§1 Basic Set-Theoretic Constructions
Every mathematical structure is built out of a few fundamental building blocks of set theory. Let's list them for reference.
Ordered Pair
Key property\(\langle x, y \rangle \stackrel{\rm def}{=} \{\{x\},\{x,y\}\}\)
Main property: \(\langle a,b\rangle = \langle c,d\rangle \Leftrightarrow a=c \wedge b=d\)
The basis for Cartesian products and all relations
Cartesian Product
Construction\(A \times B \stackrel{\rm def}{=} \{\langle x,y\rangle \mid x\in A,\; y\in B\}\)
\(n\)-ary: \(A_1 \times \cdots \times A_n\); homogeneous \(A^n\)
Builds "coordinate" spaces and graphs of functions
Function
MappingA total, single-valued relation \(f \subseteq A\times B\)
Bijection = injection + surjection
Image \(f[X]\), preimage \(f^{-1}[Y]\), composition \(g\circ f\)
Equivalence Relation
PartitionReflexive + symmetric + transitive
Class: \([x]_{\sim} = \{y\in X \mid x\sim y\}\)
Quotient set: \(X/{\sim} = \{[x]_\sim \mid x\in X\}\)
Constructs \(\mathbb{Z}, \mathbb{Q}, \mathbb{R}\) via equivalence classes
Kleene Star
Infinite union\(X^* = \bigcup_{n=0}^{\infty} X^n = X^0 \cup X \cup X^2 \cup \cdots\)
The set of all finite words over the alphabet \(X\)
Strings: \(\Sigma^*\); polynomials: \(K[x]\) on the carrier \(K^*\)
Orders
Hierarchy
Preorder: refl. + trans.
Partial order: antisymmetric preorder
Linear order: a total (connected) partial order
Well-order: \(\forall X\neq\emptyset\;\exists\text{min}\)
Ordinals: \(\omega, \omega^2, \varepsilon_0,\ldots\)
Properties of Binary Relations
| Property | Formula | Example |
|---|---|---|
| Reflexivity | \(\forall x:\; xRx\) | \(\leq\) on \(\mathbb{R}\) |
| Symmetry | \(xRy \to yRx\) | \(=\), \(\sim\) |
| Transitivity | \(xRy \wedge yRz \to xRz\) | \(<\), divisibility |
| Antisymmetry | \(xRy \wedge yRx \to x=y\) | \(\subseteq\) on \(\mathcal{P}(X)\) |
| Totality (connectedness) | \(\forall x,y:\; xRy \vee yRx \vee x=y\) | \(\leq\) on \(\mathbb{Z}\) |
| Single-valuedness | \(xRy \wedge xRz \to y=z\) | A function \(f\) |
§2 Analytic Structure
A set (a carrier) is amorphous by itself. To turn it into a mathematical space, structure is layered onto it — from the bottom up.
An analytic structure is a tuple \[\mathcal{M} = \langle M,\; \Sigma,\; \mathcal{T},\; \Phi \rangle,\] where \(M\) is the domain (carrier), \(\Sigma\) is the algebraic signature, \(\mathcal{T}\) is the topological layer (systems of subsets), and \(\Phi\) is the metric layer (functionals into a numeric field).
0. Domain \(M\)
The underlying set — the carrier of the structure. Amorphous by itself.
1. Algebraic Layer \(\Sigma\)
Internal operations \(f:M^n\to M\), external operations \(R\times M\to M\), relations \(P\subseteq M^n\), constants.
2. Topological Layer \(\mathcal{T}\)
Systems of subsets \(\mathcal{T}\subseteq\mathcal{P}(M)\): a topology \(\tau\), a \(\sigma\)-algebra, filters.
3. Metric Layer \(\Phi\)
Functionals \(F:X\to\mathbb{K}\) (\(\mathbb{R}\) or \(\mathbb{C}\)): a metric, a norm, a measure, an inner product.
As an algebraic structure: the vector space \(\langle\mathbb{R}^3,+,\cdot\rangle\). There are only vectors and operations on them — no closeness, no length.
As an analytic structure: the Banach space \(\langle\mathbb{R}^3,+,\cdot,\|\cdot\|\rangle\). Adding a norm generates a metric, and with it — continuity, differentiability, integrability.
