User:Dfeuer/sandbox

Generalized sums

 * User:Dfeuer/Increasing Net Converges to Point iff Supremum
 * User:Dfeuer/Increasing Net in Completely Totally Ordered Set Converges iff Bounded Above
 * User:Dfeuer/Net of Finite Sums of Positive Elements in Ordered Abelian Group is Increasing
 * User:Dfeuer/Generalized Sums of Positive Elements are Increasing
 * User:Dfeuer/Generalized Sum in Completely Totally Ordered Abelian Group is Convergent iff Absolutely Convergent
 * User:Dfeuer/Convergent Generalized Sum in First-Countable Totally Ordered Archimedian Abelian Group has Countably Many Non-Zero Terms

NBG, Gödel-style
Point of very substantial annoyance: Gödel and PW have opposite senses of function. That is, he says $f(x) = y$ iff $\langle y, x \rangle \in f$. He similarly inverts the sense of the "domain" of a relation, and its image (which he calls its "domain of values"). I'm not sure if it makes sense to do as he did, or if it is better to flip everything around as needed to match the PW way. This runs into all sorts of things, such as the "axiom of domain", where either the axiom or its name would have to change to match PW's approach. Simply using PW names for the concepts would be entirely unworkable, with everything he calls a function being called instead a one-to-many right-total relation and confusing the heck out of everyone. Obviously, this whole matter is ultimately trivial, but the details are not pretty.

Primitive predicates:
 * $\mathfrak{Cls}(A)$: $A$ is a class.
 * $\mathfrak{M}(A)$: $A$ is a set.
 * $X \in Y$, $X \in y$, $x \in Y$, $x \in y$: express membership

Convention: capital letters are variables whose range consists of all classes; lower case letters are variables ranging over all sets. That is, $\forall x: P(x)$ should be read as $\forall x: (\mathfrak M(x) \implies P(x)$.

Gödel uses the convention that previously unmentioned free variables are considered universally quantified. Not sure how I want to handle that&mdash;it's sometimes convenient, but it can get confusing.

Axioms:

Group A
Every set is a class:

$1 \quad \mathfrak{Cls}(x)$

An element of a class is a set:

$2 \quad X \in Y \implies \mathfrak{M}(X)$

Axiom of Extensionality:

$3 \quad \left({ \forall u: (u \in X \iff u \in Y) }\right) \implies X = Y$

Axiom of Pairing:

$4 \quad \forall x: \forall y: \exists z: (u \in z \iff u = x \lor u = y)$

Definition 1:


 * $\mathfrak {Pr}(X) \iff \lnot \mathfrak M(X)$: $X$ is a proper class.

Theorem:

Pairing is unique, by extensionality. That is:


 * $\forall x: \forall y: \exists! z: (u \in z \iff u = x \lor u = y)$

Definition 1.1:

Given sets $x$ and $y$, the unique $z$ proven to exist by axioms 3 and 4 is the non-ordered pair of $x$ and $y$, denoted $\{x,y\}$ (actually, he denotes it $\{xy\}$, but I don't think we will be following that.

Definition 1.11:

$\{x\} := \{x, x\}$ (singleton, but not named that here)

Definition 1.12:

$\langle x, y \rangle = \bigl\{ \{x\}, \{x, y\} \bigr\}$ ordered pair

Theorem 1.13:

$\langle x, y \rangle \implies x = u \land y = v$

Proof (omitted, but we already have one or two).

Definition 1.14:

$\langle x, y, z \rangle = \langle x, \langle y, z \rangle \rangle$ Ordered triple

Definition 1.15:

Inductively: $\langle x_1, x_2, \dots, x_n \rangle := \langle x_1, \langle x_2, \dots, x_n \rangle \rangle$

Theorem 1.16:

NOTE: Gödel uses "ordinal" and "ordinal number" in precisely the same way that Kelley used them 15 years later.

