Second-order arithmeticFrom Wikipedia, the free encyclopedia

Contents

1 Axiom of determinacy 11.1 Types of game that are determined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Incompatibility of the axiom of determinacy with the axiom of choice . . . . . . . . . . . . . . . . 11.3 Innite logic and the axiom of determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Large cardinals and the axiom of determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Equiconsistency 42.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Consistency strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Primitive recursive arithmetic 63.1 Language and axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Logic-free calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Reverse mathematics 94.1 General principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.1.1 Use of second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.2 The big ve subsystems of second order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2.1 The base system RCA0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2.2 Weak Knigs lemma WKL0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2.3 Arithmetical comprehension ACA0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2.4 Arithmetical transnite recursion ATR0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2.5 11 comprehension 11-CA0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.3 Additional systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.4 -models and -models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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4.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Second-order arithmetic 155.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.1.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.1.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.1.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.1.4 The full system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.2 Models of second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.3 Denable functions of second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.4 Subsystems of second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.4.1 Arithmetical comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.4.2 The arithmetical hierarchy for formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.4.3 Recursive comprehension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.4.4 Weaker systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.4.5 Stronger systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.5 Projective Determinacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.6 Coding mathematics in second-order arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6 ZermeloFraenkel set theory 226.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226.2 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.2.1 1. Axiom of extensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.2.2 2. Axiom of regularity (also called the Axiom of foundation) . . . . . . . . . . . . . . . . 236.2.3 3. Axiom schema of specication (also called the axiom schema of separation or of restricted

comprehension) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236.2.4 4. Axiom of pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.2.5 5. Axiom of union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246.2.6 6. Axiom schema of replacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256.2.7 7. Axiom of innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.2.8 8. Axiom of power set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.2.9 9. Well-ordering theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

6.3 Motivation via the cumulative hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276.4 Metamathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.4.1 Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.5 Criticisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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6.9 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 316.9.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.9.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.9.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Chapter 1

Axiom of determinacy

The axiom of determinacy (abbreviated as AD) is a possible axiom for set theory introduced by Jan Mycielski andHugo Steinhaus in 1962. It refers to certain two-person games of length with perfect information. AD states thatevery such game in which both players choose natural numbers is determined; that is, one of the two players has awinning strategy.The axiom of determinacy is inconsistent with the axiom of choice (AC); the axiom of determinacy implies thatall subsets of the real numbers are Lebesgue measurable, have the property of Baire, and the perfect set property.The last implies a weak form of the continuum hypothesis (namely, that every uncountable set of reals has the samecardinality as the full set of reals).Furthermore, AD implies the consistency of ZermeloFraenkel set theory (ZF). Hence, as a consequence of theincompleteness theorems, it is not possible to prove the relative consistency of ZF + AD with respect to ZF.

1.1 Types of game that are determinedNot all games require the axiom of determinacy to prove them determined. Games whose winning sets are closed aredetermined. These correspond to many naturally dened innite games. It was shown in 1975 by Donald A. Martinthat games whose winning set is a Borel set are determined. It follows from the existence of sucient large cardinalsthat all games with winning set a projective set are determined (see Projective determinacy), and that AD holds inL(R).

1.2 Incompatibility of the axiom of determinacy with the axiom of choiceThe set S1 of all rst player strategies in an -game G has the same cardinality as the continuum. The same is trueof the set S2 of all second player strategies. We note that the cardinality of the set SG of all sequences possible in Gis also the continuum. Let A be the subset of SG of all sequences which make the rst player win. With the axiomof choice we can well order the continuum; furthermore, we can do so in such a way that any proper initial portiondoes not have the cardinality of the continuum. We create a counterexample by transnite induction on the set ofstrategies under this well ordering:We start with the set A undened. Let T be the time whose axis has length continuum. We need to consider allstrategies {s1(T)} of the rst player and all strategies {s2(T)} of the second player to make sure that for every strategythere is a strategy of the other player that wins against it. For every strategy of the player considered we will generatea sequence which gives the other player a win. Let t be the time whose axis has length 0 and which is used duringeach game sequence.

1. Consider the current strategy {s1(T)} of the rst player.2. Go through the entire game, generating (together with the rst players strategy s1(T)) a sequence {a(1), b(2),

a(3), b(4),...,a(t), b(t+1),...}.3. Decide that this sequence does not belong to A, i.e. s1(T) lost.

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2 CHAPTER 1. AXIOM OF DETERMINACY

4. Consider the strategy {s2(T)} of the second player.

5. Go through the next entire game, generating (together with the second players strategy s2(T)) a sequence{c(1), d(2), c(3), d(4),...,c(t), d(t+1),...}, making sure that this sequence is dierent from {a(1), b(2), a(3),b(4),...,a(t), b(t+1),...}.

6. Decide that this sequence belongs to A, i.e. s2(T) lost.

7. Keep repeating with further strategies if there are any, making sure that sequences already considered do notbecome generated again. (We start from the set of all sequences and each time we generate a sequence andrefute a strategy we project the generated sequence onto rst player moves and onto second player moves, andwe take away the two resulting sequences from our set of sequences.)

8. For all sequences that did not come up in the above consideration arbitrarily decide whether they belong to A,or to the complement of A.

