where it exists,
is modeled by all extension models, for example,
generic extensions (or upward Louwenheim/Skolem).

Then, it looks that I'm positing that
"number theory's integers" doesn't have
a standard model (for that it has many
models).

Here "number theory's integers" is inclusive
of somewhat more reasoning as for example
would be outside the usual Zahlen Z, the
integers.

[Peter Percival]

Ross A. Finlayson wrote:

> Then, it looks that I'm positing that
> "number theory's integers" doesn't have
> a standard model (for that it has many
> models).

Posit away, old bean. Yes, they do.

[Ross A. Finlayson]

It appears then that I am positing
that undecideable whether the integers
have a standard model, and there are
stipulations of them where they do,
and stipulations of them where they don't
(have a standard model).

Here this isn't specific to the Zahlen
or naturals, about the variety of usual
considerations of countably and uncountably
trans-finite models of the integers, "trans-
finite" to avoid mentionining "non-standard"
models, which may be so "standard" as to that
which they model, that "Archimedean-nonstandard
integers" aren't necessarily not a standard model.

Then, where the integers "do and don't" have a
standard model, that might seem an interesting
feature of them that I haven't seen discussed
so much before or researched in detail and
established in the canon.


Mitch, now we might still be discussing
"dimensionality and the Continuum Hypothesis",
or only peripherally, or here perhaps actually
rather directly.

[khongdo...@gmail.com]

On Sunday, 9 October 2016 10:44:15 UTC-6, Peter Percival wrote:
> khongdo...@gmail.com wrote:
>
> > When can we once for all get rid of the Hilbert-era mistake which is
> > basically that _anything_ could be proven - it's just a matter of time?
>
> What makes you think that the Hilbert-era mistake, as you call it, was
> not recognized as a mistake years ago?

What makes you think you have a clue what the raised issue here is?

[Peter Percival]

Do you think that you are failing to make yourself clear?

[khongdo...@gmail.com]

On Sunday, 9 October 2016 14:57:33 UTC-6, Peter Percival wrote:
> khongdo...@gmail.com wrote:
> > On Sunday, 9 October 2016 10:44:15 UTC-6, Peter Percival wrote:
> >> khongdo...@gmail.com wrote:
> >>
> >>> When can we once for all get rid of the Hilbert-era mistake which is
> >>> basically that _anything_ could be proven - it's just a matter of time?
> >>
> >> What makes you think that the Hilbert-era mistake, as you call it, was
> >> not recognized as a mistake years ago?
> >
> > What makes you think you have a clue what the raised issue here is?
>
> Do you think that you are failing to make yourself clear?

No. I think you're unable to have a clue or are trolling - as always.

[khongdo...@gmail.com]

On Sunday, 9 October 2016 12:47:34 UTC-6, Peter Percival wrote:
> Ross A. Finlayson wrote:
>
> > Now, what's relevant to whether a given
> > Goldback conjecture could be undecideable
> > (independent of number theory) here is
> [...]
> > So, it _is_ in that sense possible that
> > a conjecture of Goldbach might be independent
> > of usual clear direct inductive naive
> > definitions of the integers, then for
> [...]
>
> If Goldbach's conjecture is undecidable then it is true in the standard
> model.

This is a classic mantra quite a few of us have been brainwashed into _believing_ !

[khongdo...@gmail.com]

Why did you mention Goldbach's conjecture here?

[Peter Percival]

You wrote "When can we once for all get rid of the Hilbert-era mistake

which is basically that _anything_ could be proven - it's just a matter

of time?" And I wondered why you thought that mistake had not been got
rid of. That's all I was asking.

[Peter Percival]

The proof, by contraposition, is elementary.

Suppose Goldbach's conjecture is false, then there are three natural
numbers x, y, z, such that x is even and > 2, and y and z are primes
such that x = y + z. But that relation on the natural numbers (i.e., x
is even and > 2, and y and z are primes such that x = y + z) is clearly
decidable.

[Peter Percival]

Peter Percival wrote:
> khongdo...@gmail.com wrote:
>> On Sunday, 9 October 2016 12:47:34 UTC-6, Peter Percival wrote:
>>> Ross A. Finlayson wrote:
>>>
>>>> Now, what's relevant to whether a given
>>>> Goldback conjecture could be undecideable
>>>> (independent of number theory) here is
>>> [...]
>>>> So, it _is_ in that sense possible that
>>>> a conjecture of Goldbach might be independent
>>>> of usual clear direct inductive naive
>>>> definitions of the integers, then for
>>> [...]
>>>
>>> If Goldbach's conjecture is undecidable then it is true in the standard
>>> model.
>>
>> This is a classic mantra quite a few of us have been brainwashed into
>> _believing_ !
>
> The proof, by contraposition, is elementary.
>
> Suppose Goldbach's conjecture is false, then there are three natural
> numbers x, y, z, such that x is even and > 2, and y and z are primes
> such that x = y + z. But that relation on the natural numbers (i.e., x
> is even and > 2, and y and z are primes such that x = y + z) is clearly
> decidable.
>

Since that proof is so simple I expect you to acknowledge its correctness.

[khongdo...@gmail.com]

On Sunday, 9 October 2016 15:18:24 UTC-6, Peter Percival wrote:
> khongdo...@gmail.com wrote:
> > On Sunday, 9 October 2016 14:57:33 UTC-6, Peter Percival wrote:
> >> khongdo...@gmail.com wrote:
> >>> On Sunday, 9 October 2016 10:44:15 UTC-6, Peter Percival wrote:
> >>>> khongdo...@gmail.com wrote:
> >>>>
> >>>>> When can we once for all get rid of the Hilbert-era mistake which is
> >>>>> basically that _anything_ could be proven - it's just a matter of time?
> >>>>
> >>>> What makes you think that the Hilbert-era mistake, as you call it, was
> >>>> not recognized as a mistake years ago?
> >>>
> >>> What makes you think you have a clue what the raised issue here is?
> >>
> >> Do you think that you are failing to make yourself clear?
> >
> > No. I think you're unable to have a clue or are trolling - as always.
>
> You wrote "When can we once for all get rid of the Hilbert-era mistake
> which is basically that _anything_ could be proven - it's just a matter
> of time?" And I wondered why you thought that mistake had not been got
> rid of. That's all I was asking.

That's not _all_ what you were asking. If you don't understand the issue
I've raised, just ask for clarification: don't retort my post with a
question showing you don't have a clue on the raised issue, like "not
recognized as a mistake years ago?".

[Peter Percival]

khongdo...@gmail.com wrote:
> On Sunday, 9 October 2016 15:18:24 UTC-6, Peter Percival wrote:
>> khongdo...@gmail.com wrote:
>>> On Sunday, 9 October 2016 14:57:33 UTC-6, Peter Percival wrote:
>>>> khongdo...@gmail.com wrote:
>>>>> On Sunday, 9 October 2016 10:44:15 UTC-6, Peter Percival wrote:
>>>>>> khongdo...@gmail.com wrote:
>>>>>>
>>>>>>> When can we once for all get rid of the Hilbert-era mistake which is
>>>>>>> basically that _anything_ could be proven - it's just a matter of time?
>>>>>>
>>>>>> What makes you think that the Hilbert-era mistake, as you call it, was
>>>>>> not recognized as a mistake years ago?
>>>>>
>>>>> What makes you think you have a clue what the raised issue here is?
>>>>
>>>> Do you think that you are failing to make yourself clear?
>>>
>>> No. I think you're unable to have a clue or are trolling - as always.
>>
>> You wrote "When can we once for all get rid of the Hilbert-era mistake
>> which is basically that _anything_ could be proven - it's just a matter
>> of time?" And I wondered why you thought that mistake had not been got
>> rid of. That's all I was asking.
>
> That's not _all_ what you were asking. If you don't understand

I don't.

> the issue
> I've raised, just ask for clarification

Please clarify.

> : don't retort my post with a
> question showing you don't have a clue on the raised issue, like "not
> recognized as a mistake years ago?".
>

[khongdo...@gmail.com]

I don't see any FOL syntactical proof here, nor have I seen any related
meta level proof!

Hint: "But that relation [...] is clearly decidable" isn't a proof of any sort.

[Peter Percival]

Neither do I. Mathemetics textbooks are full of proof that aren't
either of those things. So what?

>
> Hint: "But that relation [...] is clearly decidable" isn't a proof of any sort.

You're quite right, it isn't a proof. It's a statement of fact, so what?

[khongdo...@gmail.com]

If one recites a mantra so many times one would tend to believe what is
being recited a fact, which is what I've just mentioned a moment ago!

[Peter Percival]

Do you know what it means to say that a relation is decidable?

Let there be three natural numbers x, y and z. Do you think it would be
impossible for a Turing machine to determine whether or not x is even
and > 2, and y and z are primes such that x = y + z?

[Ross A. Finlayson]

Here it's clearly one or the other in the
"usual" model where its negation would have
either an existence proof or an example, as
you note. Reticence involves that the "ground"
model, as it were, might have a non-Archimedean
value for some given example or as via a proof
by existence instead of proof by example.

Rather like an example of the well-ordering of
the reals, some have an existence result as
axiomatic but there is no proffered example.

(Of course, I point out that it would be
what I call "sweep()" or the "Equivalency
Function" but it suffices here that there
isn't a "usual" example.)

So, one would expect there to be an object
in the model so to witness the case as
an example, here this tentative outlook
is that it's not so clear that there is
that, when some alternate ground model
could see the negation as via an existence
result sans example, or, some (extension of)
an alternate ground model would have the
example in "nonstandard" elements.

Then here a Goldbach Conjecture is here
just an example of an unresolved integer
conjecture, here with clearly delineated
acceptor/rejector for each, as decideable,
then that as possibly some "infinity-point-
independent" conjecture more generally,
these alternatives result.

[mitch]

On 10/09/2016 03:06 PM, Ross A. Finlayson wrote:

< snip >

>
>
> Mitch, now we might still be discussing
> "dimensionality and the Continuum Hypothesis",
> or only peripherally, or here perhaps actually
> rather directly.
>

It seems as if you are discussing Nam's
interpretation of my concession to the
independence of the continuum hypothesis.

mitch

[khongdo...@gmail.com]

In your challenging me "Since that proof is so simple I expect you to
acknowledge its correctness", what did you mean by _that proof_ ? Is it a
familiar FOL syntactical proof, of which the definition doesn't mention
"Turing"?

