Chap 5: Outline
Abacus computability and recursive functions
The concept of abacus machines
All abacus computable functions are Turing computable
All recursive functions are abacus computable
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Where are we?
We have only shown that
Chap1-2: Outline
Distinguish between two kinds of innite sets, the enumerable and
the nonenumerable (q8q8)
The concepts
Examples of enumerable sets
Examples of nonenumerable sets
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An intuitive denition
An
Chap 3-4: Outline
Turing computability and uncomputability
The concept of Turing machines
The concept of Turing computability, i.e., a function is
computable by a Turing machine
Examples of Turing uncomputable functions: the halting
function, et al.
Mathe
Chap 8: Outline
Equivalent denitions of computability
All Turing-computable functions are recursive (8.1)
Corollaries of the equivalence of the three denitions of
computability (8.2)
Semirecursive relations (7.2)
Recursively enumerable sets (8.3)
Mathemat
Chap6-7: Outline
Recursive functions and recursive relations
Recursive functions
Recursive sets and relations
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Recursive functions
Denition.
A function is recursive if it can be obtained from the basic
functions through compositio
Chap14: Outline
Proofs and completeness
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The general picture
The semantic notion of implication: |= D
A formal deduction of D from is some kind of nite
sequence of symbols such that there are denite, explicit rules
to check if a n
Chap 13: Outline
Proof of the Lwenheim-Skolem and Compactness theorems
o
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Proof of Compactness theorem
Compactness theorem: If every nite subset of a set of
sentences has a model, then the whole set has a model.
Let L be a languag
Chap15-16: Outline
Chap 15: Recursive / semi-recursive set of sentences
Chap 16: Representing recursive functions by formulas
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A brief review of computability theory
Functions obtainable from the basic functions by composition,
re
Chap17-18: Outline
Chap 17: Indenability, Undecidability, Incompleteness
Chap 18: The unprovability of consistency
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Overview
Central negative results of logic
Tarskis theorem: true arithmetic is not arithmetically denable
Churchs
Chap 12: Outline
The size and number of models
Equivalence relations
The Lwenheim-Skolem and Compactness theorems
o
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The size of models
A model of a sentence D or a set of sentences is an
interpretation M such that D is true in M
Chap 9-10: Outline
Syntax and semantics of rst-order logic
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Syntax and semantics
Throughout our treatment of formal logic, it is important to
distinguish between syntax and semantics.
Syntax is concerned with the structure of stri