Interpretation of Formal Systems

Question 1

What is an axiom schema?

Answer 1

How many axioms is this? Infinitly many

Axiom Schema: Gives a decision procedure for axoims. Therefore, the set of axioms must be decidable

Ex: pq-System

Question 2

Interpretation:

How to give meaning to a formal system

Answer 2

Mathematical Structure: <ℕ, +; =>
Typographical Structure: <{-, -, –, …}; p, q>

• All well-formed string become meaningful true/false statements
• An interpretation in which all axioms and theorems come out as being true is a model
• Hofstadter calls relation between two structures in an interpretation an isomorphism

Question 3

What is a model?

Answer 3

• An interpretation in which all axioms and theorems come out as being true is a model

Question 4

What is a mathematical isomorphism?

Answer 4

• Bijective function f between two domains,
• Such that if R(a, b, …) is true, then Rf(af, bf, …) is also true for any relation R

Question 5

Are < ℕ; +> and <Evens; +e> isomorphic?

Answer 5

Yes, by the function f(x) = 2x
Whenever three natural numbers a, b, and c are related by a + b = c, then the corresponding even numbers are also related by +e: f(a) + ef(b) = f(c)

E.g.,
5 + 7 = 12
f(5) + f(7) = f(12)

Question 6

Are < ℕ; +> and < ℕ; x> isomorphic?

Answer 6

No. Consider the bijection f(x)=x

Then, 3+4=7, but 3x4 not 7!

Question 7

Define intended meaning:

Answer 7

Different meaningful interpretations of a system is possible. Thus, it is misleading to speak of the meaning of a formal system (like there is only one)

“Intended Meaning to identify what they meant in the first place, but it does not exclude other possiblities

Note: What counts as a theorem is only determined by the axoims and inference rules, not the interpretation

Question 8

Define Soundness:

Answer 8

Every theorem is a true statement