Can you even write down or summarize _that proof_ ? (If you can't of course
nobody would care taking the challenge.)

[mitch]

Since the partition from Davies' paper is merely
another equivalence deriving from the continuum
hypothesis, there is nothing that has been given
beyond a comparison with mathematical practice
which would actually justify consideration.

As I had mentioned to Mr. Percival in another
thread, Russell's theory of knowledge had been
based upon treating demonstratives as generating
elementary propositions in relation to pragmatic
language acts (ostensively pointing to some
element of immediate experience -- what Strawson
would call a paradigmatic material object).
Russell's theory of knowledge does not exactly
coincide with modern notions of a semantic
theory. That would have to wait for Kaplan's
logic of demonstratives.

Kaplan describes contexts which include "agents",
"times", and "locations" in their description.
After giving a logic and a semantics for his system,
he differentiates between "validity in all
circumstances" and "validity in all contexts".
Indeed, every context is a circumstance, but not
every circumstance is a context. Kaplan refers to
circumstances as what is correlated with classical
logic. I believe he is referring to an S5 modal
logic, although that is not clear to me.

For his temporal component, Kaplan naturally
uses the natural numbers as indexes. By doing
this, Kaplan differentiates "static" from
"invariant". If an agent is witnessing an
unchanging environment, what is being witnessed
is "invariance" because the temporal index is
non-constant. What is static is the representation
at each temporal index.

Davies partition may be compared with this.

It is a countable partition with each partition
class having the "same" unique distances given
through line segments. The distances are invariant
even though the line segments are different in
each partition class.

While logic concerns itself with truth and
provability, mathematicians often seek out
various invariants that characterize whatever
system with which they may be working. In this
respect, then, the continuum hypothesis may be
understood as an invariant needed for applied
mathematics. It arises because one witnesses
change rather than the algebraic dimension of
time.

mitch

[Ross A. Finlayson]

No, not so much: what was of note was the
analogy of the Continuum Hypothesis, where
there are no intermediate cardinals dense
in cardinal exponentiation as "distinct",
and, Not CH, where in a sense the cardinals
(or their initial ordinals) would become
"dense", where the first sees some discrete
progression of orders of magnitude of that
which they model, and the second sees some
continuous progression of orders of magnitude.

This then is for some use of cardinals in
application, with the use of expansion and
collapse per Skolem, Louwenheim, and Levy.


The second point then is about some "Integer
Hypothesis" about whether the integers have
a standard model (or that they do and don't
as some particular prototype, eg as an
inductive set).

This is where theories of sets, categories,
geometries, numbers, etcetera might have
sufficient structure to be equi-interpretable
and furthermore support each other, then
that this is a notion of an "Integer Continuum
Hypothesis" of sorts.

[Peter Percival]

You supposedly did a degree in mathematics, were all proofs in your
course FOL syntactical proofs? Note that FOL syntactical proofs prove
almost nothing - validities get proved and nothing else.

> Can you even write down or summarize _that proof_ ? (If you can't of course
> nobody would care taking the challenge.)
>

[Peter Percival]

It is difficult to know what kind of stupid you are - too stupid to
recognize the validity of my proof, or too stupid to admit that you are
wrong.

[khongdo...@gmail.com]

Idiotic ad-hominem attack from the crank Peter Percival, as usual.

[khongdo...@gmail.com]

Can you at least tell sci.logic what sort of proof what you claimed
to have had (your "that proof") is?

[George Greene]

On Friday, October 7, 2016 at 12:27:38 PM UTC-4, mitch wrote:
> If, in fact, the continuum hypothesis is a
> second-order truth that cannot be proven,

Given that it cannot be proven (from the usual axioms),
it is NOT possible that the continuum hypothesis BE ANY
kind of "truth", 2nd-order OR OTHERwise, since it follows
from its unprovability that there exist models of the axioms
IN WHICH IT IS *FALSE*.

You would have to claim the non-second-order-ness AND the
non-STANDARD-ness of ALL of these models to make the continuum
hypothesis "true" IN ANY sense, and that's just not happening.

Equivalently, you would need (but, the point is, you DON'T HAVE)
some "vision of THE UNIQUE INTENDED"
model for set theory (in which case the truth-value of the continuum
hypothesis would be its truth value in THAT model).

[mitch]

On 10/12/2016 02:54 PM, George Greene wrote:
> On Friday, October 7, 2016 at 12:27:38 PM UTC-4, mitch wrote:
>> If, in fact, the continuum hypothesis is a
>> second-order truth that cannot be proven,
>
> Given that it cannot be proven (from the usual axioms),
> it is NOT possible that the continuum hypothesis BE ANY
> kind of "truth", 2nd-order OR OTHERwise, since it follows
> from its unprovability that there exist models of the axioms
> IN WHICH IT IS *FALSE*.

There is a game associated with the continuum hypothesis.

Either you do not understand it, or, you choose to ignore
it.

The idea is to formulate reasonable axioms that do not
assume the actual proposition. So, there is a reasonableness
criterion that mathematicians, and others, must be asked
to consider.

Dimension theory is an accepted branch of mathematics. The
result that I cited correctly collapses the transfinite
sequence such that there are only two notions of infinity.
The continuum hypothesis would have to be true.

What is true of dimensions is that they are preserved under
continuous transformations. That set theory has a more
expansive notion of function is not being challenged. But,
in research from the homotopy type theory community,
Awodey and Kishida write,

< begin quote >

"To sum up: In FOS4, a necessary description defines
a name, which then has a continuous denotation, whereas
a contingent description need not have a corresponding
description."

< end quote >

This is from their paper "Topology and modality: The topological
interpretation of first-order modal logic"

In general, people tend to believe that mathematical
expositions carry necessity. However, any belief succumbs
to scepticism. What Awodey and Kishida are pointing out
is that necessity and continuity have a subtle relation
to one another with regard to descriptions. And, since
so many of my remarks talk about definitions and descriptions,
much of what has motivated me seems summarized in their
statement.

I admittedly no longer have the mathematics skills to
whip off a paper relating their results to the naming
of natural numbers in relation to dimension theory. But
if such a paper were written, someone might begin looking
at the theorem I cited in the context of the continuum
hypothesis. That is because the theorem statement meets
the "reasonableness" condition of the game.

WM recently made a remark dismissing topology. His adversary
in the conversation pointed out that with one sentence he
dismissed a significant portion, if not a majority, of
twentieth century mathematics. Although you may not
recognize it, many of the views you express do the same.

>
> You would have to claim the non-second-order-ness AND the
> non-STANDARD-ness of ALL of these models to make the continuum
> hypothesis "true" IN ANY sense, and that's just not happening.
>

I have repeatedly pointed out a restricted second-order
language that coincides with first-order semantics on
discrete topologies. So, your assertion that something
has to be "non-second-order" is simply incorrect. It
is true that I have no development of that language in
this direction.

And, as you are about to point out, if there is a "true"
model, then all of the other models are non-standard.

> Equivalently, you would need (but, the point is, you DON'T HAVE)
> some "vision of THE UNIQUE INTENDED"
> model for set theory (in which case the truth-value of the continuum
> hypothesis would be its truth value in THAT model).
>

Curiously, I just said something like that to Mr. Di Egidio.

mitch

[Peter Percival]

Well, do you?

[George Greene]

On Wednesday, October 12, 2016 at 9:40:06 PM UTC-4, mitch wrote:
> There is a game associated with the continuum hypothesis.
>
> Either you do not understand it, or, you choose to ignore
> it.

FUCK you.
YOU DO NOT teach this class. YOU DO NOT understand this stuff well enough to
be advising me. You certainly do not get to TELL ME what *I* am doing.

> The idea is to formulate reasonable axioms that do not
> assume the actual proposition.

Given that the independence proofs exist, it goes WITHOUT saying that
the canonical/extant/usual axioms do NOT decide the question. It therefore
follows THAT EVERY axiom via which you might choose to extend, is in the following trichotomy: it either leaves the question undecided, decides it true, or decides it false. The problem is of course that it is not always immediately
obvious, for any given extending axiom, which of these is the case.

> So, there is a reasonableness criterion that mathematicians,
> and others, must be asked to consider.

There IS ALWAYS, IN ALL contexts, " a reasonableness criterion ", YOU IDIOT!
Researching the continuum hypothesis has nothing whatSOEVER to do with THAT!
ANYthing you propose CAN ALWAYS be dismissed with "that's not reasonable", if
in fact it is not reasonable. What you MEAN to be doing here is articulating
THE PARTICULAR "reasonableness criterion" that SHOULD be applied TO THIS case.
And that would be simply that the any proposed new axiom NOT clearly or TRIVIALLY decide the question. In particular, CH itself, and its denial, are
not "reasonable" extensions in this context.

On the other hand, however, you are, WRONGLY, talking about "truth".
Obviously IF THERE IS ANY "truth" of the matter then ALL the systems
in which the answer comse out the other way are just WRONG and THEREFORE UNreasonable. If you are going to defend some middle way of leaving the
question undecided then YOU MUST BEGIN BY CONCEDING that all undecided
questions in this context are undecided because MODELS EXIST BOTH WAYS of
deciding them. Your reasonableness criterion IS NOT AT ALL COMPATIBLE WITH
ANY NOTION WHATSOEVER of "truth", NOT EVEN "second-order truth", which you
almost certainly have NO IDEA of "the actual meaning" of!

[mitch]

On 10/15/2016 12:21 PM, George Greene wrote:
> On Wednesday, October 12, 2016 at 9:40:06 PM UTC-4, mitch wrote:

< snip >

>
>> So, there is a reasonableness criterion that mathematicians,
>> and others, must be asked to consider.
>

< snip >

> ANYthing you propose CAN ALWAYS be dismissed with "that's not reasonable"

In case you have not noticed, I have repeatedly
qualified statements with the fact that there
is no defense against scepticism.

Your penchant with ignoring the fact that there
is a body of literature -- a "corpus" in the fancy
pants language with which you were raised -- from
which citations and references are taken does not
go unnoticed. There would be no reason for bibliographies
if your methodology were uniformly implemented in
academia.

