Proposition

4 Deducibility

4.6. Proposition

(a) The formulas

t=t; t1=t2⇒t2=t1; t1=t2∧t2=t3⇒t1=t3; x=t⇒(P(x, x)⇒P(x, t))

areφ-true for any interpretation of L in which φ(=) is equality.

(b) All the formulas in(a)are deducible from the set AxL∪ {x=x|xis a variable}

∪ {x=y⇒(P(x, x)⇒P(x, y))|Pis an atomic formula}. The formulas in this list, except forAxL, are called the axioms of equality.

(c) Letφbe any interpretation ofLin a setM for which the axioms of equality are true. Then φ(=) is an equivalence relation in M that is compatible with the interpretations of all the relations and operations of L in M. Ifφ denotes the obvious interpretation of Lin the quotient set M =M/φ(=), thenφ(=) is equality, andT_φL=T_φL.

Proof (Sketch)

(a) The φ-truth is easily established. We illustrate this by showing that the last formula is φ-true. Suppose it were false at a point ξ ∈ M. Then

|x = t|(ξ) = 1,|P|(ξ) = 1 and |P(x, t)|(ξ) = 0. The first assertion means that x^ξ =t^ξ. But then|P|(ξ) =|P(x, t)|(ξ) by Proposition 2.10, contradicting the second and third assertions.

(b) Deduction of t = t : x = x (axiom of equality); ∀x(x = x) (Gen);

∀x(x=x)⇒t=t (logical axiom of specialization);t=t (MP).

Deduction oft₁=t₂⇔t₂=t₁:

(1) x=y⇒(x=x⇒y=x) (axiom of equality with = for P);

(2) Q⇒((P ⇒(Q ⇒R))⇒(P ⇒R)), whereP is x=y, Q is x=x, R is y=x(tautology);

(3) x=x(axiom of equality);

(4) (P ⇒(Q⇒R))⇒(P ⇒R) (MP is applied to (2) and (3));

(5) x=y⇒y=x(MP applied to (1) and (4)).

We then twice apply Gen, the axiom of specialization, and MP, in order to deduce the formulat₁=t₂⇒t₂=t₁from (5); we replacet₁ byt₂andt₂ byt₁ to deduce t2 = t1 ⇒ t1 = t2; we use Lemma 4.4 to deduce the conjunction of these two formulas; and, finally, the tautology (t1 =t2 ⇒ t2 =t1)∧(t2 = t1 ⇒ t1 = t2) ⇒ (t1 = t2 ⇔ t2 =t1), together with MP, gives the required formula.

The deduction of the third and fourth formulas in (a) will be left to the reader. The existence of a deduction of the fourth formula can be proved by induction on the number of connectives and quantifiers in P. P is represented in the form¬Q, Q1∗Q2,∀x Q,or∃x Q; we assume that the formula withQ, Q1, andQ2in place ofP has already been deduced, and we complete the deduction forP (see Mendelson, Chapter 2, Proposition 2.25).

(c) If the axioms of equality areφ-true, then so are the formulas in (a), since they are deducible. The first three formulas in (a), applied to three different variablesx, y, andz, then show that the relationφ(=) onM is reflexive, sym- metric, and transitive. In fact, letX, Y, andZ be any three elements ofM, let ξ ∈ M be a point such that x^ξ = X, y^ξ =Y; and z^ξ = Z and let ∼ be the relationφ(=) onM. Theφ-truth of the formulas in (a) means that

X ∼X; X ∼Y ⇔Y ∼X; X∼Y; andY ∼Z⇒X ∼Z.

By definition, to say that ∼is compatible with the φ-interpretation of all relations and operations on M means the following. Let p be a relation, and let φ(p) ⊂ M be its interpretation. If X₁, . . . , X_r ∈ φ(p) and X_i ∼ X_i, then X1, . . . , X_i, . . . , Xr ∈ φ(p). Now let f be an operation, and let φ(f) : M^r ⇒ M be its interpretation. If φ(f)(X1, . . . , Xr) = Y and X_i ∼ Xi, then φ(f)(X1, . . . , X_i, . . . , Xr) =Y ∼Y.

