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Notes for MATH/COSC 1056 Mathematical Induction
August 5, 2005 B.G.Adams
An important proof technique is mathematical induction. A proof by induction begins with a propositional function S(n) defined for all integers n m for some smal
My Rendition of Hunter's Strong Inductive Proof of The Deduction Theorem for PS Branden Fitelson
02/13/07
Theorem. Let be an arbitrary set of formulas of P, and let A and B be arbitrary formulas of P. If {A}
PS
B, then
PS
A B.
If there is a
The Logical Roots of Indeterminacy
101
The Logical Roots of Indeterminacy
GILA SHER
of entire domain B [ t k ~iverse an "LS model" of set theory] can already beenumerated by means of the fin^;^ positive integers? (Skolem 1922, p. 295) However, the
A R G U M E N T S C O N C E R N I N G SCIENTIFIC REALISM
7
Arguments Concerning Scientific Realism
The rigour of science requires that we distinguish well the undraped figure of nature itself from the gay-coloured vesture with which we clothe i t a
Preface
My main aim is to make accessible to readers without any specialist training in mathematics, and with only an elementary knowledge of modern logic, complete proofs of the fundamental metatheorems of standard (i.e. basically truth-functional)
Symposi.urn:
ON TH; ONTOLOGICAL SIGNIFICANCE O F T H E LoWENHEIM-SKOLEM THEOREM
I share with the previous speaker the conviction that the Lowenheim-Skolem theorem has no direct philosophical implications. This phrase should be clarified. What is im
Gdel's Metatheorem (45.17) and the Strong Completeness Theorem for FOTs (46.2) Branden Fitelson
04/12/07
Before getting to the salient proofs, it's important to understand Hunter's terminology "consistent set of WFFs of a first order theory K". For
Henkin's Model and Metatheorem 45.14 Branden Fitelson
04/10/07
Henkin's Model. Let T be a consistent, negation-complete, and closed first order theory. Henkin's model M is a a denumerable interpretation for T such that for each WFF A of T , A is tru
Completeness in Propositional vs Predicate Logic Branden Fitelson
04/03/07
In Section 2 of Hunter, we examined a Henkin-type proof of the completeness of the formal system PS of propositional logic. In section 3, completeness of the system QS of pre
A Proper Inductive Proof of the Interpolation Theorem for P Branden Fitelson
02/14/07
Theorem. Let A and B be formulas of P , such that (1) they share at least one propositional symbol in common, and (2) P A B. For any two such formulas of P , ther
Details of Hunter's "Informal" Proof of Craig's Interpolation Theorem for P Branden Fitelson
02/06/05
Hunter's proof of Craig's Interpolation theorem for P is a bit opaque. Here's a more detailed version of his proof, which I sketched in class on Fr
74
Chapter 3: Semantics for Sentential Logic
7 Expressive completeness
At the end of 1 in Chapter 2 we claimed that our five sentential connectives `~', `', `&', `' and `' are all we need in sentential logic, since other sentential connectives are
Some Remarks and Extra-Credit Exercises Concerning the Deductive System of Hi z Branden Fitelson
03/04/05
Hi 's system (H) for propositional logic consists of the following axiom and inference rule schemata for P: z Axiom schemata for H: (HA1) (HA2)
k-Validity vs Validity in Q: A formula of Q that is k-valid for all k, but not valid Branden Fitelson
03/13/07
Consider the following three formulas of Q [where, as always, p q p x x x [(F x x F x x ) F x x ]
(p q), and
p
p]:
In more s
258
APPENDIX
.
Case 1: All of 'zl', ' a ' , . without end turn up in sequents of S. It is evident from our general method of generating sequents that, given any sequent Q (of S) whose first.quantifier is universal, an instantiation ofQ with respec
Notes on Isomorphism, Elementary Equivalence, the Generalized Lwenheim-Skolem-Tarski Theorem, Categoricity, Non-Standard Models, etc. Kenny Easwaran
05/10/05
1 Preliminaries and Setup
I will use the notation ", rather than "F 1 ", for the two-place
134
Axiomatization of Truth-Functional Logic
28.
Metatheory of System P (I)
135
that the lemma holds for any wff A b in which b occurs. And if b does not occur in A b, we know that the lemma holds for A b, which then falls under the special case
Henkin's Strong Completeness Theorem for PS: The Big Picture of Hunter's Proof of 32.14
P
A
32.14
PS
A
Contraposition in metatheory.
PS
A
32.14
P
A
P.
32.7
Definition of
{A} is p-consistent {A} is m-consistent
32.12 {
Notes on "Skolem's Paradox" and its Philosophical Implications Branden Fitelson
04/18/07
1
1.1
Hunter on the "Paradox" and Its Implications
Hunter on the Upward LST and Number-Theoretic Concepts
I begin with some of the material from pages 205208
HISTORY AND PHILOS('t'HY
O F LOGIC, 6 (1985). 75-89
special constructions always stay within logicism? Whitehead was as pat as Russell in his lecture: 'The whole of mathematics is here.', he announced confidently, 'it is mathematics, neither more n
Reflections on Skolem's Relativity of Set-Theoretical Concepts!
IGNACIO JANE"
From 1922 onwards Skolem maintained that set-theoretical concepts are relative (in a sense of 'relative' that we must discern). In 1958 he viewed all mat.homatlcal notions
Part (a) of Hunter's Proof of Henkin's Completeness Theorem for P S Branden Fitelson
02/21/07
The first part of Henkin's completeness proof involves proving the following seven theorem schemas. 1. 2. 3. 4. 5. 6. 7.
PS PS PS PS PS PS PS
AA A (B A)
Philosophy 140A Take-Home Mid-Term Branden Fitelson
03/20/07
You are to answer all six (6) exercises on this take-home exam. Your solutions are due on Tuesday, April 10 at 4pm. You may work in groups on this exam (with the usual rules and procedures
A More Straightforward Proof of Metatheorem 40.12 Michael Caie
03/20/07
40.12: Let I be an arbitrary interpretation with domain D. Let A be an arbitrary wff. Let s and s be two sequences such that, for each free variable v in A, if v is the kth vari
My Rendition of Hunter's Proof of Metatheorem 45.12 Branden Fitelson
03/22/07
Theorem. Let K be a consistent first order theory. And, let K = K+{}, where is an arbitrary (particular) well-formed formula of the following form (added as a new proper
Some History Surrounding our Deductive Apparatus for P Branden Fitelson
02/07/07
In the Begriffsschrift, Frege gives the following deductive apparatus (PS ) for P : Six (6) Axiom Schemata: (PS1 ) A (B A) (PS2 ) (A (B C) (A B) (A C) (PS3 ) (
Part (d) of Hunter's Proof of Henkin's Completeness Theorem for P S Branden Fitelson
02/21/07
The Lindenbaum Construction. We assume that we have an enumeration A1 , A2 , . . . An , . . . of all the formulas Ai of P . [This is part (c) of Hunter's p