There are approximately seven billion people on the
planet. Sceptical philosophy would have that each
individual's belief must be addressed separately. That
is the consequence of asserting that *any* reference
to existing literature constitutes a fallacy of
authority.

Yes. I remember the nature of your flames well.

And, I remember you telling me that you believe what
you do about the continuum hypothesis because you
believe what your teachers believed.

We call that the pan calling the kettle black.
Whatever legitimate criticisms others may have of
me, they are minuscule in comparison to your misuse
of a fine education. And, since it is very clear
that what I am posting is not found in any published
materials, you are both the pan and the kettle.

One way to circumvent what you choose to *not*
believe is to eliminate "not" from the language
of mathematics.

There is, for example, no locus for "unary
negation" in the free Boolean algebra on
two generators:

https://en.wikipedia.org/wiki/File:Hasse2Free.png

Here's one where the expression "logical connectives"
forms part of the name,

https://commons.wikimedia.org/wiki/File:Logical_connectives_Hasse_diagram.svg

But, with the exception of the bottom of that
lattice representation, all of those logical
connectives fit nicely into the geometric relations
describing the cell topology of a tetrahedron,

http://www.iue.tuwien.ac.at/pdf/ib_2009/CP2009_Heinzl_1.pdf

in figure 4 on page 2. Moreover, if I label the
exterior of the tetrahedron with an empty set,
I can extend the diagram at the bottom to be
an order isomorphism with the Boolean algebra.

So, what apparently could never be empirically
demonstrated for the critics of mathematics
(geometric forms), is precisely what they base
their sceptical claims upon. But, it takes an
understanding of order theory and topology to
recognize it.

Laughably, the word "tetrahedron" may also
be applied to planar figures with four points
and six lines. So, you should carefully read
the abstract to Davies paper,

https://www.researchgate.net/publication/231947429_Partitioning_the_plane_into_denumerably_many_sets_without_repeated_distances

What you believe, George, is that because
mathematicians use proofs to organize their
statements, one can simply study logic. The
notion of "logical priority" does not hold
once one understands that the symbols logicians
use fit nicely into a geometric diagram.

Feel free to believe what you want. I am not
a teacher. Every time I read about a mathematician
with "followers" it makes me want to puke. I do
not view mathematics like a religion. That is for
people who want to win debates in divinity school.
And, if that is thinking "wrongly" about truth,
I am all for it.

What you engage in, George, is rhetoric -- not
logic. It is the language of politicians and
priests.

mitch

[Peter Percival]

Your lack of reply must mean that you agree with my proof (which is a
triviality) that if Goldbach's conjecture is undecidable then it is true
in the standard model. It's very gracious of you to admit it.

[khongdo...@gmail.com]

Have you replied, answering my Q2 question in the other post?

[Ross A. Finlayson]

One might quibble "if Goldbach's conjecture is
undecideable then it is true in _a_ standard
model...", for whether the integers do or don't
have a standard model.

This is logic, one is free to quibble, and,
in a sense, quibbles are total (that a "quibble"
is generally inconsequential yet still a
quite technical formal matter).

Rhetoricians aren't necessarily formalists.

[Peter Percival]

Ross A. Finlayson wrote:

> One might quibble "if Goldbach's conjecture is
> undecideable then it is true in _a_ standard
> model...

Implying that there's more than one. Explain yourself.

> ", for whether the integers do or don't
> have a standard model.

[Ross A. Finlayson]

Euh, indicating the indefinite over definite
article here was contingent the existence of
the thing, not the multiplicity.

Yes I know that generally the definitions of
integers as inductive follow, via induction,
that there is a standard model, because
induction itself is fundamental that generally
or rather "often", constructive inference is
solely via that.

Then there's a notion that more generally,
while induction is sound as a method, it's
not necessarily reliable as the method.

This is about things like the "the integers
are finite" and "the integers are infinite".

[George Greene]

On Sunday, October 16, 2016 at 12:51:12 PM UTC-4, Peter Percival wrote:
> Ross A. Finlayson wrote:
>
> > One might quibble "if Goldbach's conjecture is
> > undecideable then it is true in _a_ standard
> > model...
>
> Implying that there's more than one. Explain yourself.

No, please don't.

And please don't encourage stupidity.

[George Greene]

On Wednesday, October 12, 2016 at 9:40:06 PM UTC-4, mitch wrote:

> WM recently made a remark dismissing topology. His adversary
> in the conversation pointed out that with one sentence he
> dismissed a significant portion, if not a majority, of
> twentieth century mathematics. Although you may not
> recognize it, many of the views you express do the same.

They DO NOT! Why are you such a LYING sack of ad hominem SHIT?!?

[George Greene]

On Wednesday, October 12, 2016 at 9:40:06 PM UTC-4, mitch wrote:

> I have repeatedly pointed out a restricted second-order
> language that coincides with first-order semantics on
> discrete topologies. So, your assertion that something
> has to be "non-second-order" is simply incorrect.

You are completely full of shit as usual.
OBVIOUSLY, SOMEthing, namely, FIRST-ORDER LOGIC, has to be non-second-order.
More to the point, since you youreslf are calling it "a restricted second-order language", the very thing YOU ARE USING ALSO *HAS* to be NON-second order --
that's what RESTRICTED *MEANS*!!!

[George Greene]

On Saturday, October 15, 2016 at 6:08:26 PM UTC-4, mitch wrote:
> And, I remember you telling me that you believe what
> you do about the continuum hypothesis because you
> believe what your teachers believed.

No, you don't. Belief isn't even relevant here.
It is a fact, not a belief, that CH is independent of the usual axiomns.

[mitch]

When you flamed me in or around 2003, you said
that there are no issues with the notion of
identity in mathematics.

And, separately, you often assert that identity
is eliminable.

With regard to the second matter, the axiom of
extension in both "Set Theory" by Jech and "Set
Theory" by Kunen the axiom of extension is not
written with a biconditional. Because of Skolem's
criticisms of Zermelo, the standard account of
set theory defers to first-order predicate
logic in such a way that identity is not eliminable.

From a purely logical view, interpreting the
sign of equality as substitutivity is not
problematic -- provided the issue of warrant
is taken into account. The warranting of such
uses had been Leibniz' identity of indiscernibles.
But, philosophers have rejected that principle
in the standard account of identity.

Whether or not you like it, topology is ubiquitous
in mathematics. And, the paradigmatic topology is
the metric space over the reals. If you look at
axiom 2 in the link,

https://en.wikipedia.org/wiki/Metric_space#Definition

You will see that it is labeled as "identity of
indiscernibles".

If you follow the link, you will arrive at

https://en.wikipedia.org/wiki/Identity_of_indiscernibles#Identity_and_indiscernibility

Note that I am not going to dispute what is
and what is not a "logical truth".

If you read about the contradiction that arises
by excluding all four of the predicates then
it is clear that there is an asymmetry in spite
of Mr. Black's natural language argument. This
asymmetry means that the notion of "two non-identical
objects" cannot be described by the symmetric
discrete topology. Rather, it must be an
asymmetric Sierpinski space,

https://proofwiki.org/wiki/Definition:Sierpi%C5%84ski_Space

https://proofwiki.org/wiki/Category:Sierpi%C5%84ski_Space

For what this is worth, Paul Taylor is studying
the foundations of mathematics starting from
work on Sierpinski spaces,

http://www.paultaylor.eu/

And, of course, the diversity relation I defined
is specifically formulated in relation to a
base point from among the pair of arguments
to the relation.

The study of metric topology led to general
topological notions that separate points on
the basis of other criteria. But, all topologies
sit in relation to the trivial topology and the
discrete topology.

Your continual insistence that mathematics is
somehow reduced to first-order logic ignores
the role of topology in general mathematics.

But, that is not the point I wish to make.

Because of a question by Mr. Percival, I am
reading "Sets in Mathematics" by Lawvere and
Rosebrugh. In section 1.7 of that book, the
authors introduce the definition for what they
call a "coseparator". The context in which
this definition is given is the explanation
for the category of constant sets. And, they
are specifically discussing how a representative
set with two elements is a coseparator for
that category.

They write,

< begin quote >

"The coseparating property of 2 is phrased in
such a way that it is really just a dual of
the separating property that 1 has, i.e. there
are enough elements to discriminate properties
just as there are enough properties to
discriminate elements."

< end quote >

Do I need to quote the passage from Frege
corresponding to this?

The claim that there are enough properties
to discriminate elements is precisely the
identity of indiscernibles which is not
thought to be a "logical truth".

The copyright date on "Sets in Mathematics"
is 2003. It arises out of the algebraic
account of mathematics promoted by Skolem
when he criticized Zermelo.

In 2003 you said there are no issues with
respect to identity in mathematics. But,
it would seem that an old argument -- or,
an argument that was never properly explained
to the satisfaction of all involved -- is
reappearing simply because it is in the
guise of "a new foundational language".

Personally, I do not care about what is
right or what is wrong here. Nor do I
care what you believe. Topology is a
reality in the general curriculum of all
respectable mathematics departments. Your
statements are as dismissive of this fact
as are WM's.

When I began posting this year, I had been
careful to say that what is taught as
mathematics in logic and philosophy is
legitimately called mathematics even if
it differs from what is taught in mathematics
departments. I did this out of respect for
the fact that we all learn from teachers
and that we do not enter into education
as a student distrusting them. We should,
perhaps. But, that is a different
question.

In particular, however, I took this position
out of respect for you. You are extremely
intelligent, and, the only way that I could
reconcile many of your remarks with what I
know to be in the literature was by
acknowledging that "mathematics" is being
taught differently by people who often
keep their distance from one another.

It is unfortunate that you will probably
be unable to understand that.

mitch

[mitch]

But, that had not been your statement.

You specifically said that you did not
believe the continuum hypothesis to
be true because the instructors you trusted
did not believe it to be true.

You are a little fallacy mongerer.

mitch

[mitch]

One can admire your fascination with syntax.

However, what makes a particular syntax
to be of interest to a human being is the
semantics that motivates its formulation.

My statement is made because the semantics
for these languages coincide in certain
circumstances directly relevant to Zermelo's
domain description.

You are undoubtedly correct that the syntax
of a second-order language will always
differ from the syntax of a first-order
language.