Once this has been done we have a game G. If you give me a strategy s1 then we considered that strategy at sometime T = T(s1). At time T, we decided an outcome of s1 that would be a loss of s1. Hence this strategy fails. Butthis is true for an arbitrary strategy; hence the axiom of determinacy and the axiom of choice are incompatible.

1.3 Innite logic and the axiom of determinacyMany dierent versions of innitary logic were proposed in the late 20th century. One reason that has been given forbelieving in the axiom of determinacy is that it can be written as follows (in a version of innite logic):8G Seq(S) :8a 2 S : 9a0 2 S : 8b 2 S : 9b0 2 S : 8c 2 S : 9c0 2 S::: : (a; a0; b; b0; c; c0:::) 2 G OR9a 2 S : 8a0 2 S : 9b 2 S : 8b0 2 S : 9c 2 S : 8c0 2 S::: : (a; a0; b; b0; c; c0:::) /2 GNote: Seq(S) is the set of all ! -sequences of S. The sentences here are innitely long with a countably innite list ofquantiers where the ellipses appear.In an innitary logic, this principle is therefore a natural generalization of the usual (de Morgan) rule for quantiersthat are true for nite formulas, such as 8a : 9b : 8c : 9d : R(a; b; c; d) OR 9a : 8b : 9c : 8d : :R(a; b; c; d) .

1.4 Large cardinals and the axiom of determinacyThe consistency of the axiom of determinacy is closely related to the question of the consistency of large cardinalaxioms. By a theorem of Woodin, the consistency of ZermeloFraenkel set theory without choice (ZF) together withthe axiom of determinacy is equivalent to the consistency of ZermeloFraenkel set theory with choice (ZFC) togetherwith the existence of innitely many Woodin cardinals. Since Woodin cardinals are strongly inaccessible, if AD isconsistent, then so are an innity of inaccessible cardinals.Moreover, if to the hypothesis of an innite set of Woodin cardinals is added the existence of a measurable cardinallarger than all of them, a very strong theory of Lebesgue measurable sets of reals emerges, as it is then provable thatthe axiom of determinacy is true in L(R), and therefore that every set of real numbers in L(R) is determined.

1.5 See also Axiom of real determinacy (ADR) AD+, a variant of the axiom of determinacy formulated by Woodin Axiom of quasi-determinacy (ADQ) Martin measure

1.6. REFERENCES 3

1.6 References Jech, Thomas (2002). Set theory, third millennium edition (revised and expanded). Springer. ISBN 3-540-44085-2.

Kanamori, Akihiro (2000). The Higher Innite (2nd ed.). Springer. ISBN 3-540-00384-3. Martin, Donald A.; Steel, John R. (Jan 1989). A Proof of Projective Determinacy. Journal of the American

Mathematical Society 2 (1): 71125. doi:10.2307/1990913. JSTOR 1990913. Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. ISBN 0-444-70199-0. Mycielski, Jan; Steinhaus, H. (1962). A mathematical axiom contradicting the axiom of choice. Bulletin

de l'Acadmie Polonaise des Sciences. Srie des Sciences Mathmatiques, Astronomiques et Physiques 10: 13.ISSN 0001-4117. MR 0140430.

Woodin,W.Hugh (1988). Supercompact cardinals, sets of reals, andweakly homogeneous trees. Proceedingsof the National Academy of Sciences of theUnited States of America 85 (18): 65876591. doi:10.1073/pnas.85.18.6587.PMC 282022. PMID 16593979.

1.7 Further reading Philipp Rohde, On Extensions of the Axiom of Determinacy, Thesis, Department of Mathematics, Universityof Bonn, Germany, 2001

Telgrsky, R.J. Topological Games: On the 50th Anniversary of the Banach-Mazur Game, Rocky Mountain J.Math. 17 (1987), pp. 227276. (3.19 MB)

Chapter 2

Equiconsistency

In mathematical logic, two theories are equiconsistent if the consistency of one theory implies the consistency of theother theory, and vice versa. In this case, they are, roughly speaking, as consistent as each other.In general, it is not possible to prove the absolute consistency of a theory T. Instead we usually take a theory S, believedto be consistent, and try to prove the weaker statement that if S is consistent then T must also be consistentif wecan do this we say that T is consistent relative to S. If S is also consistent relative to T then we say that S and T areequiconsistent.

2.1 ConsistencyIn mathematical logic, formal theories are studied as mathematical objects. Since some theories are powerful enoughto model dierent mathematical objects, it is natural to wonder about their own consistency.Hilbert proposed a program at the beginning of the 20th century whose ultimate goal was to show, using mathematicalmethods, the consistency of mathematics. Since most mathematical disciplines can be reduced to arithmetic, theprogram quickly became the establishment of the consistency of arithmetic bymethods formalizable within arithmeticitself.Gdel's incompleteness theorems show that Hilberts program cannot be realized: If a consistent recursively enumer-able theory is strong enough to formalize its own metamathematics (whether something is a proof or not), i.e. strongenough to model a weak fragment of arithmetic (Robinson arithmetic suces), then the theory cannot prove its ownconsistency. There are some technical caveats as to what requirements the formal statement representing the meta-mathematical statement The theory is consistent needs to satisfy, but the outcome is that if a (suciently strong)theory can prove its own consistency then either there is no computable way of identifying whether a statement is evenan axiom of the theory or not, or else the theory itself is inconsistent (in which case it can prove anything, includingfalse statements such as its own consistency).Given this, instead of outright consistency, one usually considers relative consistency: Let S and T be formal theories.Assume that S is a consistent theory. Does it follow that T is consistent? If so, then T is consistent relative to S. Twotheories are equiconsistent if each one is consistent relative to the other.

