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