From Newsgroup: sci.math

On 05/17/2026 09:14 AM, Ross Finlayson wrote:

On 05/17/2026 07:50 AM, Ross Finlayson wrote:

On 05/17/2026 07:17 AM, wm wrote:

Am 16.05.2026 um 11:47 schrieb Mikko:

On 15/05/2026 19:44, WM wrote:

Am 15.05.2026 um 08:03 schrieb Mikko:

On 14/05/2026 18:00, WM wrote:

Am 14.05.2026 um 10:54 schrieb Mikko:

On 13/05/2026 23:50, wm wrote:

No, you did not. You presented some claims about what he said but >>>>>>>> you did not quote anytihing.

Can't you read? Below is the quote and its source.

Therefore you shoulsd know. But in case you have a short memory, >>>>>>>>> here it is again:
"One occasionally hears the argument rCo let us call it the
math-tea argument, for perhaps it is heard at a good math tea rCo >>>>>>>>> that there must be real numbers that we cannot describe or
define, because there are only countably many definitions, but >>>>>>>>> uncountably many reals. Does it withstand scrutiny? [...]
Question 1. Is it consistent with the axioms of set theory >>>>>>>>> that every real is definable in the language of set theory
without parameters?
The answer is Yes. Indeed, much more is true: if the ZFC >>>>>>>>> axioms of set theory are consistent, then there are models of ZFC >>>>>>>>> in which every object, including every real number, every
function on the reals, every set of reals, every topological >>>>>>>>> space, every ordinal and so on, is uniquely definable without >>>>>>>>> parameters. [J.D. Hamkins et al.: "Pointwise definable models of >>>>>>>>> set theory", arXiv (2012)]
Complete nonsense.

What he says here is perfectly correct and uses the words in their >>>>>> usual meanings.

It is nonsense since there are only countably many definitions.

How is the claim that there are only countably many definitions made
nonsense by the fact that there are only countably many dfinitions?

Hamkins' claim that uncountably many reals are nameable is nonsense.

REgards, WM

Well, "non-classical logic" is rather non-sense.

When one reads Smullyan and Fitting and
the authors are like "so what?", is,
along the lines of: "because, that's why".


For a strict definition of "because".


So, that's "forcing" for you, breaking usual accounts of
descriptive set theory after axiomatic set theory in
at least three places.

Of course that's due
mostly Skolem, not so much Louwenheim,
and Levy, not so much Mostowski.

When Smullyan and Fitting say Cohen really
wanted to keep things in "classical" logic,
is that he did, then the later account of
forcing (which isn't axiomatic a la Cohen
instead "non-classical" a la "so what?"),
is kind of like Banach-Tarski on equi-decomposability
except that it belongs to Vitali-Hausdorff,
about "measure" not "lack thereof".

So, since Goedel showed CH consistent with ZF one way,
and von Neumann showed Not CH consistent with ZF an other,
then forcing is sort of a lie that later accounts
hide in shame.

Never noticed this article on SEP before, "Inconsistent Mathematics".

https://plato.stanford.edu/entries/mathematics-inconsistent/

These days there's what's called "synthetic logic" where they take
two different theories, give some of the terms the same name
yet having different definitions, then talk like they don't know
the difference, and nobody else does either ("synthetic logic:
the abuse thereof").

File somewhere under "oxymoronic oxymorons".

As KL put it one time, "oxymoron is he who puts zit cream in eye".

(Back when "oxy" was zit cream not hillbilly-pills.)

From the article:
"[ ....] But it was noticed in the later twentieth century that there is another way, namely accept the contradiction and develop mathematical
theories containing both sides of the contradiction. This would be
impossible if the contradictory theory was erected on a logical
foundation containing the Boolean principle Ex Contradictione Quodlibet
ECQ, from a contradiction everything follows. So ECQ has to be
abandoned, but fortunetely that proves possible, indeed mathematically straightforward. What remains is a rich field, of novel mathematical applications interesting in their own right, which sidestep the vexing questions of which foundational principles to adopt, by developing contradictions in areas of mathematics such as number theory or analysis
which are far from foundations. This is inconsistent mathematics."

If oxymoron is he who puts zit cream in eye,
oxymoronic forcing puts zit cream in your eye, too.

Which stings and obscures vision, ..., and doesn't clear zits.


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From Newsgroup: sci.math

On 05/18/2026 12:48 AM, Mikko wrote:

On 17/05/2026 17:17, wm wrote:

Am 16.05.2026 um 11:47 schrieb Mikko:

On 15/05/2026 19:44, WM wrote:

Am 15.05.2026 um 08:03 schrieb Mikko:

On 14/05/2026 18:00, WM wrote:

Am 14.05.2026 um 10:54 schrieb Mikko:

On 13/05/2026 23:50, wm wrote:

No, you did not. You presented some claims about what he said but >>>>>>> you did not quote anytihing.

Can't you read? Below is the quote and its source.

Therefore you shoulsd know. But in case you have a short memory, >>>>>>>> here it is again:
"One occasionally hears the argument rCo let us call it the math- >>>>>>>> tea argument, for perhaps it is heard at a good math tea rCo that >>>>>>>> there must be real numbers that we cannot describe or define,
because there are only countably many definitions, but
uncountably many reals. Does it withstand scrutiny? [...]
Question 1. Is it consistent with the axioms of set theory >>>>>>>> that every real is definable in the language of set theory
without parameters?
The answer is Yes. Indeed, much more is true: if the ZFC
axioms of set theory are consistent, then there are models of
ZFC in which every object, including every real number, every
function on the reals, every set of reals, every topological
space, every ordinal and so on, is uniquely definable without
parameters. [J.D. Hamkins et al.: "Pointwise definable models of >>>>>>>> set theory", arXiv (2012)]
Complete nonsense.

What he says here is perfectly correct and uses the words in their
usual meanings.

It is nonsense since there are only countably many definitions.

How is the claim that there are only countably many definitions made
nonsense by the fact that there are only countably many dfinitions?

Hamkins' claim that uncountably many reals are nameable is nonsense.

First you must prove that he made such claim.

However, there is a problem with the quoted text. The proof that a
consistent theory has a finite or countable model is restricted to
first order theories. But the concepts of infinity and countability
are second order concepts. Therefore one must be careful when these
are discussed at the same time.

Furthermore, usual accounts after Smullyan and Fitting (1980's,
"Set Theory and the Continuum Problem") which one figures that
Hamkins alludes to in "Pointwise Definable Models of Set Theory",
as comments from the blog post indicate relating it to forcing,
makes for so that it's a point of great confusion to many because
the Smullyan and Fitting's non-classical logic basically _admits_
that it's contradictory.


I happen to have been reading a copy of "Set Theory and the
Continuum Problem" the other day, here in this video essay
at about 16:08 (t=968) begins a reading into it.


For example, it introduces that "set theory" in the usual account
has that always at least one of the axioms of ZF is a _schema_,
and it's never a first-order account.

https://www.youtube.com/watch?v=A3V6byu8GDo

https://www.youtube.com/watch?v=A3V6byu8GDo&t=968

Then, it talks about the Continuum Hypothesis and cardinals and
ordinals and counting and numbering and Goedel and von Neumann
and Cohen and about Cohen forcing or class forcing, yet really
about Skolem.


Damn right one must be careful when talking about them at the same time.


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