You can be proud of how correct you have
been with a trivial classification.

mitch

[mitch]

On 10/12/2016 02:54 PM, George Greene wrote:

> On Friday, October 7, 2016 at 12:27:38 PM UTC-4, mitch wrote:
>> If, in fact, the continuum hypothesis is a
>> second-order truth that cannot be proven,
>
> Given that it cannot be proven (from the usual axioms),
> it is NOT possible that the continuum hypothesis BE ANY
> kind of "truth", 2nd-order OR OTHERwise, since it follows
> from its unprovability that there exist models of the axioms
> IN WHICH IT IS *FALSE*.
>

Have you ever actually read the generic model
theorem?

There may now be other methods for generating
outer models of set theory. It has been some
time since I have focused on the literature, but
when I first read Jech, it had been implied that
all outer models essentially depended on this
theorem.

First of all, there is an *assumption* of
partiality. It is not possible to apply forcing
to the universe.

The reason it is not possible to apply forcing
to the universe is because generic sets
*converge* to a point outside of the ground
model. There is no "outside" of the universe.

Did you see the word "converge"?

The generic model theorem depends upon a
generalization of Dedekind cuts to describe
a *topology* based on regular open sets.

Did you see the word "topology"?

In his "Discourse on Metaphysics" Leibniz
explained for all of posterity that systems
of partial information are necessarily
modal.

So, unless you devise axioms which describe
a system to which the generic model theorem
cannot be applied, you will always have
models whose truths may not hold in the
actual universe.

You are on record as distrusting model
theory. And, you give little credence
to topology. Yet, you expect me to take
your "exist models" that is based on those
very subjects as somehow meaningful in
this discussion.

mitch

[Ross A. Finlayson]

But, I am not a fallacy monger.

If you prefer your conversations personal,
please keep them to yourself.

Your maintenance of regard and reserve
serves both your readers and yourself.

Good looking, out.

[Ross A. Finlayson]

About the first-order and higher-order
in the first order, the opinion of some
(or just my opinion) is that the higher
order is but a convenience for book-
keeping and notation, representing of
course yet only the "concrete numerical
resources" of the first order, then that
expressitivity can be increased with
organized in these higher orders, but
that again eventually in some theory
with truly primitive and pure primary
objects, it's all first-order again.

It's almost a joke that this isn't a
part of the curriculum, so people
don't know it. It's almost a joke,
but not very funny. OK, it's not
entirely without humor and the
sardonic and laconic. OK, by now
this is rather droll. Some people
are so stupid you can't beat sense
into them, so invincibly ignorant
you can't beat sense out of them.


That said then, and it is merely
an aside because there are goals
to establish both the heights and
the depths and here for these foundations
the very most extreme, as simple,
plain, primitive, then pure.

So, you'll certainly find people
willing to point out that there's
a justification of warrant for
various modes of these equivalences
for various modes of these quantifications
for various modes of these inferences,
for that what we call "reason".

That is to say, a conscientious
logician will appreciate what it is,
and, all the question words about it,
in the context then of the canon,
which is relevant. This is where
the established canon is the Western
philosophical literature, and
correspondingly the development
through modern computer science
and formal methods in computing.

[George Greene]

On Wednesday, October 19, 2016 at 9:59:03 PM UTC-4, mitch wrote:
> But, that had not been your statement.
>
> You specifically said that you did not
> believe the continuum hypothesis to
> be true because the instructors you trusted
> did not believe it to be true.
>
> You are a little fallacy mongerer.

You're a lying sack of shit.
QUOTE ME OR SHUT THE FUCK UP.

NObody believes the continuum hypothesis to be "true" BECAUSE THERE ARE MODELS WHERE IT COMES OUT FALSE.
I certainly never accused ANYone (except you, you being a lying dumbass)
of believing anything else. Belief IS NOT RELEVANT IN ANY case UNTIL AFTER
you privilege SOME STANDARD INTENDED model of the theory! Truth IN THAT
model can be reduced to "truth simpliciter".
The rest of the time, the notion is just irrelevant, and I certainly didn't blame that on any instructors.
I certainly would never tar anyone else with the brush of MY opinions.

[George Greene]

On Wednesday, October 19, 2016 at 10:17:40 PM UTC-4, mitch wrote:
> The reason it is not possible to apply forcing
> to the universe is because generic sets
> *converge* to a point outside of the ground
> model. There is no "outside" of the universe.

Of course there is. The universe, if it exists, if it is a thing,
IS OUTSIDE the universe. IT CAN'T BE IN the universe because this is a unvierse
of things that ARE NOT SELF-MEMBERED.
If you want to have a universe then it HAS to be self-membered and you HAVE
to toss the axiom of foundation FOR STARTERS. NOBODY IS TALKING about doing that.

[George Greene]

On Wednesday, October 19, 2016 at 10:17:40 PM UTC-4, mitch wrote:

> So, unless you devise axioms which describe
> a system to which the generic model theorem
> cannot be applied, you will always have
> models whose truths may not hold in the
> actual universe.

That's MY POINT, NOT YOURS, YOU IDIOT!!
More to the point, YOU HAVE NO CONCEPT OF ANY "actual universe"
because WE STILL HAVE AN AXIOM OF FOUNDATION which means the universe
IS NOT IN itself!
The best you can have, AS OPPOSED to "an actual universe", is a proper class of
all sets, and there's no obvious reason why one couldn't treat a hyper-class of
THAT class (other than that the axioms don't happen to).

You don't see actual mathematicians talking about "the actual universe".
It's models ALL THE WAY DOWN.

[Ross A. Finlayson]

There are a wide variety of considerations
of extra-ordinary set theories where well-
foundedness is ordinary and the objects
of the theory are not.

So, that's not so much about "tossing the
axiom of foundation" as that it's a
non-logical axiom and a restriction of
comprehension that other more inclusive
theories don't have.

You just said "universe" five times.

[Ross A. Finlayson]

From a model, that would be bounded fragments
all the way down, and, models all the way up.

[George Greene]

On Wednesday, October 19, 2016 at 10:04:25 PM UTC-4, mitch wrote:
> One can admire your fascination with syntax.

It's not MY fascination.
GODEL proved the completeness theorem as HIS dissertation.
It's not a question of "fascination". THE POINT of the completeness
theorem is that for the specific case of vanilla standard classical first-
order logic, IT DOES NOT MATTER whether you are talking syntax as opposed to semantics. The allegedly separate notions of logical consequence in the two allegedly separate arenas TURN OUT TO BE EQUIVALENT. In particular, all logical
consequences are consequent UNDER ALL POSSIBLE semantics, so semantics
does NOT actually MATTER in THAT context.
This is NOT MY RESULT and NOT MY fascination!

> However, what makes a particular syntax
> to be of interest to a human being is the
> semantics that motivates its formulation.

Since you just deined my humanity, fuck you.

[George Greene]

On Wednesday, October 19, 2016 at 10:04:25 PM UTC-4, mitch wrote:

> My statement is made because the semantics
> for these languages coincide in certain
> circumstances directly relevant to Zermelo's
> domain description.

OF COURSE THEY DO because SET THEORY IS A FOUNDATION OF MATHEMATICS, YOU IDIOT.
YOU CAN USUALLY, for ANY non-set theory, MODEL IT WITHIN THE CLASS OF ALL SETS.

At the extreme you can invoke the downward Lowenheim-Skolem theorem and model it (if it is a usual -- i.e. over a denumerable signature with finite formula-
lengths -- theory) in just plain old N, but of course that requires a lot of
unnatural encoding that people would rather not be bothered with.

But doing it with sets is just NORMAL. Not translating it into ZFC but just assuming we can have collections of things, and only caring about set-
theoretic aspects when this foundation/embedding winds up brushing against
indepenence-related/undecidable set-theoretical questions.

[George Greene]

On Monday, October 17, 2016 at 10:23:13 PM UTC-4, mitch wrote:
> Whether or not you like it, topology is ubiquitous
> in mathematics.

Whether you like it or not, "ubiquitous" means occurring everywhere, and
if there And, the paradigmatic topology is not occurring, then you're just
a lying sack of shit. And there is, so you are.

And what makes you a lying sack of shit is not that some branches of math
happen not to give a shit about topology, BUT RATHER, that YOU CHOSE TO BEGIN
A SENTENCE WITH "Whether you like it or not".

That just has no place here.

> If you look at
> axiom 2 in the link,
>
> https://en.wikipedia.org/wiki/Metric_space#Definition
>
> You will see that it is labeled as "identity of
> indiscernibles".
>
> If you follow the link, you will arrive at
>
> https://en.wikipedia.org/wiki/Identity_of_indiscernibles#Identity_and_indiscernibility

You have NO POINT. WE ALL KNOW what "identity of indiscernibles" means.
We are not going to follow any links. The extremely relevant point that you
SEEM not to be grasping is that the axio of extensionality in set theory
IS identity of indiscernibles. The language only has one predicate.
If two allegedly different terms always yield the same truth-values in all
contexts then the can safely be identified. The axiom of extensionality
says that if they are indiscernible from the bottom up then we will necessarily
insist that they are also indiscernible from the top down. After this
has happened, they are in fact totally/completely indiscernible.
If you are in first-order logic WITH equality then you need an axion
to match this set of affairs defined purely in terms of the language's
only non-logical predicate (membership) with logical/built-in equality.
BUT IF YOU ARE NOT in first-order logic with equality (but are instead
in first-order logic WITHUOT equality) then YOU CAN JUST DEFINE = as an
abbreviating symbol MEANING "having the same sets as members". It will then
follow from the axiom of extensionality that = ALSO means "being members of the same sets" and that things that are = therefore ARE indiscernible.

Obviously I agree with you that there is no point in discussing whether
identity of indiscernbles is or isn't "a logical truth" -- non-logicians
trying to talk about such things are comically entertaining like babies.

[George Greene]

> On 10/17/2016 04:55 PM, George Greene wrote:
> > Belief isn't even relevant here.
> > It is a fact, not a belief, that CH is independent of the usual axiomns.
> >

On Wednesday, October 19, 2016 at 9:59:03 PM UTC-4, mitch wrote:

> But, that had not been your statement.

