Soundness can be expressed as:
Γ, φ) If Γ ⊢ φ in proof system S then Γ ⊨ φ.Σ) If Σ satisfiable then Σ consistent in proof system S.Completeness can be expressed as:
Γ, φ) If Γ ⊨ φ then Γ ⊢ φ in proof system S. (When Γ is non-empty, sometimes called “Strong Completeness.”)Σ) If Σ consistent in proof system S then Σ satisfiable.Compactness for classical first-order predicate logic (FOL), with or without functors or equality, can be expressed as:
Γ, φ) If Γ ⊨ φ, then for some finite Γ′ ⊆ Γ, Γ ⊨ φ.Σ) If for every finite Σ′ ⊆ Σ, Σ′ is satisfiable, then Σ is satisfiable. (Converse is straightforward.)Goldrei discusses Completeness and Compactness in §3.2 (for sentential logic) and §5.5 (for predicate logic). In both cases his proofs assume the language is countable. Completeness and Compactness also hold for “uncountable languages,” that is languages with uncountably many non-logical symbols so that the sets Γ and Σ may be uncountable. But proving these theorems for such languages is harder. (See Goldrei §6.4.)
Goldrei’s proof in §5.5 temporarily assumes that the = symbol is interpreted so as to respect the appropriate axioms, but not necessarily by the relation of numerical identity. See §5.4 for more discussion of this.
Although Goldrei defines ⊨ so as to hold between arbitrary formulas, not just closed formulas/sentences (p. 188), the Completeness proof he gives is only for entailments between sentences. Lifting this restriction makes the proof harder.
Talk of a model’s cardinality = measuring its domain
Talk of a language’s cardinality = measuring its signature (its non-logical symbols: sentence constants if any, predicates, functors, term constants). Language is finite if this set is finite; countable if this set is finite or countably infinite. Otherwise has larger cardinality (for example, if each real number has its own constant).
(Downward L-S, usually proven using materials developed in proof of Completeness) If a theory Σ has a model, it has a model whose cardinality is either countable, or ≤ the cardinality of the theory’s language, if that’s larger. (So if the language is finite, this doesn’t guarantee that there’s a finite model, only a countable one.) In the usual case where the language is countable, this says: every satisfiable theory has a countable model.
(Upward L-S, proven using Compactness, ironic that Skolem’s name is attached to this because he didn’t believe in “larger infinities”) If a theory Σ has any infinite models, then it has models of every cardinality ≥ that of its language. In the usual case where the language is countable, this says: every satisfiable theory which can infinite/arbitrarily large models, has models of every cardinality.
It’s provable that any theory that has models of arbitrarily large finite size also has infinite models (see Goldrei pp. 197–8 and 276), so the Upward L-S theorem would apply to such theories.
The L-S theorems have some counter-intuitive consequences:
(ℝ, 0, 1, +, ⋅, <) also has a countable model!We’ve commented before about the difference between saying an argument vs a sentence/formula is “valid”
We’ve commented before about the difference between saying an argument vs a proof system is “sound”
We’ve commented before, but want now to stress/remind you, about the difference between saying a proof system vs a theory is “complete”. For a proof system, this is a matter of whether it catches all the semantic entailments. For a theory, this is a matter of whether for every sentence in its language, it includes either that sentence or its negation.
Another bit of overused terminology is “decidable.” We’ve talked before about what it is for a theory (or any set) to be decidable. There’s also a usage of “decidable” applied to sentences, in relation to theories. A sentence is decidable by a theory when the theory proves either the sentence or its negation.
Relations between these concepts:
a and b, but only a single unary predicate F, and the axioms consist solely of Fa; entailment for this language is decidable; but the theory won’t prove either Fb or ~Fb).Language(s) of arithmetic:
Z (for zero, more usually written as 0 in a special font)S (for successor; the term SSSZ of the formal language can be abbreviated in the metalanguage as S³Z or 3̅)+, ⋅ (for multiplication), sometimes ↑ (or another symbol for exponentiation)<Here are a collection of axioms/axiom schemas. Different “theories of arithmetic” each include a subset of these. In the strongest such theories, the others listed here then follow as theorems; in other theories they don’t.