2.2 Consistency strengthIf T is consistent relative to S, but S is not known to be consistent relative to T, then we say that S has greaterconsistency strength than T. When discussing these issues of consistency strength the metatheory in which thediscussion takes places needs to be carefully addressed. For theories at the level of second-order arithmetic, thereverse mathematics program has much to say. Consistency strength issues are a usual part of set theory, since thisis a recursive theory that can certainly model most of mathematics. The usual set of axioms of set theory is calledZFC. When a set theoretic statement A is said to be equiconsistent to another B, what is being claimed is that in themetatheory (Peano Arithmetic in this case) it can be proven that the theories ZFC+A and ZFC+B are equiconsistent.Usually, primitive recursive arithmetic can be adopted as the metatheory in question, but even if the metatheory is

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2.3. SEE ALSO 5

ZFC (for Ernst Zermelo and Abraham Fraenkel with Zermelos axiom of choice) or an extension of it, the notion ismeaningful. Thus, the method of forcing allows one to show that the theories ZFC, ZFC+CH and ZFC+CH are allequiconsistent.When discussing fragments of ZFC or their extensions (for example, ZF, set theory without the axiom of choice, orZF+AD, set theory with the axiom of determinacy), the notions described above are adapted accordingly. Thus, ZFis equiconsistent with ZFC, as shown by Gdel.The consistency strength of numerous combinatorial statements can be calibrated by large cardinals. For example,the negation of Kurepas hypothesis is equiconsistent with an inaccessible cardinal, the non-existence of special !2 -Aronszajn trees is equiconsistent with aMahlo cardinal, and the non-existence of!2 -Aronszajn trees is equiconsistentwith a weakly compact cardinal.[1]

2.3 See also Large cardinal property

2.4 References[1] Kunen, Kenneth (2011), Set theory, Studies in Logic 34, London: College Publications, p. 225, ISBN 978-1-84890-

050-9, Zbl 1262.03001

Akihiro Kanamori (2003). The Higher Innite. Springer. ISBN 3-540-00384-3

Chapter 3

Primitive recursive arithmetic

Primitive recursive arithmetic, or PRA, is a quantier-free formalization of the natural numbers. It was rstproposed by Skolem[1] as a formalization of his nitist conception of the foundations of arithmetic, and it is widelyagreed that all reasoning of PRA is nitist. Many also believe that all of nitism is captured by PRA,[2] but othersbelieve nitism can be extended to forms of recursion beyond primitive recursion, up to 0,[3] which is the proof-theoretic ordinal of Peano arithmetic. PRAs proof theoretic ordinal is , where is the smallest transnite ordinal.PRA is sometimes called Skolem arithmetic.The language of PRA can express arithmetic propositions involving natural numbers and any primitive recursivefunction, including the operations of addition, multiplication, and exponentiation. PRA cannot explicitly quantifyover the domain of natural numbers. PRA is often taken as the basic metamathematical formal system for prooftheory, in particular for consistency proofs such as Gentzens consistency proof of rst-order arithmetic.

3.1 Language and axiomsThe language of PRA consists of:

A countably innite number of variables x, y, z,.... The propositional connectives; The equality symbol =, the constant symbol 0, and the successor symbol S (meaning add one); A symbol for each primitive recursive function.

The logical axioms of PRA are the:

Tautologies of the propositional calculus; Usual axiomatization of equality as an equivalence relation.

The logical rules of PRA are modus ponens and variable substitution.The non-logical axioms are:

S(x) 6= 0 ; S(x) = S(y) ! x = y;

and recursive dening equations for every primitive recursive function as desired. For instance, the most commoncharacterization of the primitive recursive functions is as the 0 constant and successor function closed under projection,composition and primitive recursion. So for a (n+1)-place function f dened by primitive recursion over a n-placebase function g and (n+2)-place iteration function h there would be the dening equations:

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3.2. LOGIC-FREE CALCULUS 7

f(0; y1; : : : ; yn) = g(y1; : : : ; yn) f(S(x); y1; : : : ; yn) = h(x; f(x; y1; : : : ; yn); y1; : : : ; yn)

Especially:

x+ 0 = x x+ S(y) = S(x+ y) x 0 = 0 x S(y) = x y + x ... and so on.

PRA replaces the axiom schema of induction for rst-order arithmetic with the rule of (quantier-free) induction:

From '(0) and '(x)! '(S(x)) , deduce '(y) , for any predicate ':

In rst-order arithmetic, the only primitive recursive functions that need to be explicitly axiomatized are addition andmultiplication. All other primitive recursive predicates can be dened using these two primitive recursive functionsand quantication over all natural numbers. Dening primitive recursive functions in this manner is not possible inPRA, because it lacks quantiers.