It does NOT MATTER what I OR ANYone EVER *SAID*!!
It's A *FACT*!! It AND its brute factuality are BOTH COMPLETELY INDEPENDENT
of ANYTHING ANYBODY EVER said!! The fact that you think that what
somebody said EVEN MATTERS in this context IS OUTRAGEOUS!!

> You specifically said that you did not
> believe the continuum hypothesis to
> be true because the instructors you trusted
> did not believe it to be true.

Quote me or shut THE FUCK up.

[Ross A. Finlayson]

Of course, maybe you automatically assigned
it that as having the self-membership already,
that the model would be an extension model of
itself already.

Though, one would expect that to be made clear.

It's turtles: it's turtles all the way down.

The universe contains itself.
The nothing contains: itself.

[Julio Di Egidio]

I am, for reasons of closure. In fact, on the same line, I'd question FOL
already. OTOH, I disagree that everything boils down to first-order (and
that semantics can be dispensed of): e.g. isn't set construction syntax
the implementation of higher-order operations?

Julio

[Ross A. Finlayson]

About that "everything boils
down to the first-order", the
very quantifications and the
establishment of distinct variables
from what is the quantifiable
and variable has that most of
the applivation is in the higher
order that it is eventually
even granular and fine. Still,
this is all first-order.

[mitch]

You're right.

Thanks.

mitch

[mitch]

... and missed the point of writing axioms
intended to introduce a self-membered
universe as the only denotation that is
self-membered.

I guess he has not looked in the other
thread.

mitch

[George Greene]

On Wednesday, October 19, 2016 at 10:17:40 PM UTC-4, mitch wrote:

> You are on record as distrusting model
> theory. And, you give little credence
> to topology. Yet, you expect me to take
> your "exist models" that is based on those
> very subjects as somehow meaningful in
> this discussion.

It is wholly meaningful to the independence results.
As for "this discussion", I didn't know you had limited the topics of
"this discussion". You thought whether I was or wasn't a fallacy mongerer
was relevant to this discussion. You thought whether I made an "actual statement" about professors who had taught me was relevant to "this discussion".
So don't expect me to accept your clarims that you know what "this discussion" is ABOUT. You have no sense of boundary. This discussion is about whatever pops onto your irritated plate. Coherence is not acheivable.

[Ross A. Finlayson]

But - the pejorative speculative
always reflects on the caller....

There are only thirteen jokes
in the world, five of those are
too dirty to tell, and we've all
known them all since third grade.

[George Greene]

On Thursday, October 20, 2016 at 12:17:15 PM UTC-4, Ross A. Finlayson wrote:
> About that "everything boils
> down to the first-order", the
> very quantifications and the
> establishment of distinct variables
> from what is the quantifiable
> and variable has that most of
> the applivation is in the higher
> order that it is eventually
> even granular and fine.

The higher-order stuff is almost always MUCH more mathematically relevant
and important (the reals are higher-order -- there is a least-upper-bound
axiom or somehing similar) BUT THE WHOLE POINT about SET theory is that you GET ALL orders at once BECAUSE YOU HAVE A POWERSET AXIOM. The whole point about FIRST-order set theory is that first-order consequence is at least SEMI-decidable (it is recursively enumerable, even if not totally recursive), so that you can USE SOME LOGIC to HELP you find things.

Second-order consequence in general is completely intracable so first-order set theory is what you use TO APPROXIMATE higher-order reasoning in a way that HAS some tractability.

[Ross A. Finlayson]

It's like they used to say
"it's all zeros and ones",
that's first-order, and far,
far above that is a usual
language. Still: it's
all zeros and ones.

[mitch]

Because of Mr. Finlayson's observation,
I put you in a kill file. I thought it
best at the time. But, I have since
decided to give you the same consideration
I had given to WM several years ago when
language became personally insulting.

I am sorry for the remarks I made.

If you care to repost responses to which
I did not reply, feel free to do so.

Should it interest you, you will find axioms

intended to introduce a self-membered

universe in the thread "mereopredicative
set theory". In order to write the system
with a non-eliminable identity (enforced
by syntax) I formulated the schemes in
the thread "semantically warranted identity".

From a post made by Virgil, I see you have
found the latter.

mitch

[George Greene]

On Saturday, October 22, 2016 at 11:54:59 AM UTC-4, mitch wrote:
> Should it interest you, you will find axioms
> intended to introduce a self-membered
> universe in the thread "mereopredicative
> set theory". In order to write the system
> with a non-eliminable identity (enforced
> by syntax) I formulated the schemes in
> the thread "semantically warranted identity".

I did not know that there was a prior relationship between those threads and this one because I wasn't reading those.
So obviously I am going to look stupid jumping into the middle of the discussion if I didn't read the preparatory framework. But this paragraph here is very much "better late than never" -- you could have STARTED with "I'm not using regular ZFC, I'm using something else; it's what's formulated in the mereopredicateive thread" -- instead of "when you flamed me in 2003, YOU said", without EVEN QUOTING what I said!!

It really shouldn't be this hard to confine ourselves to talking about the math and not about each other.

[George Greene]

On Thursday, October 20, 2016 at 8:58:10 PM UTC-4, mitch wrote:
> ... and missed the point of writing axioms
> intended to introduce a self-membered
> universe as the only denotation that is
> self-membered.
>
> I guess he has not looked in the other
> thread.

EXACTLY.

But you could have invited me to look at the other thread instead of personally insulting me for something from 13 years ago. GOOD GRIEF!!

[Ross A. Finlayson]

Mr. A., you'll be happy to remain above that
pettiness, that poor feeling and ill will can
so dissipate, with the general notion that
there are experts in logic here and that we're
interested in modern foundations and the outlook
for future foundations.

This is logic and there's a place for strength
in rhetoric as plain strength in logic. Also,
gentlemanly behavior (and not to be exclusive
but that it embodies comportment conducive to
both expression and restraint) will, over time,
see that your corpus and samples thereof reflect
the body of your work instead of your deftness
at or susceptibility to insult.

That said and you can excuse me and all f' off if
you don't care, logic is part of the foundations
for mathematics and sciences and as well rhetoric
(and other sports), logic is part of the foundation
for reason as an extension of the technical
philosophy for reason and as of the roots of reason.

Then, we have both a canon and modern dogma (and,
dogma is always modern, and, it always changes)
that then the ultimate goal for researchers may
include both understanding of the foundations,
then extension for the extra and new applications.

Indeed, then, mathematics as king or queen of
sciences is yet only its servant, here, then,
mathematics _owes_ modern physics a reason to be,
not just support of the empiricist (which is the
only support of the empiricist) but the extensions
for completion as we know there is a place in
physics for logic about objects of mathematics
that modern dogma doesn't yet explain.

Modern physics _needs_ real infinities from
modern mathematics, that it doesn't yet have.

Then, as that basically reflects continuum
mechanics which is a common interest, some
here (or I do) promote a poly-dimensional
perspective that above definitions for
continuity as "line continuity" and "signal
continuity" that so precede and follow
"field continuity", that the conscientious
logician should find these and soar.

Because, that is beyond modern dogma already.

[George Greene]

On Sunday, October 9, 2016 at 3:00:50 AM UTC-4, mitch wrote:
> While there may be some syntactic notions of forcing
> in the literature, the generic forcing theorem in
> Jech uses an analogue to Dedekind's completion in
> order to form a complete Boolean algebra. The topology
> obtained in this way is based upon regular open sets.
> And, in his book "Simplified Independence Proofs",
> Rosser begins with a statement of topological axioms
> to be used. Clearly, while I have not seen the
> expression "topological metamathematics" in the
> literature, I have not invented it.

Well I certainly do not claim any relevant prior expertise in metamathematics.
If a topological approach is already occurring in forcing and you have come across other prior uses of one as well, then I am certainly not going to object.

My only prior acquaintance with metamathematics was the received/asserted
relevance of Primitive Recursive Arithmetic to Proof Theory.
That allegedly arose out of the stressed importance of making sure that
this metamathematics was "finitist", since formulas and proofs both HAVE to be finite.

[mitch]

The problem is that we have had such very different
views of "math" on the basis of our educational
backgrounds. I now understand your statements better
than before.

You did not look stupid at all. To the contrary,
you immediately recognized that what I had been
talking about would require the universe to be an
element of itself.

If you really want to know what I am trying to represent,
you can look in sections 2.3 and 2.5 of the paper,

http://www.filozof.uni.lodz.pl/bulletin/pdf/40_34_3.pdf

In section 2.3 a system characterized by closure
operations without complementations is described.
Back in the 1980's I recognized that the transitive
closure operation in ZFC had this form.

In section 2.5 the axiom W1 talks about a well-connected
closure algebra without qualifications. To achieve it,
intersection over the empty set must have a target. That
target, of course, is the universe of discourse.

My idea, in general, is that the only exception to
the axiom of foundation will be the universe. And,
since it will not be admissible as a Quine atom, what
is obtained through the power set for the cumulative
hierarchy will always be partial.

mitch

[Ross A. Finlayson]

About ZF there are various other "proper classes"
that are defined by their elements but can't be
well-founded sets. Would you assert that they
are your one exception?

Is that expansion of comprehension,
or restriction again?

What I call the "group noun game" is about
non-set classes as typing sets, or for example
Russell's types ramified and stratified then
within sets. There's an aspect of losing
because it just off-puts the resolution of
the cognition that led there needing to be
a "different" group noun. Here it's similar
with "admitting" the universe (that to a
platonist might already exist already,
for basically that there is no restriction
of comprehension but only definition),
for admitting the universe then there are
other proper classes as so result from
quantification (or the tools of the
language of set theory).

[mitch]

In spite of my anger about it, I am indebted to you for that
flame in 2003. I have had to do an immense amount of study
to place common statements into contexts that they do not
have when made in textbooks.

Hilbert's "Foundations of Mathematics" is just now being
translated into English. The chapter in which I am most
interested has not yet been done. But, in the opening chapter,
he concedes that no completed infinity can be demonstrated.
This leads him, in turn, to pursue "metamathematics".

I surmise that the reason "metamathematics" is arithmetical
is because he rewrote Euclidean geometry in a way that
would motivate one to focus on arithmetic. In addition,
"arithmetization" had been a response to "the crisis
in geometry".

