While it is mainly accepted that the sentence is true, we shall first try to show that it is so beyond doubt in the original interpretation (Bojadziev 1995). In this sense we shall have to face one of the most fundamental of all denials presented by Sloman (1992). He attacked the core meaning of Gödel's theorem: Gödel's sentence does not mean what it seems to mean and Penrose cannot see the truth of G(F) since there are models in which it is true and those in which it is false.

The usual meta-arithmetical meaning of Gödel sentence is that it says of itself that it is not provable. In his first argument, Sloman rejects this meaning by pointing out that, when analysing Gödel numbers, that "k is, after all, just a numeral: it denotes a number, not a formula' where k is the k in P_k(k) (p. 386). He repeats analogous statements for predicates and sentences: the predicate in the Gödel sentence 'is a complex arithmetical predicate about numbers, not a predicate concerned with the derivability of formulas', and the Gödel sentence 'is merely an assertion about numbers' (p. 386). This denial of the role of the Gödel code, besides being uncalled for, is also not very carefully stated: discussing the relationship between arithmetical and meta-arithmetical meaning, Sloman sometimes represents the meta-arithmetical meaning as being additional to the arithmetical one, sometimes as being separate from it, and non-existent, and sometimes as excluding the arithmetical one. Sloman mentions the first, standard variant, when he considers the predicate in the Gödel sentence as 'expressing not just a property of numbers but a syntactic property of formulas in F' (p. 385), but replaces it with the second variant, present in the quotations above. The third, most unusual variant, appears when Sloman considers the statement that the numeral "k" denotes not the number k but the corresponding formula (p. 385). This variant, and the intermediate one according to which the meta-arithmetical meaning is the primary one, would make G(F) paradoxical and actually prevent the proof of Gödel's theorem, as Varga (1990) claims. Therefore, some of objections by Sloman contradict what has been well accepted in various books. No need to waste too much time here

But Sloman's most fundamental claim is that "argument that G(F) does not have the meaning it is commonly taken to have depends on the fact that because neither G(F) nor its negation can be derived in F, F will have some models in which G(F) is true and some in which it is false' (p. 385). The claim is based on two models: (F,G(F)) and (F,ØG(F)). This is legitimate since neither G(F) nor ØG(F) is provable in F if F is consistent. Now, nobody can see the truth of G(F) in (F,ØG(F)), Penrose included.

The argument seems to be that G(F) means something else in those models of F in which it is false. Strictly speaking, this is true, but the difference in the meaning of G(F) in those models in which it is false comes from the difference in those models themselves, namely the difference in the domain of the quantification through which G(F) says of itself that it has no proof. This basic meta-arithmetical meaning remains constant across models, and it is this meaning which enables us to understand why G(F) is false in some models. It thus makes much more sense to turn Sloman's thesis around and say that it is because G(F) does have the meaning it is commonly taken to have that it is false in some models of F.

The basic argument about different models of F is already introduced , e.g. in (Hofstadter 1979), to which Sloman refers. There, Hofstadter discusses the possibility of extending F with the negation of G(F) (Hofstadter 1979, pp. 452-6). This possibility is better understood if the deductive structure of F and the logical form of G(F) are seen in some more detail, using a couple of omega-notions. First, a system is called w-incomplete if, for some predicate P, P(n) is provable for all n, where n is the numeral of the number n, although " P(x) is not provable. Gödel's theorem involves this kind of incompleteness, with the predicate of unprovability in the role of P: G(F), which is equivalent to VxØpr(x,g), where g is the numeral of its Gödel number, is not provable, if the system is consistent, but all its particular instances Øpr(n,g) are provable, for all n. The second omega-notion is that of w-inconsistency: a system is w-inconsistent if P(n) is provable for all n, and yet $xØP(x) is also provable, for some predicate P; if there is no such predicate, the system is w-consistent. It follows from Gödel's theorem that if F is w-consistent, then ØG(F), equivalent to $x pr(x,g), is also not provable in F. Since w-consistency entails simple consistency, neither G(F) nor ØG(F) can be proved if F is w-consistent. In that case, F can be consistently extended by adding ØG(F) as a new axiom. But since a consistent F is w-incomplete, the new system F+ØG(F) is w-inconsistent: all instances of Øpr(n,g) are provable, along with the new axiom $x pr(x,g). If such a system is to have a model, that model must contain, in addition to the natural numbers, some other, non-finite number, which would make $x pr(x,g) true. Such a number could be thought of as the number of some infinite proof of G(F).