3.2 Logic-free calculusIt is possible to formalise PRA in such a way that it has no logical connectives at all - a sentence of PRA is just anequation between two terms. In this setting a term is a primitive recursive function of zero or more variables. In 1941Haskell Curry gave the rst such system.[4] The rule of induction in Currys system was unusual. A later renementwas given by Reuben Goodstein.[5] The rule of induction in Goodsteins system is:F (0)=G(0) F (S(x))=H(x;F (x)) G(S(x))=H(x;G(x))

F (x)=G(x) :

Here x is a variable, S is the successor operation, and F, G, and H are any primitive recursive functions which mayhave parameters other than the ones shown. The only other inference rules of Goodsteins system are substitutionrules, as follows:F (x)=G(x)F (A)=G(A)

A=BF (A)=F (B)

A=B A=CB=C :

Here A, B, and C are any terms (primitive recursive functions of zero or more variables). Finally, there are symbolsfor any primitive recursive functions with corresponding dening equations, as in Skolems system above.In this way the propositional calculus can be discarded entirely. Logical operators can be expressed entirely arith-metically, for instance, the absolute value of the dierence of two numbers can be dened by primitive recursion:P (0) = 0 P (S(x)) = x

x _0 = x x _S(y) = P (x _y)jx yj =(x _y) + (y _x):

Thus, the equations x=y and |x-y|=0 are equivalent. Therefore the equations jxyj+juvj = 0and jxyjjuvj = 0express the logical conjunction and disjunction, respectively, of the equations x=y and u=v. Negation can be expressedas 1 _jx yj = 0 .

3.3 See also Elementary recursive arithmetic Heyting arithmetic

8 CHAPTER 3. PRIMITIVE RECURSIVE ARITHMETIC

Peano arithmetic Second-order arithmetic Primitive recursive function

3.4 References[1] Thoralf Skolem (1923) The foundations of elementary arithmetic in Jean van Heijenoort, translator and ed. (1967) From

Frege to Gdel: A Source Book in Mathematical Logic, 1879-1931. Harvard Univ. Press: 302-33.

[2] Tait, W.W. (1981), Finitism, Journal of Philosophy 78:524-46.

[3] Georg Kreisel (1958) Ordinal Logics and the Characterization of Informal Notions of Proof, Proc. Internat. Cong.Mathematicians: 289-99.

[4] Haskell Curry, A Formalization of Recursive Arithmetic. American Journal of Mathematics, vol 63 no 2 (1941) pp 263-282

[5] Reuben Goodstein, Logic-free formalisations of recursive arithmetic, Mathematica Scandinavica vol 2 (1954) pp 247-261

See also

Rose, H.E., On the consistency and undecidability of recursive arithmetic, Zeitschrift fr mathematischeLogik und Grundlagen der Mathematick Volume 7, pp. 124135.

3.5 External links Feferman, S (1992)What rests on what? The proof-theoretic analysis of mathematics. Invited lecture, 15th int'lWittgenstein symposium.

Chapter 4

Reverse mathematics

Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required toprove theorems of mathematics. Its dening method can briey be described as going backwards from the theoremsto the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. The reversemathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom ofchoice and Zorns lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to studypossible axioms of ordinary theorems of mathematics rather than possible axioms for set theory.Reverse mathematics is usually carried out using subsystems of second-order arithmetic, where many of its denitionsand methods are inspired by previous work in constructive analysis and proof theory. The use of second-orderarithmetic also allows many techniques from recursion theory to be employed; many results in reverse mathematicshave corresponding results in computable analysis.The programwas founded by Harvey Friedman (1975, 1976). A standard reference for the subject is (Simpson 2009).

4.1 General principlesIn reverse mathematics, one starts with a framework language and a base theorya core axiom systemthat is tooweak to prove most of the theorems one might be interested in, but still powerful enough to develop the denitionsnecessary to state these theorems. For example, to study the theorem Every bounded sequence of real numbers hasa supremum it is necessary to use a base system which can speak of real numbers and sequences of real numbers.For each theorem that can be stated in the base system but is not provable in the base system, the goal is to determinethe particular axiom system (stronger than the base system) that is necessary to prove that theorem. To show that asystem S is required to prove a theorem T, two proofs are required. The rst proof shows T is provable from S; thisis an ordinary mathematical proof along with a justication that it can be carried out in the system S. The secondproof, known as a reversal, shows that T itself implies S; this proof is carried out in the base system. The reversalestablishes that no axiom system S that extends the base system can be weaker than S while still proving T.

4.1.1 Use of second-order arithmeticMost reverse mathematics research focuses on subsystems of second-order arithmetic. The body of research inreverse mathematics has established that weak subsystems of second-order arithmetic suce to formalize almost allundergraduate-level mathematics. In second-order arithmetic, all objects can be represented as either natural numbersor sets of natural numbers. For example, in order to prove theorems about real numbers, the real numbers can berepresented as Cauchy sequences of rational numbers, each of which can be represented as a set of natural numbers.The axiom systems most often considered in reverse mathematics are dened using axiom schemes called compre-hension schemes. Such a scheme states that any set of natural numbers denable by a formula of a given complexityexists. In this context, the complexity of formulas is measured using the arithmetical hierarchy and analytical hierar-chy.The reason that reverse mathematics is not carried out using set theory as a base system is that the language ofset theory is too expressive. Extremely complex sets of natural numbers can be dened by simple formulas in the

