Post by Ross FinlaysonPost by Ross FinlaysonPost by FromTheRaftersPost by David ChmelikPost by David ChmelikIs the universe set called univrset?
'universet'
Domain of Discourse. Usually a blackboard bold (or doublestruck) D is
the symbol.
See for exampler Forster's "Set Theory with a Universal Set".
The idea that a universal set exists is called "Domain Principle"
or "Domainprinzip".
The domain of discourse is a usual term.
See for example Finsler and Boffa, Kunen inconsistency,
set of all sets, order type of ordinals, group of all groups,
infinite-dimensional space, "Continuum", sometimes just
"the world".
http://youtu.be/aHS0VKOM09U
"Thomas Forster - Recent developments in Set Theory with a Universal Set"
I don't vouch for this yet it's part of the study, about
things like "New Foundations with Ur-Elements" or
"New Foundations with Universes" and so on.
Here's it's "Null Axiom Theory" or "Universal Axiom Theory",
for example.
It's often called part of the "extra-ordinary" for Mirimanoff,
as it relates to Kant's "Sublime", about in reflection the
Leibnitz' Monadology as about the Ding-an-Sich.
Let's see, Forster starts talking about Ramsey Theory,
which in a sense is about law(s) of large numbers and
things like whether the existence of long arithmetic
progressions is independent standard theory, like
various conjectures of Goldbach are independent standard
number theory, about that Ramsey theory and about Birkhoff
and the lacunary, is actually about the _applied_ and the
fact that the guaranteed existence of non-standard models,
actually has application, about law(s) of large numbers.
It's like when you hear about Groethendieck universes for
ALGEBRAIC geometry, which is not to be confused with
algebraic GEOMETRY, and various conjectures are actually
independent standard theory.
Forster mentions that a new way to talk about "equi-interpretability"
is to call then "in synonymy", "up to the logical", then as with
regards to quite particular definitions of logical and non-logical,
where for example some have that any restriction of comprehension,
results non-logical objects ("properly" logical). He also mentions
that that means in model theory they model each other or "have
the same models". Then in seems that the Ackermann trick is
sort of presuming that a usual encoding of a set of naturals
can be encoded in a natural, as sort of like a bit of Cohen
forcing or that Hausdorff and "constructible, countable, ...".
I.e., that Peano Arithmetic, has after Goedel a completeness
theorem, in case you've never heard of Goedel's completeness
about learning about Goedel's incompleteness, there's that
PA being "standard" isn't actually decided when of course
there are non-standard models, and extra-ordinary.
Then these weaker "strictly finite" bits are looking roundabout
wrong.
"The thought is that the noumena that they are describing
is the same thing."
Kant's noumena basically reflect on the objects of though,
reason particularly, here the platonic objects of mathematics.
"NF says there's a universal set, and ZF says there isn't."
"They're just two different ways of describing the same mathematics."
That's sort of disagreeable.
"The universe is a boolean algebra." Euh, .... What you get
is that the object is that the universe minus the object is
the context is the same object, "the content is the context".
15:20 = 920s
http://youtu.be/aHS0VKOM09U
This passage from Forster really helps illustrate
that NF and ZF are different.
I'll disagree that PA is tight, simply as that there
are non-standard models of it, while, "ubiquitous ordinals"
is tight.
Alright then if you'd like to know more about universal sets,
there's a usual gentle sort of introduction to some examples.
Mikhail Katz talks about infinitesimals in ZF, and not hyper-reals,
reflecting on the Internal Set theory as with regards to
infinitesimals and Nelson's results that IST and ZFC are co-consistent.