**TABLE OF CONTENTS**

# minimum limit for axioms in a system?

If we have an axiom system with a finite number of axioms, we can always reduce them to only one, replacing the set of original axioms with their conjunction.

Thus, every non-trivial axiom system that is finitely axiomatized can be formulated in an equivalent form with a single axiom.

Gödel's Incompleteness Theorems apply to systems that (in addition to other conditions) have a set of axioms that is finite or at least decidable; Robinson arithmetic, for example, is finitely axiomatized and it is enough for G's Theorem.

# Robinson arithmetic

**1. Sx ≠ 0**

0 is not the successor of any number.

**2. (Sx = Sy) → x = y**

If the successor of x is identical to the successor of y, then x and y are identical. (1) and (2) yield the minimum of facts about N (it is an infinite set bounded by 0) and S (it is an injective function whose domain is N) needed for non-triviality. The converse of (2) follows from the properties of identity.

**3. y=0 ∨ ∃x (Sx = y)**

Every number is either 0 or the successor of some number. The axiom schema of mathematical induction present in arithmetics stronger than Q turns this axiom into a theorem.

**4. x + 0 = x**

**5. x + Sy = S(x + y)**

(4) and (5) are the recursive definition of addition.

**6. x·0 = 0**

**7. x·Sy = (x·y) + x**

(6) and (7) are the recursive definition of multiplication.

# Travelling salesman problem - Wikipedia

- and cognitive psychology https://en.wikipedia.org/wiki/Travelling_salesman_problem#Human_performance
- Geometric measure theory - Wikipedia

# General number field sieve - Wikipedia

# links, notes and resources

- Peano axioms - Wikipedia
- Cardinality - Wikipedia
- Zermelo–Fraenkel set theory - Wikipedia
- Hilbert's paradox of the Grand Hotel - Wikipedia
- Galileo's paradox - Wikipedia
- Pigeonhole principle - Wikipedia
- Public-key based on roots of polynomial - Cryptography Stack Exchange
- Chapter 1 The Gödel Phenomena in Mathematics - IAS School of ... www.math.ias.edu › ~avi › BOOKS › Godel_Widgerson_Text
- Gödel numbering - Wikipedia
- Mathematics, Cryptology, Security.
- Crypto Aspects of The Jacobian Conjecture | Gödel's Lost Letter and P=NP
- The Gödel Letter | Gödel's Lost Letter and P=NP
- P versus NP problem - Wikipedia
- "An answer to the P = NP question would determine whether problems that can be verified in polynomial time can also be solved in polynomial time."