Natural Numbers

 * User:Dfeuer/Axiom of Infinity/Peano Structure
 * User:Dfeuer/Zero Precedes Every Natural Number
 * User:Dfeuer/Ordering on Natural Numbers/Peano

NBG Set Theory

 * User:Dfeuer/Definition:Class
 * User:Dfeuer/Definition:Set
 * User:Dfeuer/Definition:Subclass
 * User:Dfeuer/Definition:First-Order Formula
 * User:Dfeuer/Axiom of Extensionality
 * User:Dfeuer/Axiom Schema of Separation
 * User:Dfeuer/Not Every Class is a Set
 * User:Dfeuer/Each Class has Subclass which is not Element
 * User:Dfeuer/Definition:Transitive Class
 * User:Dfeuer/Definition:Swelled Class
 * User:Dfeuer/Definition:Supercomplete Class
 * User:Dfeuer/Set is Class
 * User:Dfeuer/Subclass of Set is Set
 * User:Dfeuer/Universal Class is Supercomplete
 * User:Dfeuer/Universal Class is not Set
 * User:Dfeuer/Definition:Empty Class
 * User:Dfeuer/Empty Class Exists and is Unique
 * User:Dfeuer/Empty Class is Subclass of Every Class
 * User:Dfeuer/Empty Class is Supercomplete
 * User:Dfeuer/Axiom of the Empty Set
 * User:Dfeuer/Nonempty Universe implies Axiom of Empty Set
 * User:Dfeuer/Definition:Singleton
 * User:Dfeuer/Set has Unique Singleton
 * User:Dfeuer/Empty Set does not Equal its Singleton
 * User:Dfeuer/Singleton Empty Set is Supercomplete
 * User:Dfeuer/Singletons are Equal iff Elements are Equal
 * User:Dfeuer/Definition:Unordered Pair
 * User:Dfeuer/Axiom of Pairing
 * User:Dfeuer/Weak Pairing implies Axiom of Pairing
 * User:Dfeuer/Singleton is Set
 * User:Dfeuer/Definition:Ordered Pair
 * User:Dfeuer/Equality of Ordered Pairs implies Equality of Elements
 * User:Dfeuer/Definion:Union
 * User:Dfeuer/Union Axiom
 * User:Dfeuer/Definition:Intersection
 * User:Dfeuer/Intersection of Non-Empty Class is Set
 * User:Dfeuer/Intersection of Empty Class is Universal Class
 * User:Dfeuer/Definition:Binary Union
 * User:Dfeuer/Definition:Binary Intersection
 * User:Dfeuer/Definition:Class Difference
 * User:Dfeuer/Union of Subclass is Subclass of Union
 * User:Dfeuer/Intersection of Subclass is Superclass of Intersection
 * User:Dfeuer/Binary Union of Sets
 * User:Dfeuer/Binary Intersection of Sets
 * User:Dfeuer/Union of Singleton
 * User:Dfeuer/Intersection of Singleton
 * User:Dfeuer/Miscellaneous Class Arithmetic
 * User:Dfeuer/Definition:Power Set
 * User:Dfeuer/Power Set Axiom

Note: these three theorems about power sets don't actually require the power set axiom.
 * User:Dfeuer/Set is Subset of Power Set of Union
 * User:Dfeuer/Union of Power Set of Set is the Set
 * User:Dfeuer/Power Set of Subset is Subclass of Power Set


 * User:Dfeuer/Definition:Cartesian Product
 * User:Dfeuer/Cartesian Product of Sets is Set


