Mathematics - Part 1: Foundations & Reason

This exploratory research work is graded A on July 10th, 2025.

What is Mathematical Logic?

Mathematical logic

is a subfield of mathematics exploring formal systems in relation to the way we reason. It provides tools to analyze the structure of arguments, statements, and proofs.

Major subareas

include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.

Propositional Logic

Table 1. Propositions and Logical Connectives in Mathematical Logic

Symbol	Name	Operation Example	Meaning
¬	Negation	¬P	"not P"
∧	Conjunction	P ∧ Q	"P and Q"
∨	Disjunction	P ∨ Q	"P or Q" (inclusive)
→	Implication	P → Q	"Whenever P then Q"
↔	Biconditional	P ↔ Q	"P if and only if Q" (iff)

What is Propositional Calculus?

Propositional calculus

is a branch of logic. It is also called propositional logic.

Mathematical logic is also the study of truths in mathematical statements. These are called propositions — a claim that is either true or false. A mix of these create propositional logic.

Example of a proposition: 2 + 2 = 4 This can only be either true or false.

Notational Variants of Connectives in propositional calculus

These are some standard notations for propositional logic.

Table 2. Logical Connectives and Operations

Operator	Name	Alternate Symbols	Truth Condition
AND	Conjunction	A ∧ B, A · B, AB, A & B, A && B	True if both A and B are true.
OR	Disjunction	A ∨ B, A + B, A ∣ B, A ∥ B	True if at least one of A or B is true.
NOT	Negation	¬A, −A, Ā, ~A, !A	True if A is false.
IMPLIES	Implication	A → B, A ⇒ B, A ⊃ B	False only if A is true and B is false.
EQUIVALENT	Biconditional	A ⇔ B, A ≡ B, A ↔ B, A ⟺ B	True if A and B are logically equivalent.
XOR	Exclusive OR	A ⊕ B, A ⊻ B, A ⩒̅ B	True if exactly one of A or B is true.
XNOR	Exclusive NOR	A ⊙ B, ¬(A ⊕ B), A ≡ B	True if both A and B are equal.
NAND	Not AND	¬(A ∧ B), A ↑ B	True unless both A and B are true.
NOR	Not OR	¬(A ∨ B), A ↓ B	True if neither A nor B is true.
NONEQUIVALENT	Negated Biconditional	A ≢ B, A ⇎ B, A ↮ B	True if A and B have different truth values.

The table above in a reference.
In my actual work I use :stem: notation instead.

Table 3. Types of Numbers in Mathematics

Symbol	Name	Description / Example
ℕ	Natural Numbers	{0, 1, 2, 3, …​} — Counting numbers.
ℤ	Integers	{…​, -2, -1, 0, 1, 2, …​} — Whole numbers and their negatives.
ℚ	Rational Numbers	Numbers that can be expressed as a/b, where a, b ∈ ℤ, b ≠ 0.
ℝ	Real Numbers	All rational and irrational numbers.
ℂ	Complex Numbers	a + bi, where i = √-1.
ℍ	Quaternions	a + bi + cj + dk — Extension of complex numbers.
ℙ	Prime Numbers	Natural numbers greater than 1 with no divisors other than 1 and itself.
ℙ*	Probable Primes	Numbers that pass a primality test but not definitively proven prime.
𝕋	Transcendental Numbers	Numbers not roots of any polynomial with integer coefficients (e.g. π, e).
𝔸	Algebraic Numbers	Solutions to polynomials with integer coefficients (includes rationals, some irrationals).
𝕌	Unit Circle Numbers	Complex numbers with modulus 1 (roots of unity).
𝕆	Octonions	8-dimensional number system generalizing quaternions.
∅	Empty Set	A set with no elements.

Table 4. ∀ Quantifiers and Set Relations

Symbol	Name	Mathematical Meaning
∃	Existential Quantifier	"There exists". Example: ∃x ∈ ℝ: x² = 4 ⇒ true.
∃!	Uniqueness Quantifier	"There exists exactly one". Example: ∃!x: x + 1 = 2
∀	Universal Quantifier	"For all". Example: ∀x ∈ ℕ, x ≥ 0
∈	Element of	x ∈ A: x is a member of set A.
∉	Not an Element of	x ∉ A: x is not in A.
⊂	Proper Subset	A ⊂ B: all A in B, A ≠ B.
⊆	Subset	A ⊆ B: all A in B, possibly A = B.
⊄	Not a Subset	A ⊄ B: A is not subset of B.
⊃	Proper Superset	A ⊃ B: A has all elements of B and more.
⊇	Superset	A ⊇ B: A contains B, possibly equal.
∪	Union	A ∪ B: Set of elements in A, B, or both.
∩	Intersection	A ∩ B: Elements common to A and B.
\	Set Difference	A \ B: Elements in A not in B.
∅	Empty Set	A set containing no elements.
℘	Power Set	Set of all subsets of a set.

Axioms

Important point: accepted without proof.

Axioms: Foundations of Logical Reasoning

An axiom, also known as a postulate or assumption, is a statement accepted as true without proof. It serves as a starting point for further reasoning and deductive processes.

In mathematics and logic, axioms are fundamental to building formal systems. They come in two main types.

Types of Axioms

Logical and non-logical.

Logical Axioms

are formulas within a formal language that are universally valid. These are tautologies—statements that are always true regardless of the truth values of their components. They form the backbone of logical systems.

Examples include:

(A ^ B) ⇒ A

A ⇒ (A v B)

Logical axioms provide the formal structure necessary for logical deductions and are typically used in propositional and predicate logic.

Non-Logical Axioms

also known as postulates or proper axioms, these are domain-specific assertions used to define particular mathematical structures. Unlike logical axioms, they are not universally valid but are accepted as true within the context of a specific mathematical theory.

Example: the Peano Axioms from arithmetic

∀x (S(x) ≠ 0)

Meaning, "for all x, the successor of x is not zero."

This axiom is non-logical because:

• It depends on the interpretation of symbols like 0 and S(x), the successor function.

• It makes a specific claim about natural numbers, not about general logical structure.

In integer arithmetic non-logical axioms establish the foundational truths from which theorems of a specific theory are derived.

In other words, our invented reasoning structures.

Common Axioms in Mathematics

Reflexive Axiom: Every entity is equal to itself.

• a = a

Transitive Axiom of Equality: Equality is transitive.

• If a = b and b = c , then a = c

Addition Axiom: Equal quantities added to equal quantities yield equal sums.

• If a = b and c = d , then a + c = b + d

  See the thruthfunctional connectives?!

Role in Mathematical Systems

Axioms form the basis of deductive reasoning in mathematics. They allow mathematicians to derive theorems and build entire structures such as geometry (e.g., Euclidean postulates) and number theory (e.g., Peano axioms).
Recall domains, i.e, fields of study, are created by non-logical axioms.

A well-chosen set of axioms ensures a consistent, complete, and useful framework for exploring mathematical truths.

• Next is Mathematics - Part 2: Set Theory

• Completed with On Sets

• Top is Mathematics - My Research

Final Grade and Comments

• ✅ Graded assignment by rdd13r on July 12th 2025. A