By contrast, it took time for Cohen's proof to be
reformulated to a general form involving topological
elements. I doubt anyone even thought to classify this
as a different sense of metamathematics as it developed.
And, I am sure that there are those who would challenge
my use. All I do know is that I needed some way to
account for how I have been applying mathematical ideas
to logical forms. In his book "Model Theory", Wilfrid
Hodges acknowledges many connections with topology, but
purposely avoids discussion of any depth. So, I eventually
concluded that "metamathematics" could be divided into
arithmetical studies and topological studies.

It probably never occurred to you that my insistence on
"defined language elements" had been precisely because I
question the liberality with which languages are expanded
with constants in model theory. Mathematics within the
context of a proof is different from mathematics outside
of a proof. So, under the guise of how I had been taught
that "set theory is the foundations of mathematics" I
sought to reformulate set theory so that its language
elements could be introduced with well-formed formulas.

Of course, I had been ignorant of what was being taught
about "undefined language primitives". Or, rather, I did
not understand how controversial such an approach might be.
I certainly knew about it. It had been one of the
two things I thought might restrict the theory to models
in which questions like the continuum hypothesis might be
decided.

mitch

[mitch]

Which proper class is in ZF?

Technically, a class is a grammatical form unless
it has a representation as an element in the theory.
There cannot be any singular terms in ZF that are
intended to denote proper classes.

It would be correct to say that I began these
investigations with a notion of "class" having
nothing to do with "falling under a concept". My
original sentences will not admit urelements because
they are formulated so that "classes" are the basis
of "topological separations" and that "classes"
are denoted in the theory because they are an element
of the universe. The system of "objects" must be
able to account for how the objects are a "plural"
within the theory itself.

What I had observed is that to say

"Plato is a philosopher"

may be represented as a "dot" in a rectangular region
denoted as "philosophers" and representing the universe
of discourse.

But, were one to say

"A is a class"

then it may be represented with a "dot" and a "circle"
in a rectangular region denoted as "classes" and
representing the universe of discourse.

So, a naive notion of "logic" applied to a typical
"object" and a typical "universe" yields a "dot" in
the universe. But, for a class, one obtains a "dot"
and a "circle".

This reminded me very much of "wave-particle duality.
I have been trying to understand set theory as a
self-contained system ever since.

At the heart of what I am doing is a second-order notion
of identity where one has

AxAy( x = y <-> Az( x in z <-> y in z ) )

So, describing a system in which extensionality
and topological inseparability coincide is
the objective.

Now, no "proper class" introduced by extensions
can be "in" anything. But, every proper class
can be "not in" everything. The sense by
which this constitutes an "individual" is
called an adjunction space.

If, however, one specifically introduces a
universe as being an element of itself, then
it is different from any extensionally-defined
proper class that satisfies the usual axiom
of foundation. So, they may be ignored just
as they are in the standard theory.

mitch

[Ross A. Finlayson]

But, these usual proper (non-set) classes
in ZFC with classes like Ord (the order type
of all ordinals, where the ordinals are
embedded in the sets and here extra-ordinary)
or V or L the universe can't be ignored when
you introduce their setting.

In an axiomless system of natural deduction,
one might deduce ubiquitous ordinals of a
sort then that pure sets as combinatoric
constructs so follow, or, alternatively:
that various sets of the same order type
are indistinguishable as ordinals. (Here
the powerset result becomes the successor
result instead of uncountability.)

This duality as the particle/wave duality,
as about physics' dimensions with time-like
(just the one), space-like (just the three)
or light-like (just the rest) dimensions
(or dimensional vector bases with a usual
orthonormal vector basis as the establishment
of dimension), this duality is about the
effect(s) or "regime" of transfer in terms
of conserved quantities under transformation.


Then, for the point and total for the local
and global in effects, here the void and
universal has that introducing the universe
(in a, measured, way, as it were) into your
theory that otherwise still starts with null
and proceeds via induction, has that then
you have a cap or model or Cohen's M as for
the establishment of usual model-theoretic
concerns inside the "set" theory, along the
lines of the Univalency of Homotopy Type Theory
and other would-be "accommodations" of these
dualities, but, then it's a stub and a model
and a toy instead of a tool. Excuse me if
that's too blunt because there are various
expressions that are much more concise and
still correct within some "guardrails" in
the language, for concise index and reference
to assertion, the "accommodation" therefore
a gainer in utility, in the language, but
the eventual deduction so follows that its
partial introduction suffers the same problems
as the vacuum of its lack.

[Ross A. Finlayson]

I'm somewhat a zealot for
this "One True Theory" bit -
and happily.

[mitch]

If contradiction is to be interpreted
as non-existence, what do you mean by
"Ord"?

Many results in set theory depend upon
the fact that one ordinal is contained
within another ordinal. This is done
specifically because one cannot simply
refer to Ord.

Moreover, the universe is not really
"known". I have not developed my work
beyond trying to connect the relations
together without inviting contradiction.
Many discussions of set-theoretic "things"
may only "exist" because the universe is
not "known". As I have stated before,
the use of inductive axioms may limit
interpretation to Cohen's minimal
model. What, then, of everything else?

I am motivated by a certain aesthetic
that "all" and "the universe" actually
correspond to a denotation. Under
interpretation, it might still be vague
and have many possibilities.

One reason my "logic" is non-standard
is because analysis of "the universe"
leads directly back to Russell's theory
of definite descriptions (I noted that
Russell, Tarski, and Lambert all have
influenced how I wrote the logical
schema. Lambert's work in free logic
may be the only paradigm that models definite
descriptions.)

Before one judges between tool and toy,
one must have something that is not contradictory.

Besides, no one is going to worry about something
written by a guy who swings a sledgehammer. If, as
a corollary, I solved world hunger, no one would
notice. Meanwhile, have some fun with it. It is
nothing but a syntactic jigsaw puzzle.

mitch

[mitch]

I understand.

Hopefully, I will read enough of "Sets in
Mathematics" to give Mr. Percival a little
sense of how it could be a "foundation" in
relation to what I wrote. Neither of us
is actually convinced of an algebraic
approach based upon presentations of universal
algebra. Similarly, the HOTT book wants to
speak of "homotopy" as some sort of primitive.
But, it is really understood through classical
algebraic topology. Trying to sort out how
advanced language can be understood as primitive
language is difficult.

Its like trying to figure out what church
to attend each Sunday. Do I stick with the
tried and true (fire and brimstone)? Or, do
I run across the street to success theology?

Maybe I just muddle along on the double yellow
in the middle of the street and hope I do not
get run over.

In any case, the term "multi-foundational" (John
Baez) seems like an oxymoron to me.

mitch

[Ross A. Finlayson]

About Ord (or ORD),
the ordinal of all
ordinals, (and for
example Grp the group
of all algebraic groups),
von Neumann illustrates
that there are many
constructions of sets
that model ordinals or
ordinal arithmetic including
as about the initial ordinals
of cardinals (this is where
the cardinal is not its
initial ordinal). There
are established ordinals
as throughout the transfinite,
that the course-of-passage
through ordinals as trans-
finite induction is the course
of trans-finite induction, as
defined for non-limit and limit
(lately, "fixed point") ordinals.

Cesar Burali-Forti observed that

the order type of all ordinals,

where the order type of an ordinal
is its successor, and of a class of
non-limit ordinals is its limit ordinal,
that the order type of all ordinals
or Ord is extra-ordinary. Then, in
theories of "ZF with Classes", Ord
is a proper class, as the collection
of the ordinals is structurally the
ordinal, as for example (and usually)
von Neumann construction (and there
are many others).


Then, about "ubiquitous ordinals",
what's key in that point is that
powerset illustrates two flip sides:
uncountability and successorship,
that the course of induction (of the
exhaustion of combinatoric possibility
as of the powerset) doesn't complete,
or that it does.

This is simple and should be clear.


Again toward a notion of an "Integer
Continuum Hypothesis", that's it's
independent of the usual axioms whether
the integers are variously standard,
nonstandard countable, or nonstandard
uncountable, it is as well with ordinals
through those: and as well whether
they are complete or incomplete and
whether they are ordinary or extra-ordinary.

[Ross A. Finlayson]

There's a usual notion that "A" "Theory of
Everything" would be "The", that it either
exists or not, then in the singular (or,
"One True Theory", "Sole Foundation", or
as I would see it "Free Foundation" as
the "Null Axiom Theory", a Platonist's
logically scientific foundation for reason).

I think those words mean
what I think they mean.
Also, it's what they say.
I think those words mean
what they say.

The ur-element or primary
element or purely logical
primitive, _is_ the constant
(of terms and definition) _and_
the variable (in course and
carriage). This is usually
Kant's Ding-an-Sich (and also
the noumenon, further conflating
terms) to the philosophy department.

Or, rather, I hope it is.

That said I certainly believe it.

[mitch]

On 10/22/2016 09:12 PM, Ross A. Finlayson wrote:
>
>
> Then, about "ubiquitous ordinals",
> what's key in that point is that
> powerset illustrates two flip sides:
> uncountability and successorship,
> that the course of induction (of the
> exhaustion of combinatoric possibility
> as of the powerset) doesn't complete,
> or that it does.
>
> This is simple and should be clear.
>

The power set of a given set is the
set of all of its subsets.

To be a subset, an individual must first
be a set.

So, every subset of a set is necessarily
a set.

To be a set, an individual must be an
element of the universe.

The sets of the cumulative hierarchy
do not have complements. So, a power
set need not be a system of complements.

Holding as close as possible to the usual
formulation of set theory, the only sets
that can be in the power set of the universe
are the sets of the cumulative hierarchy.

This cannot be because the axiom which
yields the cumulative hierarchy also
provides for well foundedness. And, the
well foundedness of sets provides for the
well foundedness of classes which are
not sets. So, Russell's paradox forces
the well-founded system to be partial with
respect to the power set.

Add a denotation for the universe by
describing it as an element of itself.

Russell's paradox no longer restricts
the definition of the power set.

Being an element of the universe, the
universe is a set. Every set of the universe
is a subset of the universe. Every subset of
the universe is a set of the universe. The
universe is its own powerset.

There is nothing outside the universe.
So, the power set operation "completes".

But, other axioms must be adjusted.
You do not want, for example, the
universe to be a Quine atom.