9

10 CHAPTER 4. REVERSE MATHEMATICS

language of set theory (which can quantify over arbitrary sets). In the context of second-order arithmetic, results suchas Posts theorem establish a close link between the complexity of a formula and the (non)computability of the set itdenes.Another eect of using second-order arithmetic is the need to restrict general mathematical theorems to forms thatcan be expressed within arithmetic. For example, second-order arithmetic can express the principle Every countablevector space has a basis but it cannot express the principle Every vector space has a basis. In practical terms, thismeans that theorems of algebra and combinatorics are restricted to countable structures, while theorems of analysisand topology are restricted to separable spaces. Many principles that imply the axiom of choice in their general form(such as Every vector space has a basis) become provable in weak subsystems of second-order arithmetic when theyare restricted. For example, every eld has an algebraic closure is not provable in ZF set theory, but the restrictedform every countable eld has an algebraic closure is provable in RCA0, the weakest system typically employed inreverse mathematics.

4.2 The big ve subsystems of second order arithmeticSecond order arithmetic is a formal theory of the natural numbers and sets of natural numbers. Many mathematicalobjects, such as countable rings, groups, and elds, as well as points in eective Polish spaces, can be represented assets of natural numbers, and modulo this representation can be studied in second order arithmetic.Reverse mathematics makes use of several subsystems of second order arithmetic. A typical reverse mathematicstheorem shows that a particular mathematical theorem T is equivalent to a particular subsystem S of second orderarithmetic over a weaker subsystem B. This weaker system B is known as the base system for the result; in order forthe reverse mathematics result to have meaning, this systemmust not itself be able to prove the mathematical theoremT.Simpson (2009) describes ve particular subsystems of second order arithmetic, which he calls the Big Five, thatoccur frequently in reverse mathematics. In order of increasing strength, these systems are named by the initialismsRCA0, WKL0, ACA0, ATR0, and 11-CA0.The following table summarizes the big ve systems Simpson (2009, p.42)The subscript 0 in these names means that the induction scheme has been restricted from the full second-orderinduction scheme (Simpson 2009, p. 6). For example, ACA0 includes the induction axiom (0X n(nX n+1X)) n nX. This together with the full comprehension axiom of second order arithmetic implies the fullsecond-order induction scheme given by the universal closure of ((0) n((n) (n+1))) n (n) for anysecond order formula . However ACA0 does not have the full comprehension axiom, and the subscript 0 is areminder that it does not have the full second-order induction scheme either. This restriction is important: systemswith restricted induction have signicantly lower proof-theoretical ordinals than systems with the full second-orderinduction scheme.

4.2.1 The base system RCA0RCA0 is the fragment of second-order arithmetic whose axioms are the axioms of Robinson arithmetic, inductionfor 01 formulas, and comprehension for 01 formulas.The subsystem RCA0 is the one most commonly used as a base system for reverse mathematics. The initials RCAstand for recursive comprehension axiom, where recursive means computable, as in recursive function. Thisname is used because RCA0 corresponds informally to computable mathematics. In particular, any set of naturalnumbers that can be proven to exist in RCA0 is computable, and thus any theorem which implies that noncomputablesets exist is not provable in RCA0. To this extent, RCA0 is a constructive system, although it does not meet therequirements of the program of constructivism because it is a theory in classical logic including the excluded middle.Despite its seeming weakness (of not proving any noncomputable sets exist), RCA0 is sucient to prove a numberof classical theorems which, therefore, require only minimal logical strength. These theorems are, in a sense, belowthe reach of the reverse mathematics enterprise because they are already provable in the base system. The classicaltheorems provable in RCA0 include:

4.2. THE BIG FIVE SUBSYSTEMS OF SECOND ORDER ARITHMETIC 11

Basic properties of the natural numbers, integers, and rational numbers (for example, that the latter form anordered eld).

Basic properties of the real numbers (the real numbers are an Archimedean ordered eld; any nested sequenceof closed intervals whose lengths tend to zero has a single point in its intersection; the real numbers are notcountable).

The Baire category theorem for a complete separable metric space (the separability condition is necessary toeven state the theorem in the language of second-order arithmetic).

The intermediate value theorem on continuous real functions. The BanachSteinhaus theorem for a sequence of continuous linear operators on separable Banach spaces. Aweak version of Gdels completeness theorem (for a set of sentences, in a countable language, that is alreadyclosed under consequence).

The existence of an algebraic closure for a countable eld (but not its uniqueness). The existence and uniqueness of the real closure of a countable ordered eld.

The rst-order part of RCA0 (the theorems of the system that do not involve any set variables) is the set of theoremsof rst-order Peano arithmetic with induction limited to 01 formulas. It is provably consistent, as is RCA0, in fullrst-order Peano arithmetic.