We verify this compatibility by using the φ-truth of the last formula in 4.6(a) at a suitable point ξ∈M. Here we take the formulasp(x1, . . . , xr) and f(x1, . . . , xr) = y, respectively, for P; we take the variable x_i for t and the variablexi forx; and we setx^ξ_i =Xi, x_i^ξ=X_i,andy^ξ =Y.

It follows from the compatibility that we can construct an interpretationφ of Lin M =M/∼such thatφ(p) =φ(p) mod ∼, φ(f) =φ(f) mod∼, and

φ(=) is equality. The last formula in 4.6(a) will then imply that all theφ-true

formulas remainφ-true, and conversely.

From now on, when we speak of the special axioms for any language inL1

having the symbol =, we shall without explicit mention always include among them the axioms of equality for =. Models in which = is interpreted as equality are called normalmodels.

Special axioms of arithmetic

4.7. Proposition.The following formulas are true in the standard interpreta- tion of L₁Ar, and are called the special axioms ofL₁Ar:

(a) The axioms of equality.

(b) The axioms of addition:

x+ ¯0 =x; x+y=y+x; (x+y) +z=x+ (y+z);

x+z=y+z⇒x=y.

(c) The axioms of multiplication:

x·¯0 = ¯0; x·1 =x; x·y=y·x; (x·y)·z=x·(y·z).

(d) The distributive axiom:

x·(y+z) =x·y+x·z.

(e) The axioms of induction:

P(0)¯ ∧ ∀x(P(x)⇒P(x+ ¯1))⇒ ∀x P(x), where P is any formula inL₁Arhaving one free variable.

The proof is trivial and will be left to the reader. We note only that the

“proof” that the induction axioms are true itself uses induction.

Remarks

(a) In (b), (c), and (d) above, we have written the usual axioms for a commutative (semi) ring in order to shorten the formal deductions; any in- formal computation that uses only these axioms can easily be transformed into a formal deduction of the result of the computation in L1Ar. In Chapter 3 of Mendelson’s textbook, he gives an apparently weaker set of axioms, and then shows how to deduce our formulas from them. This takes up 5–6 pages of text, and is basically a tribute to a historical tradition going back to Peano.

(b) The induction axioms are a countable set of formulas in L1Ar; it is customary to say that 4.7(e) is an axiom schema. The corresponding fact in intuitive mathematics is stated as follows; “For any propertyP of nonnegative integers, if 0 has the property P, and, whenever xhas the property P, x+ 1 also has the property P, then all nonnegative integers have the property P.”

Here “property of nonnegative integers” means the same as “any subset of the nonnegative integers.”

However, in the means of expression of L1Ar there is no way to say “any subset.” Neither is there any way to say “all properties”; we can only list one by one the properties that are definable by formulas in the language. We recall that there are only countably many such properties, while the intuitive interpretation refers to a continuum of properties. Thus, the formal axiom of induction is weaker than the informal one, and is also weaker than the version of this axiom that is obtained by embedding L1Ar in L1Set.

Special axioms of Zermelo–Fraenkel set theory

(see the description of V in the appendix to ChapterII)

4.8. Proposition.The following formulas are true in the standard interpreta- tion of L₁Setin the von Neumann universe V:

(a) Axiom of the empty set: ∀x¬(x∈∅).

(b) Axiom of extensionality: | ∀z(z∈x⇔z∈y)⇔x=y.

(c) Axiom of pairing: ∀u∀w∃x∀z(z∈x⇔z=u∨z=w).

(d) Axiom of the union: ∀x∃y∀u(∃z(u∈z∧z∈x)⇔u∈y).

(e) Axiom of the power set:∀x∃y∀z(z⊂x⇔x∈y),wherez ⊂xis abbrevi- ated notation for the formula∀u(u∈z⇒u∈x).

(f) Axiom of regularity:∀x(¬x=∅⇒ ∃y(y∈x∧y∩x=∅)),wherey∩x=∅ is abbreviated notation for¬∃z(z∈y∧z∈x).

Proof and explanations. This is not a complete list of the axioms of Zermelo–Fraenkel; the axiom of infinity, axiom of replacement, and also the axiom of choice, which are more subtle, will be discussed in the next subsection.

