mathematical logic - basic

7/24/2019 Mathematical Logic - Basic

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Mathematical Logic.

Proof Theory.

Recursion Theory.

Model Theory.Set Theory.

Core Logic 2005/06-1ab p. 2/


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Model Theory (1).

Syntax. Symbols Formal Proof

Semantics. Interpretations Truth |=

Gdels Completeness Theorem. = |=for first orderlogic.

More precisely: IfT is any first-order theory and anysentence, then the following are equivalent:

1. T , and

2. for all Msuch that M|=T, we have that M|=.



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Model Theory (2).

IfTis any fi rst-order theory and any sentence, then the following are equivalent:

1. T, and

2. for allM such thatM |=T, we have thatM |=.

For a set of sentencesT, letMod(T)be the class of modelsofT. For a class of structuresMletThy(M)be the class ofsentences true in all structures inM.

Then:

Thy(Mod(T))is the deductive closure ofT.

It is not true thatMod(Thy(M)) =M: letM:={N}, thenthere are models N|= Th(N)such that N= N.



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Products (1).

LetL={fn, Rm; n, m}be a first-order language andSbe aset.

Suppose that for everyiS, we have anL-structure

Mi=Mi, fin, R

im; n, m.

LetMS :=

iSMi. ForX0,...,Xk M, we let

fSn(X0,...,Xk)(i) :=fin(X0(i),...,Xk(i))and

RS

m(X0,...,Xk) : iS(Ri

m(X0(i),...,Xk(i)).



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Products (2).

In general, classes of structures are not closed underproducts:

LetLF:={+, , 0, 1}be the language of fields andFbethe field axioms. LetS={0, 1}and M0= M1 = Q. ThenMS= QQ is not a field: 1, 0 QQ doesnt have aninverse.

Theorem(Birkhoff, 1935). If a class of algebras isequationally definable, then it is closed under products.

Garrett Birkhoff(1884-1944)

GarrettBirkhoff, On the structure of abstract algebras,Proceedings of the Cambridge

Philosophical Society31 (1935), p. 433-454



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Ultraproducts (1).

SupposeSis a set, Mi is anL-structure andU is anultrafilter onS.

DefineU onMSby

XU Y : {i ; X(i) =Y(i)} U,

and letMU :=MS/U.

The functionsfSn and the relationsRSm are welldefined on

MU (i.e., ifXU Y, thenfSn(X)U f

Sn(Y)), and so they

induce functions and relationsfUn andRUm onMU.

We call

MU:= Ult(Mi; iS, U) :=MU, fUn, R

Um; n, m

theultraproduct of the sequenceMi; iSwithU.Core Logic 2005/06-1ab p. 8/


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Ultraproducts (2).

Theorem(os.) LetMi; iSbe a family ofL-structuresandUbe an ultrafilter onS. Let be anL-sentence. Thenthe following are equivalent:

1. MU |=, and

2. {iS; Mi|=} U.

Applications.If for all i S,Mi is a fi eld, thenMUis a fi eld.

LetS= N. Sets of the form{n ; N n}are calledfi nal segments. An ultrafi lter Uon

N is callednonprincipalif it contains all fi nal segments. If Mn; n Nis a family of

L-structures,Ua nonprincipal ultrafi lter, and an (infi nite) set of sentences such that

each element is eventually true, thenMU |= .

Nonstandard analysis(Robinson). LetL be the language of fi elds with an additional

0-ary function symbol c. LetMi |= Th(R) {c= 0 c < 1

i}. ThenMUis a model of

Th(R)plus there is an infi nitesimal.



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Tarski (1).

Alfred Tarski1902-1983

Teitelbaum(until c. 1923).

1918-1924. Studies in Warsaw. Student of Lesniewski.

1924. Banach-Tarski paradox.

1924-1939. Work in Poland.

1933. The concept of truth in formalized languages.

From 1942 at the University of California at Berkeley.

Students. 1946. Bjarni Jnsson (b. 1920). 1948. Julia Robinson (1919-1985). 1954.

Bob Vaught (1926-2002). 1957. Solomon Feferman (b. 1928).1957. Richard

Montague (1930-1971). 1961. Jerry Keisler. 1961. Donald Monk (b. 1930). 1962.

Haim Gaifman. 1963. William Hanf.



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Tarski (2).

Undefinability of Truth.If a language can correctly refer to its own sentences,then the truth predicate is not definable.

Limitative Theorems.Provability Truth Computability

1931 1933 1935

Gdel Tarski Turing



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Tarski (2).

Undefinability of Truth.

Algebraic Logic.

Leibnizcalled for an analysis of relations (Plato istaller than Socrates Plato is tall in as much asSocrates is short).

Relation Algebras: Steve Givant, Istvn Nmeti,

Hajnal Andrka, Ian Hodkinson, Robin Hirsch,Maarten Marx.

Cylindric Algebras: Don Monk, Leon Henkin, Ian

Hodkinson, Yde Venema, Nick Bezhanishvili.



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Tarski (2).

Undefinability of Truth.

Algebraic Logic.

Logic and Geometry.A theoryTadmits elimination of quantifiersif everyfirst-order formula isT-equivalent to a quantifier-freeformula (Skolem, 1919).

1955. Quantifier elimination for the theory of realnumbers (real-closed fields).

Basic ideas of modernalgebraic model theory.

Connections to theoretical computer science:running time of the quantifier elimination algorithms.



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Back to Set Theory for a while:

Applications of Set Theory in the foundations ofmathematics:(Remember Hausdorffs question in pure set theory: are there regular limit cardinals?)

Vitalis construction of a non-Lebesgue measurable set (1905).

Hausdorffs Paradox: theBanach-Tarski paradox(1926).