4.2.2 Weak Knigs lemma WKL0The subsystemWKL0 consists of RCA0 plus a weak form of Knigs lemma, namely the statement that every innitesubtree of the full binary tree (the tree of all nite sequences of 0s and 1s) has an innite path. This proposition,which is known as weak Knigs lemma, is easy to state in the language of second-order arithmetic. WKL0 can alsobe dened as the principle of 01 separation (given two 01 formulas of a free variable n which are exclusive, thereis a class containing all n satisfying the one and no n satisfying the other).The following remark on terminology is in order. The term weak Knigs lemma refers to the sentence which saysthat any innite subtree of the binary tree has an innite path. When this axiom is added to RCA0, the resulting sub-system is called WKL0. A similar distinction between particular axioms, on the one hand, and subsystems includingthe basic axioms and induction, on the other hand, is made for the stronger subsystems described below.In a sense, weak Knigs lemma is a form of the axiom of choice (although, as stated, it can be proven in classicalZermeloFraenkel set theory without the axiom of choice). It is not constructively valid in some senses of the wordconstructive.To show that WKL0 is actually stronger than (not provable in) RCA0, it is sucient to exhibit a theorem of WKL0which implies that noncomputable sets exist. This is not dicult; WKL0 implies the existence of separating sets foreectively inseparable recursively enumerable sets.It turns out that RCA0 and WKL0 have the same rst-order part, meaning that they prove the same rst-ordersentences. WKL0 can prove a good number of classical mathematical results which do not follow from RCA0,however. These results are not expressible as rst order statements but can be expressed as second-order statements.The following results are equivalent to weak Knigs lemma and thus to WKL0 over RCA0:

The HeineBorel theorem for the closed unit real interval, in the following sense: every covering by a sequenceof open intervals has a nite subcovering.

The HeineBorel theorem for complete totally bounded separable metric spaces (where covering is by a se-quence of open balls).

A continuous real function on the closed unit interval (or on any compact separable metric space, as above) isbounded (or: bounded and reaches its bounds).

A continuous real function on the closed unit interval can be uniformly approximated by polynomials (withrational coecients).

12 CHAPTER 4. REVERSE MATHEMATICS

A continuous real function on the closed unit interval is uniformly continuous.

A continuous real function on the closed unit interval is Riemann integrable.

The Brouwer xed point theorem (for continuous functions on a nite product of copies of the closed unitinterval).

The separable HahnBanach theorem in the form: a bounded linear form on a subspace of a separable Banachspace extends to a bounded linear form on the whole space.

The Jordan curve theorem

Gdels completeness theorem (for a countable language).

Determinacy for open (or even clopen) games on {0,1} of length .

Every countable commutative ring has a prime ideal.

Every countable formally real eld is orderable.

Uniqueness of algebraic closure (for a countable eld).

4.2.3 Arithmetical comprehension ACA0ACA0 is RCA0 plus the comprehension scheme for arithmetical formulas (which is sometimes called the arith-metical comprehension axiom). That is, ACA0 allows us to form the set of natural numbers satisfying an arbitraryarithmetical formula (one with no bound set variables, although possibly containing set parameters). Actually, it suf-ces to add to RCA0 the comprehension scheme for 1 formulas in order to obtain full arithmetical comprehension.The rst-order part of ACA0 is exactly rst-order Peano arithmetic; ACA0 is a conservative extension of rst-orderPeano arithmetic. The two systems are provably (in a weak system) equiconsistent. ACA0 can be thought of as aframework of predicative mathematics, although there are predicatively provable theorems that are not provable inACA0. Most of the fundamental results about the natural numbers, and many other mathematical theorems, can beproven in this system.One way of seeing that ACA0 is stronger thanWKL0 is to exhibit a model ofWKL0 that doesn't contain all arithmeti-cal sets. In fact, it is possible to build a model of WKL0 consisting entirely of low sets using the low basis theorem,since low sets relative to low sets are low.The following assertions are equivalent to ACA0 over RCA0:

The sequential completeness of the real numbers (every bounded increasing sequence of real numbers has alimit).

The BolzanoWeierstrass theorem.

Ascolis theorem: every bounded equicontinuous sequence of real functions on the unit interval has a uniformlyconvergent subsequence.

Every countable commutative ring has a maximal ideal.

Every countable vector space over the rationals (or over any countable eld) has a basis.

Every countable eld has a transcendence basis.

Knigs lemma (for arbitrary nitely branching trees, as opposed to the weak version described above).

Various theorems in combinatorics, such as certain forms of Ramseys theorem.

4.3. ADDITIONAL SYSTEMS 13

4.2.4 Arithmetical transnite recursion ATR0The system ATR0 adds to ACA0 an axiom which states, informally, that any arithmetical functional (meaning anyarithmetical formula with a free number variable n and a free class variable X, seen as the operator taking X to theset of n satisfying the formula) can be iterated transnitely along any countable well ordering starting with any set.ATR0 is equivalent over ACA0 to the principle of 11 separation. ATR0 is impredicative, and has the proof-theoreticordinal 0 , the supremum of that of predicative systems.ATR0 proves the consistency of ACA0, and thus by Gdels theorem it is strictly stronger.The following assertions are equivalent to ATR0 over RCA0:

Any two countable well orderings are comparable. That is, they are isomorphic or one is isomorphic to a properinitial segment of the other.

Ulms theorem for countable reduced Abelian groups.

The perfect set theorem, which states that every uncountable closed subset of a complete separable metricspace contains a perfect closed set.

Lusins separation theorem (essentially 11 separation).

Determinacy for open sets in the Baire space.