The first such adjustment must be made
to the axiom of foundation. A single
exception to well foundedness must
be supported.

But, before that one must be able
to qualify statements with respect
to "x in x"...

... without contradiction.

mitch

[George Greene]

On Saturday, October 22, 2016 at 2:28:01 PM UTC-4, mitch wrote:
> It probably never occurred to you that

STOP!!

GOOD GRIEF!!!

STOP!!!!

Subjective value judgments about what is going on in somebody ELSE'S mind are
ALWAYS inappropriate if the other person IS RIGHT THERE!!


Any sentence you start that way is not likely to end well.
I know what occurred to me (I am THE ONLY POSSIBLE authority on that subject) and I am only going to flame you for getting it wrong.

> my insistence on
> "defined language elements" had been precisely because I
> question the liberality with which languages are expanded
> with constants in model theory.

DUH.
I completely SHARED that FROM DAY ONE.
We have sort of a core difference in temperament in that it never motivated
me to do any research; I could just go "that's obviously bullshit" and just
NOT ENGAGE with that thread. The reason you don't publicly assert that is
that in this room (sci.logic) and under this paradigm, there is ONLY ONE
warrant for waxing PUBLICLY dismissive OF ANYthing and that is inconsistency.
Until and unless you can prove that, you just endure it.

> Mathematics within the
> context of a proof is different from mathematics outside
> of a proof.

Again, room.
Again, paradigm.
The dogma will be that it's NOT math UNTIL it's "proof".
Obviously people investigate conjectures that they haven't proved YET so
a LOT of math is "immature" in this sense; the proofs come "last".
But they are despite that what it's all ABOUT. The analogy that leaps unbidden
to mind is pro-life/pro-choice "is a fetus a person? and if, so, when?"
But the point is you are ALWAYS in the ARENA of proof. You ought always to
know or care WHICH axioms you've insisted upon SO FAR.
In other words, even in the context where the proof hasn't been achieved yet,
or the context where the proof will (provably!) NEVER be achieved (i.e.
independence results), you ARE ALWAYS in THE CONTEXT OF *proof*, if it's math.

> So, under the guise of how I had been taught
> that "set theory is the foundations of mathematics" I
> sought to reformulate set theory so that its language
> elements could be introduced with well-formed formulas.

But PLEASE! I'm not trying to hurt your feelings, but this is just
a foolish aspiration! ZERMELO ALREADY DID THIS!!!
Worse, it's circular! Set theory already comes WITH AXIOMS!
The axioms ALREADY DEFINE e as a language-element, as a two-place
predicate! There is a prior paradigmatic notion OF a WFF and OF
a first-order language! Did you really feel that the prior specifications
of THOSE were not set-theoretic enough?? But you couldn't be doing set
theory AT ALL UNTIL AFTER you had those!
If all you were looking for was some way to justify a ban on "adding uncountably many new constants" then I think my approach was always going
to be better.
But now I am too far inside your mind to be relevant.

[George Greene]

On Saturday, October 22, 2016 at 4:48:19 PM UTC-4, mitch wrote:
> Which proper class is in ZF?

See, this is one of those shocking kinds of questions that
cause the reader to marvel. Shocking because it's basic -- it's completely elementary -- yet you have read so much advanced/esoteric research WITHOUT knowing it! Did it really not occur to you to read THE EASY STUFF first?!??


Your main problem here is that "in" is ambiguous.

There are 4 relevant kinds of "in" in a logical treatment of set theory.

Kind #1 is the epsilon/membership predicate itself; the thing on the left
is "in" the thing on the right. In that sense, by definition, NO proper
class is "in" ANYthing in ZF.

Kind #2 is "in" the domain of discourse -- it would make sense to say, in
common language, that some object occurred "in" a first-order theory if it occurred in the domain over which that theory was quantifying. In this sense,
again, by definition, NO proper class occurs in ZF -- ZF is mono-sortal -- EVERYthing in ZF is a set, in the same way that EVERYthing in PA is a "number".

Kind #3 is "in" or among the various SUBCOLLECTIONS OF things already-conceded-to-be "in" the theory. In particular, the domain of discourse -- the whole domain -- is indispensable to the theory; you can't plausibly argue that it's not an important part of the theory. And the universe -- the domain of discourse -- the class of ALL sets -- is proper. The domain of quantification is in fact THE ARCHEtypical example of a proper class. Von Neumann proves later that all other classes that are equinumerous to the whole domain are also proper, basically for that sae size-related reason; there is a treatment with his "limitation of size principle" that equates being proper with being as big as the whole domain. Obviously there are a great many smaller subcollections (of things from the domain) that can leave out humongously many elements of it in humongously complicated patterns (or apparently randomly) and still be equinumerous to it; all of these are proper classes that are in some sense "in" ZF. Even though they are not individually elements of the domain, they are collections of individual elements of the domain; they are proper subclasses of the domain.

Kind #4 is "in" the list of LINGUISTIC components of the theory.
The usual axiom schema of separation has a unary definable-predicate-function
embedded in it; it takes one argument from the domain. Once you have
conceded the existence of this definable-predicate unary function, you unavoidably have two classes of things, namely, those that satisfy it and those that don't. These classes are not guaranteed to be sets until you bound both
of them by some larger set; if you already have a set then the defined predicate will separate it into two more, but if you are just applying it to the whole domain then you might not be so lucky. The axiom schema of replacement has a binary schema and you can think of that as defining a function, again, IF YOU START FROM a set that is known to be the domain. The range of the "definable proper class as a relation" WHEN RESTRICTED TO A DOMAIN that is a known set, will also be a set. But prior to such restriction, the definably binary predicate, which is indisputably "in" the theory, can be satisfied by a proper class of ordered pairs.

But the one thing you are DEFINITELY NOT going to get from among these linguistic components is a proper class OF COMPONENTS -- there are ONLY COUNTABLY many different definable unary and binary predicates since the formulas defining them are finite and the alphabet over which they are defined is finite. The notion of even ADDING CONSTANTS AT ALL to this theory is problematic because you don't even need the empty set as a constant; the existence of the empty set follows from the separation schema with a contradictory predicate.

[George Greene]

On Sunday, October 23, 2016 at 9:19:59 AM UTC-4, mitch wrote:
> The power set of a given set is the
> set of all of its subsets.

Fine so far, but SEZ WHO? In the usual case, sez THE POWERSET *AXIOM*.
Whenever you say "must", you need to be clear "on whose authority".
WHO SAYS this or that "must" be the case,and whey need THEY be respected?

> To be a subset, an individual must first be a set.
> So, every subset of a set is necessarily a set.

Fine, natural-language grammar, re the usual working of prefixes in English.

> To be a set, an individual must be an element of the universe.

This is getting dangerous.
You have to pick a paradigm here. In the FOL paradigm, to BE, PERIOD,
you "must be" an element of the universe. THERE IS NO OTHER KIND of
JUST *be*ing! The "set" thing is just irrelevant. The universe by definition
is just the class of all&only "those things that exist" and EVERY such
thing "must be an element of the universe", sets and non-sets both.

> The sets of the cumulative hierarchy
> do not have complements. So, a power
> set need not be a system of complements.

Again, this is in THE USUAL treratment. In the usual treatment OF the cumulative hierarchy, the cumulative hierarchy DOES NOT EXIST because IT
cannot fit "inside" itself.

> Holding as close as possible to the usual
> formulation of set theory, the only sets
> that can be in the power set of the universe

No, No, NO! As soon as you say "holding as close as possible to the usual formulation of set theory", you GIVE UP the right to speak of "the power set of the universe"! The universe IS NOT a set and DOES NOT HAVE a powerset!
The question of whether the universe is CLOSED under powerset (or not) IS COMPLICATED! EVERY set has MORE subsets than members! YOU CAN'T PUT ALL the subsets of the universe into the universe as individuals! Well, you can put all its subSETS there, but NOT all its subCOLLECTIONS or subclasses! ONLY the subSETS and NOT the sub-PROPER-classes CAN HAVE powersets!!

[mitch]

On 10/23/2016 09:59 AM, George Greene wrote:
> On Sunday, October 23, 2016 at 9:19:59 AM UTC-4, mitch wrote:
>> The power set of a given set is the set of all of its subsets.
>
> Fine so far, but SEZ WHO? In the usual case, sez THE POWERSET
> *AXIOM*. Whenever you say "must", you need to be clear "on whose
> authority". WHO SAYS this or that "must" be the case,and whey need
> THEY be respected?
>

I believe this would still be the power set
axiom. What is problematic for the model
theory is "all".

>> To be a subset, an individual must first be a set. So, every subset
>> of a set is necessarily a set.
>
> Fine, natural-language grammar, re the usual working of prefixes in
> English.
>
>> To be a set, an individual must be an element of the universe.
>
> This is getting dangerous. You have to pick a paradigm here. In the
> FOL paradigm, to BE, PERIOD, you "must be" an element of the
> universe. THERE IS NO OTHER KIND of JUST *be*ing! The "set" thing
> is just irrelevant. The universe by definition is just the class of
> all&only "those things that exist" and EVERY such thing "must be an
> element of the universe", sets and non-sets both.
>

Thank you. You have just made the argument that
philosophical treatments of set theory have an
underlying metaphysical basis even when the use
of uninterpreted symbols keeps one from making
ontological commitment.

To admit this position, however, would require a
change to the accepted axioms for Zermelo-Fraenkel
set theory to include urelements. Jech calls this
ZFA where the "A" stands for atoms. Among the
changes to set theory which may be attributed to
Skolem, the idea that the theory of pure sets
sufficed as a foundation for mathematics arose.
So, the "set" thing is relevant in some literature
and not in other literature.

Unfortunately, Jech does not write the axioms
out formally. What he does say is this:

< begin quote >

"Set Theory With Atoms (ZFA)

In this modified set theory we have not only
sets, but also additional objects, *atoms*. These
atoms do not have any elements themselves but can
be collected into sets. Obviously, we have to
modify the axiom of extensionality, for any two atoms
have the same elements -- none.

The language of ZFA has, in addition to the
predicate "in", a constant "A". The elements of
A are called *atoms*, all other objects are called
*sets*.