Generally, the axioms should be understood as prefixing the following with a ∀x for each free variable x in the formula. For one theory we’ll discuss, they’ll be understood differently.
These first three groups are like recursive definitions of +, ⋅, and ↑.
A1. (x + Z) = x
A2. (x + Sy) = S(x + y)
M1. (x ⋅ Z) = Z
M2. (x ⋅ Sy) = ((x ⋅ y) + x)
E1. (x ↑ Z) = SZ
E2. (x ↑ Sy) = ((x ↑ y) ⋅ x)
Any theory whose language includes S will have these axioms:
S1. Sx = Sy ⊃ x = y (S is injective)
S2. Sx ≠ Z (Z has no predecessor)
(Additionally, it’s implicit in the functor notation that everything has a unique successor.)
After that, things are more mixed. The prefixes of the following axioms indicate which theories include them.
R3. x ≠ Z ⊃ ∃y Sy = x
R4. x < y ⊂⊃ ∃z (Sz + x) = y (sometimes this is used as a defintion of < rather than an axiom)
H/B3. ~(x < Z) (Z is minimal for <)
H/B4. y < Sx ⊂⊃ (y = x ∨ y < x)
H5. (x < y ∨ x = y ∨ y < x) (< is weakly connected)
B5. x ≠ Z ⊂⊃ Z < x
B6. Sx < y ⊂⊃ (x < y ∧ Sx ≠ y)
First-order Induction schema: (φ[x ← Z] ∧ ∀x(φ ⊃ φ[x ← Sx])) ⊃ ∀xφ
Second-order Induction: ∀F((FZ ∧ ∀x(Fx ⊃ FSx) ⊃ ∀xF)
Here are the theories of interest. The “strongest” arithmetics (ones with the most theorems) are called “Peano Arithmetic” (though they’re also following the work of Dedekind).
Second-order Peano Arithmetic: language includes at least Z and S; background logic is second-order and axioms include Second-order Induction, as well as S1 and S2. Other operations (+, ⋅, ↑, etc) can be “defined” inductively. The relation < can be defined using R4.
In FOL, we can’t appeal to quantification over predicates as in Second-order Induction. Instead First-order Peano Arithmetic has only the First-order Induction schema, for any predicate φ expressible in its language. (This schema is a pattern yielding infinitely many axioms.) First-order Peano Arithmetic also includes axioms S1 and S2; and A1 and A2 to specify the behavior of +; and M1 and M2 to specify the behavior of ⋅. Henceforth, when we say “Peano Arithmetic” (or “PA”) we mean this first-order version.
All the remaining axioms listed (perhaps with the exception of E1 and E2) are theorems of Peano Arithmetic.
Peano Arithmetic captures enough of arithmetic to prove Gödel’s First and Second Incompleteness Theorems. These have the consequences that: (a) If Peano Arithmetic is consistent, its language has “undecidable sentences,” that is, sentences such that neither they nor their negation are theorems/provable from the Peano axioms. (b) This will also be true of any consistent strengthening of Peano Arithmetic. (c) Peano Arithmetic cannot prove its own consistency (unless it’s inconsistent, and thus proves everything).
Some weaker arithmetics don’t have these properties. They are “complete” with respect to their language (which as we said above entails, but isn’t entailed by, their being decidable theories). One example is Presburger arithmetic, whose language includes only Z, S and +, and whose axioms include S1, S2, A1, A2, and the First-order Induction schema (for its restricted language). Another example is Skolem arithmetic, whose language includes only ⋅ (that’s enough to define notions like divisbility and the constant 1).
Some “mathematically rich/complex” theories are also complete (and so decidable). This includes the first-order theory of real numbers (including +, ⋅, <).
Smith discusses a “Baby Arithmetic” that (like Peano Arithmetic) has Z, S, +, and ⋅ in its language, and versions of S1, S2, A1, A2, M1 and M2. But this Baby Arithmetic lacks any Induction axioms and it also doesn’t use any quantifiers. Instead, the “axioms” like S1 are treated as schemas, where the variables can be replaced by any term (such as Z, SSSZ, (SZ + SSZ), and so on). This system is also complete.
What makes the Gödel results possible is the combination of + and ⋅ and quantifiers, and also that the system has other axioms that give us natural-number-like behavior, rather than mathematical structures like the reals. Peano Arithmetic gets the last feature from its Induction axioms.