4.2.5 11 comprehension 11-CA011-CA0 is stronger than arithmetical transnite recursion and is fully impredicative. It consists of RCA0 plus thecomprehension scheme for 11 formulas.In a sense, 11-CA0 comprehension is to arithmetical transnite recursion (11 separation) as ACA0 is to weakKnigs lemma (01 separation). It is equivalent to several statements of descriptive set theory whose proofs make useof strongly impredicative arguments; this equivalence shows that these impredicative arguments cannot be removed.The following theorems are equivalent to 11-CA0 over RCA0:

The CantorBendixson theorem (every closed set of reals is the union of a perfect set and a countable set).

Every countable abelian group is the direct sum of a divisible group and a reduced group.

4.3 Additional systems Weaker systems than recursive comprehension can be dened. The weak system RCA*0 consists of elementary function arithmetic EFA (the basic axioms plus 00 induction in the enriched languagewith an exponential operation) plus 01 comprehension. Over RCA*0, recursive comprehension as dened earlier (that is, with 01 induction) is equivalent to the statement thata polynomial (over a countable eld) has only nitely many roots and to the classication theorem for nitelygenerated Abelian groups. The system RCA*0 has the same proof theoretic ordinal 3 as EFA and is conservative over EFA for 02 sentences.

Weak Weak Knigs Lemma is the statement that a subtree of the innite binary tree having no innite pathshas an asymptotically vanishing proportion of the leaves at length n (with a uniform estimate as to how manyleaves of length n exist). An equivalent formulation is that any subset of Cantor space that has positive measureis nonempty (this is not provable in RCA0). WWKL0 is obtained by adjoining this axiom to RCA0. It isequivalent to the statement that if the unit real interval is covered by a sequence of intervals then the sum oftheir lengths is at least one. The model theory of WWKL0 is closely connected to the theory of algorithmicallyrandom sequences. In particular, an -model of RCA0 satises weak weak Knigs lemma if and only if forevery set X there is a set Y which is 1-random relative to X.

14 CHAPTER 4. REVERSE MATHEMATICS

DNR (short for diagonally non-recursive) adds to RCA0 an axiom asserting the existence of a diagonallynon-recursive function relative to every set. That is, DNR states that, for any set A, there exists a total functionf such that for all e the eth partial recursive function with oracle A is not equal to f. DNR is strictly weakerthan WWKL (Lempp et al., 2004).

11-comprehension is in certain ways analogous to arithmetical transnite recursion as recursive comprehen-sion is to weak Knigs lemma. It has the hyperarithmetical sets as minimal -model. Arithmetical transniterecursion proves 11-comprehension but not the other way around.

11-choice is the statement that if (n,X) is a 11 formula such that for each n there exists an X satisfying then there is a sequence of setsXn such that (n,Xn) holds for each n. 11-choice also has the hyperarithmeticalsets as minimal -model. Arithmetical transnite recursion proves 11-choice but not the other way around.

4.4 -models and -modelsThe in -model stands for the set of non-negative integers (or nite ordinals). An -model is a model for a fragmentof second-order arithmetic whose rst-order part is the standard model of Peano arithmetic, but whose second-orderpart may be non-standard. More precisely, an -model is given by a choice S2 of subsets of . The rst ordervariables are interpreted in the usual way as elements of , and +, have their usual meanings, while second ordervariables are interpreted as elements of S. There is a standard model where one just takes S to consist of all subsetsof the integers. However there are also other -models; for example, RCA0 has a minimal -model where S consistsof the recursive subsets of .A model is an model that is equivalent to the standard -model for 11 and 11 sentences (with parameters).Non- models are also useful, especially in the proofs of conservation theorems.

4.5 References Ambos-Spies, K.; Kjos-Hanssen, B.; Lempp, S.; Slaman, T.A. (2004), Comparing DNR and WWKL, Jour-

nal of Symbolic Logic 69 (4): 1089, doi:10.2178/jsl/1102022212.

Friedman, Harvey (1975), Some systems of second order arithmetic and their use, Proceedings of the Inter-national Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, Canad. Math. Congress, Montreal,Que., pp. 235242, MR 0429508

Friedman, Harvey; Martin, D. A.; Soare, R. I.; Tait, W. W. (1976), Meeting of the Association for SymbolicLogic, The Journal of Symbolic Logic (Association for Symbolic Logic) 41 (2): 557559, ISSN 0022-4812,JSTOR 2272259 |chapter= ignored (help)

Simpson, StephenG. (2009), Subsystems of second order arithmetic, Perspectives in Logic (2nd ed.), CambridgeUniversity Press, ISBN 978-0-521-88439-6, MR 2517689

Solomon, Reed (1999), Ordered groups: a case study in reverse mathematics, The Bulletin of Symbolic Logic5 (1): 4558, doi:10.2307/421140, ISSN 1079-8986, JSTOR 421140, MR 1681895