Axiom A. If a in A, then there is no x such that
x in A

The axiom of extension takes this form:

(21.1) If two sets X and Y have the same elements,
then X=Y

All other axioms of ZF remain unchanged. In particular,
the axiom of regularity [foundation] states that every
nonempty set has an in-minimal element. This minimal
element may be an atom.

The effect of atoms is that the universe is no
longer obtained by iterated power set operation from
the empty set. In ZFA, the universe is built up from
atoms.

Ordinal numbers are defined as usual except that
one has to add that an ordinal does not contain any
atom."

< end quote >

Jech goes on to explain how a cumulative hierarchy
for an arbitrary set is generated and then defines
the universe to be the cumulative hierarchy generated
from A.

In a very real sense, Russell's paradox is what
makes the "set" thing irrelevant. Where you told
Mr. DiEgidio that first-order logic carries existential
import by virtue of definition, what had been required
for modern compositional logics to be accepted had
been a representation for the square of opposition
in the new logic. So, in the translations,

All A are B
Ax( A(x) -> B(x) )

Some A are B
Ex( A(x) /\ B(x) )

No A are B
Ax( A(x) -> ~B(x) )

Some A are not B
Ex( A(x) /\ ~B(x) )

the predicate "A(x)" is what names the universe of
discourse as "number" or "set" or "philosophers".
For example,

All men are animals.

But, Russell's paradox showed that one could not
name "sets" in this manner if, by set, one meant
sets for which "x in x" does not hold.

Now, I happen to agree with your statement concerning
the paradigm and understand why you said it. But,
that does not mean that it is applied that way
uniformly in the literature.

Although I found Jech last night, I have not yet
found "Set Theory and the Continuum Hypothesis".
From that book I can quote the author interpreting
a "bare quantification" as "meaning" what I have
just said.

For what this is worth, I am not the author of
these conflicting views.

>> The sets of the cumulative hierarchy do not have complements. So,
>> a power set need not be a system of complements.
>
> Again, this is in THE USUAL treratment. In the usual treatment OF
> the cumulative hierarchy, the cumulative hierarchy DOES NOT EXIST
> because IT cannot fit "inside" itself.
>

The problem then are all of the references in
the literature using the expression,

"the universe"

or

"the set universe"

Without knowledge of the history of foundations,
I simply applied the notion of definite description
to this expression. More precisely, I asked
how I might be able to apply that notion. In any
case, the "epistemic warrant" of definite description
goes immediately to a syntactic version of Leibniz'
identity of indiscernibles. And, this is what has
been excluded from the standard account of identity
in the received paradigm of first-order logic.

If one is supposed to surmise non-existence from
paradox, why is the literature replete with such
uses that ignore the problem?

I am not trying to account for reality. A problem
in which I am interested is bound up in a great
deal of "vague" literature. Presumably the problem
is mathematical. Mathematical does not mean
metaphysical. But influential authors like Quine
have pursued mathematics as if it is.

I have no problem with your statement that the
cumulative hierarchy cannot fit inside of itself.
I have explicitly stated that Russell's paradox
forces the cumulative hierarchy to be partial.
But, that does not mean that the axioms cannot
be altered to add a denotation to the language
that acts as a container for the cumulative
hierarchy. It then becomes, like Ord of Card,
a class that is not representable in the
theory.

>> Holding as close as possible to the usual formulation of set
>> theory, the only sets that can be in the power set of the universe
>
> No, No, NO! As soon as you say "holding as close as possible to the
> usual formulation of set theory", you GIVE UP the right to speak of
> "the power set of the universe"!

What I mean by "as close as possible" is that the sets
which are not elements of themselves still be recursively
generated with respect to the axiom of foundation.

> The universe IS NOT a set and DOES
> NOT HAVE a powerset! The question of whether the universe is CLOSED
> under powerset (or not) IS COMPLICATED!

The latter statement is a somewhat trivial
observation.

> EVERY set has MORE subsets
> than members!

Have you ever tried to prove the theorem for
strict subsets without first invoking the
power set axiom and removing one element?

Because there is an ambiguity between singletons
and elements, my original first two sentences have
parallel syntax,

AxAy( x psub y <-> ( Az( y psub z -> x psub z ) /\ Ez( x psub z /\ ~( y
psub z ) ) ) )

AxAy( x in y <-> ( Az( y psub z -> x in z ) /\ Ez( x in z /\ ~( y psub z
) )

When I first tried to normalize the relationship I
needed a power set axiom,

AxEyAz( z in y <-> z psub x )

My approach had been naive and followed what is described
in Quine using a defined identity. But, that naive approach
included an axiom that violated pairing. More precisely,
only one singleton could exist in any model. But, any limit
ordinal could model the axioms.

In effect, the two predicates were being taken to be
equivalent. The power set was fixed for every element
of any model. Yet, all three of these sentences are
"true" in set theory.

Syntactic forcing is based upon introducing "truths" in
a particular order. So, once a formula is introduced,
its negation is excluded. In looking at these sentences,
I am not doing anything extraordinary.

But, since what I am doing is based on "defining one's terms",
it is very different from "undefined language primitives".

> YOU CAN'T PUT ALL the subsets of the universe into the
> universe as individuals! Well, you can put all its subSETS there,
> but NOT all its subCOLLECTIONS or subclasses! ONLY the subSETS and
> NOT the sub-PROPER-classes CAN HAVE powersets!!
>

Since I found Jech last night, I now have access
to the definition about which I questioned Rupert.
It reads as follows:

< begin quote >

"We say that a class M is *almost universal* if every
subset X of M is included in some Y in M. (Note that
M has to be a proper class.)

Theorem 31. Let M be a transitive, closed, almost
universal class. Then M is a model of ZF"

< end quote >

My question to Rupert lies with the word "included".
It it means "included as a subset", then would it
mean "included as an element" by virtue of the
power set axiom. Otherwise, it must mean "included
as an element" in some other element Y of M.

So, an "almost universal" class is precisely one
which contains all of its subsets as elements.

Of course, in the typical language, it is a proper
class. What I am trying to do is to create a
denotation for such a class by describing it as
one that is an element of itself.

Unless it can be included as a subset which does
not then appear in a power set, you are holding
me responsible for what is already in accepted
literature.

mitch

[mitch]

On 10/23/2016 09:49 AM, George Greene wrote:
> On Saturday, October 22, 2016 at 4:48:19 PM UTC-4, mitch wrote:
>> Which proper class is in ZF?
>
> See, this is one of those shocking kinds of questions that cause the
> reader to marvel. Shocking because it's basic -- it's completely
> elementary -- yet you have read so much advanced/esoteric research
> WITHOUT knowing it! Did it really not occur to you to read THE EASY
> STUFF first?!??

< snip >

>
> Kind #2 is "in" the domain of discourse -- it would make sense to
> say, in common language, that some object occurred "in" a first-order
> theory if it occurred in the domain over which that theory was
> quantifying. In this sense, again, by definition, NO proper class
> occurs in ZF -- ZF is mono-sortal -- EVERYthing in ZF is a set, in
> the same way that EVERYthing in PA is a "number".
>

Apparently, you understood me perfectly
well.

And, again, you hold me accountable for
reams of incompatible views from the
literature that I describe as "extra-mathematical"
because they have nothing to do with interpreting
the universal quantifier of a mathematical
theory.

Professor Rubin once shocked Aatu on sci.math
when a poster asked about mathematical logic.
Whereas Aatu felt that this included the many
and varied views that have filled many, many
books, Professor Rubin declared that it go no
further than the deductive calculus.

Up until recently, I have held the very position
of Professor Rubin. That is why my work is always
expressed in well-formed formulas that may be
included in a formal proof. What changed that had
been the distinction between an eliminable and
a non-eliminable identity.

Even then, I used a standard variation of
first-order logic formulated for algebraic
representations to address the problem.

If paradoxes are supposed to be interpreted
as proof of non-existence, how is it that
many-sorted logic suddenly brings such things
into existence?

Frege had been very careful to distinguish
between a "concept" and "the extension of
a concept". Russell had been very careful
to treat definite descriptions as a form of
quantifier in order to preserve a classical
notion of logic.

The problem with "easy stuff" is that such
careful deliberations are ignored by mediocre
textbook writers who believe that anything
they denote can exist.

mitch

[mitch]

On 10/23/2016 09:27 AM, George Greene wrote:
> On Saturday, October 22, 2016 at 2:28:01 PM UTC-4, mitch wrote:
>> It probably never occurred to you that
> STOP!!
>
> GOOD GRIEF!!!
>
> STOP!!!!
>
> Subjective value judgments about what is going on in somebody ELSE'S mind are
> ALWAYS inappropriate if the other person IS RIGHT THERE!!
>
>
> Any sentence you start that way is not likely to end well.
> I know what occurred to me (I am THE ONLY POSSIBLE authority on that subject) and I am only going to flame you for getting it wrong.
>
>> my insistence on
>> "defined language elements" had been precisely because I
>> question the liberality with which languages are expanded
>> with constants in model theory.
>
> DUH.
> I completely SHARED that FROM DAY ONE.
> We have sort of a core difference in temperament in that it never motivated
> me to do any research; I could just go "that's obviously bullshit" and just
> NOT ENGAGE with that thread. The reason you don't publicly assert that is
> that in this room (sci.logic) and under this paradigm, there is ONLY ONE
> warrant for waxing PUBLICLY dismissive OF ANYthing and that is inconsistency.
> Until and unless you can prove that, you just endure it.
>
>

I suppose I had been referring to that
difference in "core temperament". And,
the use of the word "probably" had been
intended to indicate that I can only
guess at such things. I do not view that
as the same as an assertion of such things.
But, I will try to avoid it since you find
it problematic. As it may be habitual, I
might require some "training".

The problem with simply ignoring the
situation in model theory is that it
generates results that other people would
believe without having read the proofs.
I am reconciled with what model theory
does. And, I acknowledge that parts of
what I am doing constitute second-order
methods with respect to first-order
model theory. But, it took a lot of
research to get to that point.

In particular, the relations defined in terms
of themselves ultimately must be interpreted
as "the least...such that...exists". This is
essentially a quantification over
first-order models. But, it is also a
restriction against arbitrary expansions
of languages.

Such methods are typical in mathematics
when an object is generated in relation
to a specification.

mitch