 * User:Dfeuer/Peano Axioms
 * User:Dfeuer/Definition:Inductive Class
 * User:Dfeuer/Universal Class is Inductive
 * User:Dfeuer/Definition:Natural Number
 * User:Dfeuer/Definition:Zero
 * User:Dfeuer/Axiom of Infinity
 * User:Dfeuer/Existence of Inductive Set implies Axiom of Infinity
 * User:Dfeuer/Existence of Set that is not a Natural Number implies Axiom of Infinity
 * User:Dfeuer/Zero is Natural Number
 * User:Dfeuer/Successor of Natural Number is Natural Number
 * User:Dfeuer/Zero has no Predecessor
 * User:Dfeuer/Principle of Mathematical Induction
 * User:Dfeuer/Natural Number is Transitive
 * User:Dfeuer/Natural Number does not Contain Itself
 * User:Dfeuer/Membership is Asymmetric on Natural Numbers
 * User:Dfeuer/Successor Mapping is Injective
 * User:Dfeuer/Natural Numbers Satisfy Peano Axioms
 * User:Dfeuer/Element of Natural Number is Natural Number
 * User:Dfeuer/Double Induction Principle/Naturals
 * User:Dfeuer/Inductive under Mapping
 * User:Dfeuer/Minimally Inductive under Mapping
 * User:Dfeuer/Double Induction Principle
 * User:Dfeuer/Progressing Function Lemma
 * User:Dfeuer/Class Minimally Inductive under Inflationary Mapping forms Nest
 * User:Dfeuer/Sandwich Principle
 * User:Dfeuer/Lemma 3.4.9.2
 * User:Dfeuer/Lemma 3.4.9.3
 * User:Dfeuer/Theorem 3.4.10
 * User:Dfeuer/Definition:Class Bounded by Set
 * User:Dfeuer/Definition:Bounded Subset of Class
 * User:Dfeuer/Class Bounded by Set is Set
 * User:Dfeuer/Non-Empty Bounded Subset of Class Minimally Inductive under Subset-Inflationary Mapping has Greatest Element
 * Definition:Fixed Point
 * User:Dfeuer/Fixed Point of Subset-Inflationary Mapping is Greatest Element of Minimally Inductive Class
 * User:Dfeuer/Lemma 3.4.15
 * User:Dfeuer/Minimally Inductive Class under Subset-Inflationary Mapping is Well-Ordered
 * User:Dfeuer/Theorem 3.4.7
 * User:Dfeuer/Definition:Ordering of Natural Numbers
 * User:Dfeuer/Natural Numbers are Totally Ordered
 * User:Dfeuer/Definition:Relation of Set Inclusion
 * User:Dfeuer/Relation of Set Inclusion is Ordering
 * User:Dfeuer/Set Difference of Universal Class and Non-Empty Set is not Transitive

Simple bits useful for various things; some left out or glossed over as trivial; some may be present but I haven't gotten to them yet.


 * Definition:Epsilon Relation
 * User:Dfeuer/Subclass is Order-Like

Order and Lattice

 * User:Dfeuer/Definition:Complete Meet Subsemilattice
 * User:Dfeuer/Definition:Complete Join Subsemilattice
 * User:Dfeuer/Intersection of Complete Meet Subsemilattices


 * User:Dfeuer/Upper Closure of Element of Complete Lattice is Complete Lattice

Group Theory

 * User:Dfeuer/Coset stuff in progress


 * User:Dfeuer/Definition:Normal Submagma
 * User:Dfeuer/Definition:Normal Subset of Group
 * User:Dfeuer/Definition:Normal Submagma of Group

Topology

 * Order Topology on Convex Subset is Subspace Topology
 * User:Dfeuer/Definition:Stone Space
 * User:Dfeuer/Stone's Representation Theorem for Boolean Algebras
 * User:Dfeuer/Stone Space is Topological Space
 * User:Dfeuer/Compact Subspace of Linearly Ordered Space (draft converse idea)
 * User:Dfeuer/Compact Subspace of Linearly Ordered Space strengthened
 * User:Dfeuer/Compact Subspace of Linearly Ordered Space/Converse Proof 2
 * User:Dfeuer/Closed Set in Linearly Ordered Space
 * User:Dfeuer/Definition:Convex Component (Order Theory)
 * User:Dfeuer/Convex Component is Closed
 * User:Dfeuer/Convex Component of Open Set in GO-Space is Open
 * User:Dfeuer/Compact Subspace of Linearly Ordered Space/Revproof3
 * User:Dfeuer/GO-Space Embeds Densely in LOTS
 * User:Dfeuer/Compact Separable Perfect Hausdorff Space Cardinality

Usual Topology

 * User:Dfeuer/Definition:Usual Topology

Properties of Compatible Relations

 * Operating on Transitive Relationships Compatible with Operation
 * Transitive Relation Compatible with Semigroup Operation Relates Powers of Related Elements


 * Properties of Relation Compatible with Group Operation
 * User:Dfeuer/CTR5


 * User:Dfeuer/Transitive Closure of Relation Compatible with Operation is Compatible

Properties of Ordered Groups

 * User:Dfeuer/Totally Ordered Group with Order Topology is Topological Group

Systematic development of positivity
The names are made up, but the stories are real.