Logicians realized that Peano Arithmetic was stronger than was needed to establish Gödel’s First Incompleteness Theorem (which gives us results (a) and (b) mentioned above). They sought out systems that were weaker than Peano Arithmetic, but still strong enough to express/prove things about what’s provable in themselves, as Gödel’s proofs involved. Here are three such systems. They all omit Peano Arithmetic’s Induction schema; and instead have only a finite number of axioms. In none of these systems are all of the above axioms provable. Generally in these systems, you can prove things like (SZ + SSZ) = (SSZ + SZ), and so on for addition of any particular numbers. But you cannot always prove generalizations like ∀x∀y (x + y) = (y + x).
Robinson Arithmetic: includes A1, A2, M1, M2, S1, S2, R3, R4 (or R4 may instead be used as a definition of <). Usually this theory is called “Q”. Was used (and called “Q”) in Boolos and Jeffrey’s 1st and 2nd editions (perhaps also 3rd).
Shoenfield Arithmetic: includes A1, A2, M1, M2, S1, S2, H/B3, H/B4, H5 (appeared in a 1967 textbook by Shoenfield). Strangely Franzén in his book calls this formal system “Robinson Arithmetic,” though it is not equivalent to the preceding system; Shoenfield’s system was used in Boolos, Burgess, and Jeffrey’s 4th edition and oddly they called it “Q”, though they discussed differences between it and Robinson Arithmetic. Seems that: R doesn’t have H/B4 or H5 as theorems; H doesn’t have either R3 or R4 as theorems.
Burgess Arithmetic: includes A1, A2, M1, M2, S1, S2, H/B3, H/B4, B5, B6 (this is the system used in Boolos, Burgess, and Jeffrey’s 5th edition, presumably because it makes some proofs easier). Burgess calls this system “Q,” though again discussing differences between it and Robinson Arithmetic. I believe this system is neither weaker nor stronger than Shoenfield Arithmetic. Seems that: R doesn’t have H/B4 or B6 as theorems; H doesn’t have B6 as theorem; B doesn’t have any of R3, R4, or H5 as theorems.
Although these three systems are weaker than Peano Arithmetic, they are also ones in which Gödel’s First Incompleteness Theorem holds. These systems are not strong enough to carry out Gödel’s Second Incompleteness proof, though its conclusion is probably true of them: they probably are too weak prove their own consistency.
These systems as discussed are strong enough to establish Gödel’s First Incompleteness Theorem; but certain results are more easily proved if the systems also include ↑ in their language and axioms E1 and E2. So sometimes one sees the systems expanded in that way.
I mentioned above how instead of taking R4 as an axiom, one could take it as a definition of < in terms of S and +. It’s also possible (instead) to define S in terms of <; or to define + in terms of S and ⋅; or to define Z and S in terms of +; or to define + and ⋅ in terms of ↑.
We’ll discuss Gödel’s results in remaining classes. Before we get to them…
Second-order Peano Arithmetic is categorical (has only one model, up to isomorphism).
Upward L-S tells us that any first-order theory with an infinite model also has models with larger domains, and so isn’t categorical. In particular, any of the first-order arithmetic theories described above also have models with domains larger than ℕ.
Compactness can be used to show that those arithmetic theories will even have countable non-standard models. These models have domains the same size as ℕ, but they aren’t isomorphic to the standard interpretations of <, +, ⋅, and so on.
If time permits, will describe the general shape of these models.
Here is the proof from Compactness: Extend the language of PA with a new constant c and consider the theory which is PA extended with the infinite set of axioms Z < c, SZ < c, SSZ < c, … Any finite subset of this theory’s axioms would be satisfied by a standard model of arithmetic that interpreted c as a number larger than any mentioned in that subset. By Compactness, then the whole set of axioms must be satisfiable. But in any model that satisfies all the axioms, c cannot be interpreted as any of the standard numbers. It must be some “non-standard” number larger than all of them (and so too c’s predecessor and successor, and their predecessor and successor, and so on). Since any model of this language and axioms will also be a model of the subset of the axioms that don’t contain the new constant c, which are just the axioms of PA, the model with these non-standard larger numbers must also be models of PA.