4.6 External links Harvey Friedmans home page Stephen G. Simpsons home page

Chapter 5

Second-order arithmetic

In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural num-bers and their subsets. It is an alternative to axiomatic set theory as a foundation for much, but not all, of mathematics.It was introduced by David Hilbert and Paul Bernays in their book Grundlagen der Mathematik. The standard ax-iomatization of second-order arithmetic is denoted Z2.Second-order arithmetic includes, but is signicantly stronger than, its rst-order counterpart Peano arithmetic. Un-like Peano arithmetic, second-order arithmetic allows quantication over sets of numbers as well as numbers them-selves. Because real numbers can be represented as (innite) sets of natural numbers in well-known ways, andbecause second order arithmetic allows quantication over such sets, it is possible to formalize the real numbers insecond-order arithmetic. For this reason, second-order arithmetic is sometimes called analysis.Second-order arithmetic can also be seen as a weak version of set theory in which every element is either a naturalnumber or a set of natural numbers. Although it is much weaker than Zermelo-Fraenkel set theory, second-orderarithmetic can prove essentially all of the results of classical mathematics expressible in its language.A subsystem of second-order arithmetic is a theory in the language of second-order arithmetic each axiom ofwhich is a theorem of full second-order arithmetic (Z2). Such subsystems are essential to reverse mathematics, aresearch program investigating how much of classical mathematics can be derived in certain weak subsystems ofvarying strength. Much of core mathematics can be formalized in these weak subsystems, some of which are denedbelow. Reverse mathematics also claries the extent and manner in which classical mathematics is nonconstructive.

5.1 Denition

5.1.1 Syntax

The language of second-order arithmetic is two-sorted. The rst sort of terms and variables, usually denoted bylower case letters, consists of individuals, whose intended interpretation is as natural numbers. The other sort ofvariables, variously called set variables, class variables, or even predicates are usually denoted by upper caseletters. They refer to classes/predicates/properties of individuals, and so can be thought of as sets of natural numbers.Both individuals and set variables can be quantied universally or existentially. A formula with no bound set variables(that is, no quantiers over set variables) is called arithmetical. An arithmetical formula may have free set variablesand bound individual variables.Individual terms are formed from the constant 0, the unary function S (the successor function), and the binary op-erations + and (addition and multiplication). The successor function adds 1 (=S0) to its input. The relations =(equality) and < (comparison of natural numbers) relate two individuals, whereas the relation (membership) relatesan individual and a set (or class). Thus in notation the language of second-order arithmetic is given by the signatureL = f0; S;+; ;=;

16 CHAPTER 5. SECOND-ORDER ARITHMETIC

5.1.2 SemanticsSeveral dierent interpretations of the quantiers are possible. If second-order arithmetic is studied using the fullsemantics of second-order logic then the set quantiers range over all subsets of the range of the number variables. Ifsecond-order arithmetic is formalized using the semantics of rst-order logic then any model includes a domain forthe set variables to range over, and this domain may be a proper subset of the full powerset of the domain of numbervariables.Although second-order arithmetic was originally studied using full second-order semantics, the vast majority of cur-rent research treats second-order arithmetic in rst-order predicate calculus. This is because the model theory ofsubsystems of second-order arithmetic is more interesting in the setting of rst-order logic.

5.1.3 AxiomsBasic

The following axioms are known as the basic axioms, or sometimes the Robinson axioms. The resulting rst-ordertheory, known as Robinson arithmetic, is essentially Peano arithmetic without induction. The domain of discoursefor the quantied variables is the natural numbers, collectively denoted byN, and including the distinguished member0 , called "zero.The primitive functions are the unary successor function, denoted by prex S; , and two binary operations, additionand multiplication, denoted by inx "+" and " ", respectively. There is also a primitive binary relation called order,denoted by inx "

5.2. MODELS OF SECOND-ORDER ARITHMETIC 17

Induction and comprehension schema

If (n) is a formula of second-order arithmetic with a free number variable n and possible other free number or setvariables (written m and X), the induction axiom for is the axiom:

8m8X(('(0) ^ 8n('(n)! '(Sn))! 8n'(n))

The (full) second-order induction scheme consists of all instances of this axiom, over all second-order formulas.One particularly important instance of the induction scheme is when is the formula n 2 X expressing the factthat n is a member of X (X being a free set variable): in this case, the induction axiom for is

8X((0 2 X ^ 8n(n 2 X ! Sn 2 X))! 8n(n 2 X))

This sentence is called the second-order induction axiom.Returning to the case where (n) is a formula with a free variable n and possibly other free variables, we dene thecomprehension axiom for to be:

8m8X9Z8n(n 2 Z $ '(n))

Essentially, this allows us to form the set Z = fnj'(n)g of natural numbers satisfying (n). There is a technicalrestriction that the formula may not contain the variable Z, for otherwise the formula n 62 Z would lead to thecomprehension axiom

9Z8n(n 2 Z $ n 62 Z)

which is inconsistent. This convention is assumed in the remainder of this article.

5.1.4 The full system

The formal theory of second-order arithmetic (in the language of second-order arithmetic) consists of the basicaxioms, the comprehension axiom for every formula , (arithmetic or otherwise) and the second-order inductionaxiom. This theory is sometimes called full second order arithmetic to distinguish it from its subsystems, denedbelow. Second-order semantics eliminates the need for the comprehension axiom, because these semantics implythat every possible set exists.In the presence of the unrestricted comprehension scheme, the single second-order induction axiom implies eachinstance of the full induction scheme. Subsystems that limit comprehension in some way may oset this limitationby including part of the induction scheme. Examples of such systems are provided below.

5.2 Models of second-order arithmeticRecall that we view second-order arithmetic as a theory in rst-order predicate calculus. Thus a modelM of thelanguage of second-order arithmetic consists of a setM (which forms the range of individual variables) together witha constant 0 (an element of M), a function S from M to M, two binary operations + and on M, a binary relation