(a) The truth of these formulas must, of course, be proved by computing the function | |using the rules in 2.4 and 2.5. We do this, for example, for the axiom of extensionality. Let ξbe any point in the interpretation class, and let X =x^ξ, Y =y^ξ. We must show that

|∀z(z∈x⇔z∈y)|(ξ) =|x=y|(ξ), i.e., that

min

Z∈V(|Z ∈X| |Z ∈Y|+ (1− |Z∈X|)(1− |Z∈Y|)) =|X =Y|, where we have written |Z ∈X| instead of|z ∈ x|(ξ) with z^ξ = Z, x^ξ = X, and so on. But the left-hand side equals 1 if and only if for everyZ ∈V either both Z ∈X andZ ∈Y, or else bothZ ∈Y andZ ∈Y, that is, if and only if X =Y.

More generally, if we replaceV by any subclassM ⊂V and restrict the standard interpretation of L1Set toM, then the same reasoning shows thatThe axiom of extensionality is true in M if and only if for any elements X, Y ∈M we have

X =Y ⇔X∩M =Y ∩M,

i.e., if and only if every element of M is uniquely determined by its elements which lie in M. This result will be used later.

The analogous computations for all the other axioms will be given sys- tematically in a much more difficult context in Chapter III. Hence, at this point we shall only explain how to translate them into argot, as in Chapter I, and why they are fulfilled inV.

(b) The axiom of the empty set does not need special comment. We only remark that if we interpret L1Set in a subclassM ⊂V, then the constant ∅ may be interpreted as any elementX ∈M with the property thatX∩M =∅, and this axiom will still hold.

(c) The axiom of pairing is true, because ifU, W ∈Vα, then{U, W} ∈ P(Vα), so that all pairs lie in V.

(d) The axiom of the union is true, because if X ∈ V, then the set Y =∪Z∈XZ also lies in V. In fact, if X ∈ V_α+1 =P(V_α), then the elements ofX are subsets ofV_α, and their union therefore lies inV_α+1.

(e) The axiom of the power set is true, because ifX ∈V, thenP(X)∈V. In fact, if X ∈ Vα, then X ⊂Vα, and hence P(X)⊂ P(Vα) =Vα+1, so that P(X)∈Vα+2.

(f) The axiom of regularity is true, because any nonempty setX ∈V has an empty intersection with at least one of its elements; in this form the axiom is proved in the appendix to this chapter.

4.9. The axioms of L1Set in Section 4.8 have one property in common: their simplest model in the standard interpretation is precisely the union Vω₀ =

∪^∞n=0Vn of the firstω0 levels of the von Neumann universe. In other words, this is the set of hereditarily finite sets X∈V, i.e., those such that ifXn∈Xn−1∈

· · · ∈X0=X then all theXi are finite.

Vω₀ is the reliable, familiar world of combinatorics and number theory.

Additional principles are needed to force us out of this world. There are two such principles: the axiom of infinity and the axiom schema of replacement.

(a)Axiom of infinity:

∃x(∅∈x∧ ∀y(y∈x⇒ {y} ∈x)).

Here {y} ∈ x is abbreviated notation for ∃z(z = {y, y} ∧z ∈ x), where the meaning of z = {y, y} was explained in 3.7 of Chapter I. This axiom re- quires that we add to Vω₀ some set containing the elements∅,{Ø},{{∅}}, . . . (a countable sequence). Then, in order to preserve the intuitive version of the axiom of the power set, we must addP(X),P²(X), . . . ,thereby hopelessly leaving the realm of finite sets, countable sets, continua, and so on.

It is a striking fact that none of this is necessary in the formal, as opposed to intuitive, version of set theory, where we can always limit ourselves to hereditarily countable submodels ofV. This important fact will be discussed in detail in§7.

(b) Axiom schema of replacement. We introduce the following convenient abbreviated notation (in any language of L1 having the notion of equality):

∃!y P(y) means ∃y P(y)∧ ∀x∀y(P(x)∧P(y)⇒x=y). Thus, this formula is read; “There exists a unique object y with the propertyP,” where we assume

that = is interpreted as equality. When other variables besidesyoccur freely in P, the formula∃!yP(y) is true precisely whenP determinesy as an “implicit function” of the other variables.