§3 Analytic Layers: A Comparative Table
Each layer adds new "semantics" on top of the previous ones. The table shows the type of object, its key properties, and characteristic examples.
| Layer | Type of Object | Key Properties / Concepts | Characteristic Examples |
|---|---|---|---|
|
Set-theoretic Domain \(M\) |
Just a set — the carrier |
Amorphous by itself: no operations, no closeness, no measurements. |
\(\mathbb{R}\), \(\mathbb{Z}\), \(\{0,1\}^\omega\), \(\mathcal{P}(X)\), Words \(\Sigma^*\) |
|
Algebraic Signature \(\Sigma\) |
Internal operations: \(f: M^n \to M\) External operations: \(R\times M\to M\) Relations: \(P\subseteq M^n\) Constants: \(c\in M\) |
Dynamics: elements generate new elements of the same domain. External operations link the structure to a numeric "scale". |
Group: \((G,\cdot)\) Ring: \((\mathbb{Z},+,\cdot)\) Vector space: \((V,+,\cdot_\mathbb{R})\) Lattice: \((\mathcal{P}(X),\cup,\cap)\) Lindenbaum–Tarski algebra: \((\mathcal{B}/{\sim},\wedge,\vee,\neg)\) |
|
Topological \(\mathcal{T}\subseteq\mathcal{P}(M)\) |
Topology \(\tau\): open sets \(\sigma\)-algebra: measurable sets Filter \(\mathcal{F}\): "most of" Ultrafilter: factorization |
The "texture" of a space — closeness without a ruler. A topology is given by a system of subsets, not by a number. |
Metric topology: open balls \(B(x,\varepsilon)\) Discrete: \(\tau=\mathcal{P}(M)\) Indiscrete: \(\tau=\{\emptyset,M\}\) Probability space: the \(\sigma\)-algebra of events \(\mathcal{F}\) Stone space: ultrafilters as points of \(\beta\mathbb{N}\) |
|
Metric \(\Phi: X\to\mathbb{K}\) |
Metric: \(d:M^2\to\mathbb{R}_{\geq 0}\) Norm: \(\|{\cdot}\|:M\to\mathbb{R}_{\geq 0}\) Inner product: \(\langle{\cdot,\cdot}\rangle:M^2\to\mathbb{K}\) Measure: \(\mu:\mathcal{T}\to\overline{\mathbb{R}}_{\geq 0}\) |
Projects the structure onto a numeric scale: distance, volume, probability, energy. |
Metric space: \((\mathbb{R}^n, d_{\rm eucl})\) Banach space: \((C[0,1],\|f\|_\infty)\) Hilbert space: \((L^2,\langle f,g\rangle)\) Lebesgue measure: \(\lambda:\mathcal{B}(\mathbb{R})\to\overline{\mathbb{R}}\) Probability: \(\mathbb{P}:\mathcal{F}\to[0,1]\) |
Real mathematical models combine all four layers. The Hilbert space \(L^2\): the carrier is classes of equivalence of functions (a factorization); the algebra is addition (a vector space); the functional is the inner product \(\langle f,g\rangle\); the topology is generated by the norm \(\|f\| = \sqrt{\langle f,f\rangle}\).
§4 Interaction of Layers: An Example
Layers are not isolated shelves but interconnected mechanisms. Let's trace how an object in one layer gives rise to new objects in another, using the notion of a path as an example.
Consider a path — a continuous motion of a point \(\gamma:[0,1]\to M\).
- The topological layer \(\mathcal{T}\): defines the very notion of continuity of \(\gamma\) — without it, a path simply does not exist as a mathematical object.
- The relational layer \(\Sigma\): identifies reparametrized paths \(\gamma \sim \gamma\circ\varphi\) — yielding a geometric shape (a trajectory) independent of the speed of motion.
- The metric layer \(\Phi\): measures the length of the path, \(\ell(\gamma) = \int_\gamma ds\).
- The algebraic layer \(\Sigma\): classes of deformable loops form the fundamental group \(\pi_1(X,x_0)\) — the space itself has become an algebraic structure.
The layers generate one another: a topological object gives rise to an algebraic superstructure.
In theoretical mechanics and field theory: the layer of functionals defines the action \(S[\gamma]\), while the layer of predicates formulates a law of nature as \(\delta S[\gamma] = 0\). A physical law is a predicate imposed on a functional: nature selects extremal trajectories.