On 02/04/2025 01:44 AM, Ross Finlayson wrote:
On 02/03/2025 06:15 PM, Jim Burns wrote:
On 2/3/2025 8:21 PM, Ross Finlayson wrote:
On 02/03/2025 11:48 AM, Jim Burns wrote:
On 2/3/2025 1:36 PM, WM wrote:
On 03.02.2025 19:06, Jim Burns wrote:
On 2/3/2025 7:41 AM, WM wrote:
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How can Peano create
the complete set by induction?
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Peano describes a set with induction.
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Without axioms
nothing must be used oin formal mathematics.
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Axioms describe the domain.
Describe a finite ordinal:
⎛ Sets of them are minimummed or empty.
⎜ Each has an immediate predecessor or is zero,
⎜ and each of the priors of each
⎝ has an immediate predecessor of is zero.
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That description is the axioms of the finite ordinals.
(There are other ways to describe them.)
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Therefore Peano, Zermelo, or v. Neumann
create ℕ as well as the set of all FISONs
for use in set theory.
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Axioms describe.
Magic spells create.
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Axioms that are not _false_, define a domain.
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Axioms describe a domain.
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If they are self.contradictory,
then what's described doesn't exist.
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If they aren't self.contradictory,
then what's described exists.
Maybe what exists isn't
what you wanted to exist. Nonetheless.
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⎛ It's a consequence of Gödel's completeness theorem
⎜ (not to be confused with his incompleteness theorems)
⎜ that a theory has a model if and only if it is consistent,
⎜ i.e. no contradiction is proved by the theory.
⎜ Therefore, model theorists often use "consistent" as
⎜ a synonym for "satisfiable".
⎝
https://en.wikipedia.org/wiki/Model_theory
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Others define a mere contingent contrivance.
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If there is a mere contingent contrivance which
describes what I'd like described,
I will use it and be grateful for it.
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What.I'd.like.described is very, very often
(I'd say, for nearly everyone, it's 'always')
NOT everything.
What.I'd.like.described differs
from time to time and from place to place.
It is, in a word, contingent.
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And it is a contrivance, too! Yes!
What else did you think it might be, Ross?
Wafted from the sky by angels?
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"Russell's retro-thesis"
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The, "fundamental question of metaphysics",
or, "why is there something rather than nothing",
has that, according to, for example, Hegel,
in Wissenschaft der Logik, or here via "A Theory",
that _strong mathematical platonism_ makes for
that it's possible to arrive at an axiomless geometry
from an otherwise pre-scientific/pre-geometric world
(of Derrida and Husserl) so that with regards to
something like Liebnitz' principle of perfection,
that Kant's Sublime reflects Duns Scotus' super-natural
and Spinoza's infinite, arrives at also a _stronger
logicist positivism_ and without the "absconditus"
or "hidden", that in the "knackbaut", that the
tradition from Heraclitus and Parmenides as down
through Aristotle, makes "amicus Plato".
It's that there's already a theory of truth at all,
with DesCartes' Deceiver put away from reflections
on Comenius language in our Coleridge language,
so, pretty much from our _canon_ and _dogma_ and
_doctrine_ of what's technical philosophy with
a merest metaphysics and resulting a mathematical
ontology with its own merited ontological status.
Then, "axiomless natural geometry" gives things
like perfect circles and metrics and norms and
classical constructions and so on, from the
plain ponderance of inversion.
So, you can find for example Goedel has that
finitely-axiomatized theories are at best incomplete,
and when they're about models of completions,
then they're simply out.
Then, Mirimanoff and the extra-ordinary,
and Skolem and fragments and extensions,
make for that infinity is _in_, and,
each of big, biggest, and bigger.
A "silver thread".
The set of necessary FISONs
Re: The set of necessary FISONs