So far I believe I've established the equivalence of the theories of transitive relations compatible with group operations and of "cones" compatible with group operations. There is more to flesh out in this realm, as below. I have just started working on the ring stuff.

Discussion point: the notion of a "cone" compatible with an operation can be defined for any magma and closed ringoid. However, without a group structure, I'm not sure if there's a way to link such to compatible relations. Does anyone else see a way to do so?

Proofs sketched out reasonably well; cleanup work appreciated

 * User:Dfeuer/Definition:Cone Compatible with Operation
 * User:Dfeuer/Cone Compatible with Group Induces Transitive Compatible Relation
 * User:Dfeuer/Transitive Relation Compatible with Group Operation Induced by Unique Cone
 * User:Dfeuer/Cone Condition Equivalent to Antisymmetry
 * User:Dfeuer/Cone Condition Equivalent to Reflexivity

Unstarted/Unfinished Ideas/Conjectures

 * User:Dfeuer/Cone Condition Equivalent to Irreflexivity
 * User:Dfeuer/Cone Condition Equivalent to Symmetry
 * User:Dfeuer/Cone Condition Equivalent to Asymmetry
 * User:Dfeuer/Cone Condition Equivalent to Trichotomy -- I'm not sure this one goes anywhere, since "trichotomy" takes on different flavors for weak and strict orders, and this may confuse matters more than clarify them. For now, the trichotomy bit is rolled into the strict and weak total cone definitions.
 * User:Dfeuer/Definition:Positive Cone
 * User:Dfeuer/Definition:Strict Positive Cone
 * User:Dfeuer/Definition:Total Positive Cone
 * User:Dfeuer/Definition:Strict Total Positive Cone
 * User:Dfeuer/Cone Condition Equivalent to Congruence


 * User:Dfeuer/Definition:Relation Compatible with Closed Ringoid with Zero (or just limit to rigs or rings for the sake of a shorter name?)
 * User:Dfeuer/Definition:Cone Compatible with Closed Ringoid
 * User:Dfeuer/Cone Compatible with Ring Induces Transitive Compatible Relation
 * User:Dfeuer/Transitive Relation Compatible With Ring Induced by Unique Cone
 * User:Dfeuer/Multiplying Compatible Relationship by Zero-Related Element

Properties of Ordered Rings

 * Product of Positive Element and Element Greater than One
 * User:Dfeuer/Strictly Positive Power of Strictly Positive Element Greater than One Not Less than Element
 * User:Dfeuer/OR1
 * User:Dfeuer/OR2
 * User:Dfeuer/OR3
 * User:Dfeuer/OR4
 * User:Dfeuer/OR5
 * User:Dfeuer/OR6
 * User:Dfeuer/OR7
 * User:Dfeuer/OR8
 * User:Dfeuer/OR9
 * User:Dfeuer/OR10
 * User:Dfeuer/OR11


 * User:Dfeuer/Totally Ordered Division Ring with Order Topology is Topological Division Ring
 * User:Dfeuer/Totally Ordered Field with Order Topology is Topological Field

Lexicographic Orderings

 * User:Dfeuer/Definition:Lexicographic Ordering on Product
 * User:Dfeuer/Definition:Lexicographic Ordering of Finite Sequences

p-norms

 * User:Dfeuer/Derivative of P-Norm wrt P
 * User:Dfeuer/P-Norm of Real Sequence is a Strictly Decreasing Function of P
 * User:Dfeuer/Bounds for P-Norm of Real Sequence

Useful links
Axiom of Foundation at NLab