We can now write the replacement axioms. In the formulaP below we list all the variables that occur freely in P:

∀z₁· · · ∀zn∀u(∀x(x∈u⇒ ∃!y P(x, y, z1, . . . , zn))

⇒ ∃w∀y(y∈w⇔ ∃x(x∈u∧P(x, y, z1, . . . , zn)))).

The hypothesis says that “P givesy as a function of x ∈ u(for given values of the parametersz₁, . . . , z_n)”; the conclusion says that “the image of the setu under this function is some setw.”

From the standpoint of the formal theory it is worthwhile to note that from this axiom and the axioms of equality are deducible the so-called separation axioms, namely

∀z1· · · ∀zn∀x∃y∀u(u∈y⇔u∈x∧P(u, z1, . . . , zn)).

This says that if we take the class of sets having a propertyP and intersect it with a setx, we obtain a set.

The replacement axioms should be looked at very carefully. They go beyond the usual, “intuitively obvious” working tools of the topologist and analyst. The axioms assert that, for example, it is impossible to “stretch” an ordinalαtoo far by means of a functionf; for anyf we choose, there is always an ordinalβ such that all the valuesf(γ), γα, lie inVβ. In other words, the universeV is incomparably more infinite than any of its levels Vα.

Even if we adopt this axiom, questions remain that are very similar in style, that are beyond the reach of our intuition, and that are not solvable using this and the other axioms. For example, do there exist so-called inaccessible cardinalsγ? One of the properties of an inaccessible cardinalγis the following:

iff is a function fromV_αto V_γ (withα < γ), then the set of values off is an element ofV_γ. In particular, there is an “upper bound” beyond which ordinals not exceeding γcannot be “stretched.” Do such infinities exist or not?

After thinking about this and related problems, many specialists on the foundations of mathematics have come to the conclusion that such languages of set theory as L1Set with a suitable axiom system are the only reality one should work with, and any attempt to make intrinsic sense out of the universe V or similar models is in principle doomed to failure. In particular, the set of formulas in L1Set that are true in the standard interpretation is not defined, and we can only talk about formulas that are deducible from the axioms.

But we shall not entirely adopt this point of view for several reasons. The simplest reason is the feeling that a language without an interpretation not only loses its intrinsic justification, but also cannot be used for anything. We cannot even play the “formal game” well unless we master the intuitive concepts that give meaning to the symbols. A language (along with the external world) helps bring order and precision to these intuitive concepts, which, in turn, make us change the language or at least revise our earlier linguistic constructions. But we can never assume that we have achieved complete clarity.

We should understand the need for certain types of self-restraint. However, intellectual asceticism (like all other forms of asceticism) cannot be the lot of many.

(a) Axiom of choice:

∀x(¬x=∅⇒ ∃y(“yis a function with domain of definitionx”

∧ ∀u(u∈x∧ ¬u=∅⇒ ∃w(w∈u∧“ u, w ∈y”)))).

That is, y chooses one element from each nonempty elementu∈x.

The belief that this axiom is true in V is at least as justified as the belief in the existence ofV itself. Over the past fifty years it has become customary for every working mathematician to accept this axiom, and the heated con- troversies about it at the beginning of the century are now all but forgotten.

The interested reader is referred to Chapter II ofFoundations of Set Theoryby Fraenkel and Bar-Hillel (North-Holland, Amsterdam, 1958).

4.10.General properties of axioms. Despite the wide variety of concepts reflected in these axioms, each of our sets of axioms for languages in L1 (tautologies;

AxL; special axioms of L1Ar and L1Set) have the following informal syntactic characteristics:

(a) An algorithm can be given that tells whether any given expression is an axiom (compare the syntactic analysis in §1 and the verification of the tautologies in Section 3.4).

(b) A finite number of rules can be given for generating the axioms.

It is clear that a priori, property (b) is less restrictive than (a). In fact, an algorithm as in (a) can be transformed into a rule for generating the axioms:

“Write out all possible expressions one by one in some order, and take those for which the algorithm gives a positive answer.”

It is actually natural to suppose that property (a) should characterize axioms, and property (b) should characterize deducible formulas, no matter how we explicitly describe the axioms and the deducible formulas in a given language. In Part III we make these intuitive ideas into precise definitions and show that (b) is strictly weaker than (a). See also the discussion in Section 11.6(c) of this chapter.