Mid-Term Review
Here I try to list all the important terms, distinctions, symbols, and results from the first half of the course. It wouldn't help much as a study guide if I listed everything we covered. You can get that by re-reading Hunter. I've tried to limit the list to what's most important. Believe it or not, I've omitted a lot.
This list covers the introductory unit on infinite sets and the unit on the metatheory of truth-functional propositional logic (TFPL).
The topics are only roughly in the order in which we encountered them. I've adjusted this order when I thought it important to cluster related topics together. I've put in some blank lines to separate clusters from one another.
I believe that all the technical terms on this list are defined in my glossary. If any are not, please let me know and I'll revise the glossary.
For a review sheet for the second half of the course, on predicate logic, see the final exam review.
- set notation
= {A, B,
C} (set gamma consists of members A, B, and C)
- A
(A is a member of set gamma)
(set delta is a subset of set gamma)
(set delta is a proper subset of set gamma)
(the union of set delta and set gamma)
- Ø (the null set)
- formal language
- alphabet / grammar (what Hunter calls the set of symbols and formation rules)
- well-formed formula, or wff
- why formal languages are "formal"
- formal language / metalanguage
- metalanguage variables that range over wffs of the formal language
- deductive apparatus
- axioms
- rules of inference (what Hunter calls transformation rules)
- formal system
- why formal systems are "formal"
- abbreviation, "iff"
- theorem / metatheorem
- proof within a system (proof of a theorem) / proof about a system (proof of a metatheorem)
- model theory / proof theory
- semantics / syntax
- effective method
- decidable set
- one-to-one correspondence of sets
- cardinality of a set
- notation, |S| (not used in Hunter)
- subset / proper subset
- power set
- notation, *S
- set / sequence
- enumeration / effective enumeration
- enumerable set / effectively enumerable set
- finite / countable / denumerable / uncountable
- natural numbers / integers / rational numbers / real numbers
- 11.2, the power set of the natural numbers is uncountable
- diagonalization
- Cantor's theorem (the cardinality of an abitrary set A is greater than the cardinality of A) (Hunter 24-25)
- notation,
0, 1 , 3 ...
- negative proof (also called indirect proof, apagogical proof)
- 14.1 - 14.3, informal proof of the incompleteness of any finitary formal system of number theory (Hunter 28-30)
- arithmetization
- why there are only countably many strings of symbols of finite length (hence countably many wffs, theorems, names, sentences, descriptions, books...)
- proof that the set of rational numbers is countable (not in Hunter)
- notation, c (for the cardinality of the continuum)
- A4, the set of real numbers is uncountable
- A7, the power set of the set of natural numbers has cardinality c
- A9, the set of points inside a square has cardinality c
- A16, the power set of the set of natural numbers has the cardinality 2^0
- continuum hypothesis / generalized continuum hypothesis
- why the claim c = 1
presupposes the continuum hypothesis
- function
- domain / range of a function
- 1-place function / n-place functions
- total / partial functions
- computable / uncomputable functions
- truth functions
- truth tables
- truth values
- connectives
- truth-functional connectives
- truth-functional propositional logic (what I've been calling TFPL)
- interepretation / model
- model of a wff / model of a set of wffs
- truth for an interpretation (or truth for I)
- why truth for I must be defined separately for every connective in the language
- tautology / contingency / contradiction
- logically valid formula
- notation,
A (A is a
tautology or logically valid formula)
- notation, A (A is a semantic
consequence of the wffs in set )
- model-theoretic consistency (m-con) / model-theoretic inconsistency (m-incon)
- disjunctive normal form (DNF)
- why every wff of language P can be expressed in DNF
- sets of connectives adequate to express all truth functions
- 21.1, that the set {~,
} is adequate (this is the set
used in language P)
- 21.5, that the dagger function alone is adequate
- 21.6, that the stroke function alone is adequate
- 21.12, that the stroke and dagger functions are the only dyadic connectives adequate by themselves
- language P / system PS
- axiom / axiom schema
- modus ponens (the only rule of inference for PS)
- proof / derivation
- why proofs and derivations are finite by definition
- why proofs and derivations are defined syntactically (without reference to truth)
- why axioms are theorems
- notation,
A (A is a
theorem)
- notation, A (A is a syntactic
consequence of the wffs in set )
- semantic consequence () / syntactic consequence
()
- why we need and
in the metalanguage
when we already have in the formal language
- m-consistency / p-consistency
- why one model is enough for m-consistency
- 23.2, if A, then A (what I've been calling
monotonicity) (variation on 20.2, for )
- 23.4, if A and AB, then B (what I've been calling meta-modus-ponens) (variation on 20.4, for )
- simple consistency / absolute consistency
- 24.1, simple con implies absolute con (for most systems)
- 24.2, absolute con implies simple con (for most systems)
- why contradictions imply everything (for most systems) (not in Hunter)
- Post's model-theoretic proof of the consistency of TFPL (Hunter 79)
- Lukasiewicz's proof-theoretic proof of the consistency of TFPL (Hunter 81)
- logical hereditary (not Hunter's term)
- dilemma for consistency proofs: circularity v. relativity (not in Hunter)
- mathematical induction
- basis / induction step / induction hypothesis
- 26.1, the deduction theorem for PS
- conditional proof (not in Hunter)
- semantic completeness
- 28.3, if A, then
A
- why truth tables are an effective method for testing tautologousness
- why, given AB, all
models of A are models of B
- why all the models of a set of wffs are models of an arbitrary subset of wffs
- lemma
- 31.14, lemma for Kalmar's proof of semantic completeness (that all compound wffs can be expressed as derivations)
- 31.15, if A, then
A (Kalmar's proof)
- 32.15, if A, then
A (Henkin's proof)
- maximal p-consistent set
- Lindenbaum's lemma for PS (that every p-con set of wffs is a subset of some maximal p-con set of wffs)
- 32.7, {~A} is p-inconsistent iff
A
- 32.11, enumeration theorem for language P
- 32.6, m-consistency implies p-consistency
- 32.13, p-consistency implies m-consistency
- 28.4, if A, then A
- 32.14, if A, then A (strong completeness)
- syntactic completeness
- decidable wff / decidable system
- decidable system / system with an effective proof procedure
- 33.1, PS is syntactically complete
- 33.2, if ~A is a wff of P and a non-theorem of PS, then A can be added to PS as an axiom without creating inconsistency (what I've been calling the non-standardness theorem for PS)
- 34.1, PS is decidable
- independence of axioms (and axiom schemata)
- standard / non-standard interpretations
- 36.1 - 36.3, the axiom schemata of PS are independent
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