Banachs generalized measure problem (1930): existence ofreal-valued measurable

cardinals.

Banach connects the existence of real-valued measurable cardinals to Hausdorffs

question about inaccessibles:

if Banachs measure problem has a solution, then Hausdorffs answer is Yes.

Ulams notion of ameasurable cardinalin terms of ultrafi lters.



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Early large cardinals.

Weakly inaccessibles (Hausdorff, 1914).

Inaccessibles (Zermelo, 1930).

Real-valued measurables (Banach, 1930).Measurables (Ulam, 1930).

measurable

real-valued meas.

GCH

inaccessible

weakly inaccessible

Question. Are these notions different? Can we prove thatthe least inaccessible is not the least measurable?



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Ultraproducts in Set Theory.

Recall:A cardinal is calledmeasurableif there is a-complete nonprincipal ultrafi lter on.

Idea: Apply the theory of ultraproducts to the ultrafilterwitnessing measurability.

Let Vbe a model of set theory and V|=is measurable.LetUbe the ultrafilter witnessing this. Define M:= Vforalland MU:= Ult(V, U).

By os, MUis again a model of set theory with ameasurable cardinal.

Theorem(Scott / Tarski-Keisler, 1961). If is measurable,then there is some < such thatis inaccessible.

Corollary.The least measurable is not the leastinaccessible.



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More on large cardinals.

Reflection.Some properties of a large cardinal reflectdown to some (many, almost all) cardinals < .

Lvy(1960);Montague(1961). Reflection Principle.Hanf(1964). Connecting large cardinal analysis to infi nitary logic.

Gaifman(1964);Silver(1966). Connecting large cardinals and inner

models of constructibility (iterated ultrapowers).



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Hilberts First Problem.

Is theContinuum Hypothesis(every set of reals is eithercountable or in bijection with the set of all reals) true?



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Gdels Constructible Universe (1).

In 1939, Gdel constructed theconstructible universe Land proved:

Theorem(Gdel; 1938). L|=ZFC + CH.

Corollary. IfZFis consistent, then ZFC + CHis consistent.

Consequences.

CHcannot be refuted in ZFC.

The system ZFC + CHcannot be logically stronger than ZF,i.e.,

ZFC + CHCons(ZF).

L is tremendously important for the investigation of logical strength. It turns out that if

there is a measurable cardinal, then L |=there are inaccessible but no measurable

cardinals.

L is aminimal model of set theory.



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Gdels Constructible Universe (2).

A new axiom?V=L. The set-theoretic universe isminimal.

Gdel Rephrased. ZF +V=LAC + CH.

Possible solutions.

Prove V=LfromZF.

Assume V=Las an axiom. (V=Lis generally notaccepted as an axiom of set theory.)

Find a different proof ofACandCHfromZF.

ProveACandCHto be independent by creating modelsofZF + AC,ZF + CH, andZFC + CH.



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Cohen.

Paul Cohen(b. 1934)

Technique of Forcing(1963). Take a modelMofZFCanda partial order P M. Then there is a model construction of

a new modelMP

, theforcing extension. By choosing Pcarefully, we can control properties ofMP.

Let > 1. If P is the set of finite partial functions from

into2, thenMP

|=CH.Theorem(Cohen). ZFCCH.

Theorem(Cohen). ZFAC.



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Solovay.

Robert Solovay1962. Correspondence with Mycielski about theAxiom of

Determinacy.

1963. Development of Forcing as a method.1963. Solves the measure problem: it is consistent with ZFthat al

sets are Lebesgue measurable.

1964. PhD University of Chicago (advisor: Saunders Mac Lane).

1975. Baker-Gill-Solovay: There are oraclesp and q such thatPp = NPp and Pq = NPq .

1976. Solovay-Woodin: Solution of the Kaplansky problem in the

theory of Banach algebras.

1977. Solovay-Strassen algorithm for primality testing.


N h i h i f h i ?


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Now, what is the size of the continuum?

Gdels Programme.1947. What is Cantors Continuum Problem?

Use new axioms (in particular large cardinal axioms) in

order to resolve questions undecidable in ZF.

Lvy-Solovay(1967).Large Cardinals dont solve the continuum problem.

More about this in two weeks.


M d l l i (2)


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Modal logic (2).

Modalities as operators.McColl (late XIXth century); Lewis-Langford (1932). asan operator on propositional expressions:

Possibly.

for the dual operator:

Necessarily.

Iterated modalities:

It is necessary thatis possible.


M d l l i (3)


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Modal logic (3).

What modal formulas should be axioms? This depends onthe interpretation ofand .Example. (axiom T).

Necessity interpretation. Ifis necessarily true, then itis true.

Epistemic interpretation. Ifpknows that, thenistrue.

Doxastic interpretation. Ifpbelieves that, thenistrue.

Deontic interpretation. Ifis obligatory, thenis true.


E l d l ti


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Early modal semantics.

Topological Semantics(McKinsey / Tarski).LetX, be a topological space andV : N (X)avaluation for the propositional variables.

X,,x,V |=if and only ifxis in the closure of{z; X , , z , V |=}.

X, |=if for allxXand all valuationsV,

X,,x,V |=.

Theorem(McKinsey-Tarski; 1944). X, |=if and only ifS4.

(S4={T, })


K i k


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Kripke.

Saul Kripke

(b. 1940)

SaulKripke, A completeness theorem in modal logic,Journal of Symbolic Logic24 (1959), p. 1-14.

Naming and Necessity.


K i k ti (1)


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Theorem(Kripke).

1. Tif and only if for all reflexive frames M, we haveM|=.

2. S4if and only if for all reflexive and transitive framesM, we have M|=.

3. S5if and only if for all frames Mwith an

equivalence relationR, we have M|=.


mathematical logic - basic

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