Let us reevaluate the basic Sloman's argument: if G(F) is false in a model of F, that model is a model of ØG(F), and so a model of F+ØG(F). In this model, G(F)=$xØpr(x,g) says that it has no proof (whatsoever, not even an infinite one), and so is false, because it does have one, since ØG(F)=$x pr(x,g) says so.

This situation may seem a little strange: "if ØG(F) says that G(F) is provable, and ØG(F) is provable, then G(F) should also be provable". And in general, "if you prove that something is provable, then it should be provable". But this is actually not so: a meta-proof is not proof enough. This point may be easier to consider if the notation is simplified a little, writing just pr(G) for ØG(F)=$x pr(x,g), and G for G(F). The argument would then be that pr(G) entails G, but it does not do so: the implication pr(G)G is not provable, because the implication pr(p)p is provable only for provable sentences p. This is a statement of Löb's theorem (Boolos, Jeffrey 1980, p. 187), which is equivalent to Gödel's second theorem; the implication pr(p)p is called the principle of reflection (of the meta-theory in the theory).

Sloman has an argument for his reduction of the meaning of the Gödel sentence to the numerical, purely arithmetical one. He suggests that 'perhaps it will turn out, in some of the "non-standard" models that make G(F) false, that the assumed mapping between complex arithmetical expressions and meta-linguistic statements about F goes awry' (p. 386). But there is no independent reason to think that the Gödel code could "go awry" in any model of F+ØG(F), or anywhere else; on the contrary, there is reason to think that it does not, since that can explain how G(F) is false in such models.

It should be added that the same observations about the falsity of the Gödel sentence in some models apply to Rosser's variation on Gödel's sentence, which Sloman does not consider. Rosser's sentence says of itself that if it has a proof, it has an even shorter refutation. For such a sentence, simple consistency is enough both for its own unprovability and for the unprovability of its negation. Sloman actually discusses Gödel's sentence as if it were Rosser's, and does not mention w-consistency (except, parenthetically, in his introductory statement of Gödel's theorem on p. 383, and once again on p. 386).

Sloman says that those models of F in which G(F) is false are sometimes called "non-standard" (p. 385). Such models are of course non-standard in the descriptive sense of the word, but not in its technical sense. Non-standard models do not change but, on the contrary, preserve the truth-values of statements in the standard model, although they are not isomorphic to it (Hunter 1971, p. 203), (Boolos, Jeffrey 1980, p. 193). So, in non-standard models of F, G(F) is actually (still) true. And as will be shown in the next paragraph, the Gödel sentence is true in the intended model.

The standard proof that G(F) is true in the domain of natural numbers goes by way of w-incompleteness: while G(F)=$xØpr(x,g) is not provable, if F is consistent, all its particular instances Øpr(n,g) are provable. These instances are thus true in the domain of natural numbers, which means that the general statement G(F) is also true in it (Boolos, Jeffrey 1980, p. 115). But Sloman is not satisfied with such a proof, and does not accept Penrose's more informal "seeing" that G(F) is true because what it says (that it is not provable) is so (if F is consistent). The reason is that Sloman questions the possibility of specifying exactly the notion of natural numbers which is used in the proof and in Penrose's seeing that G(F) is true: 'how are we supposed to grasp which model we are talking about? How can we unambiguously identify this infinite set?' (p. 387). The reason for this insistence on uniqueness seems to be Sloman's belief that 'unless the semantic properties of F somehow uniquely determine which (infinite) model is being talked about ... they cannot uniquely determine the truth-values of all the formulas expressible in F' (p. 385). But this is not so, and Sloman is much too strict in this insistence on uniqueness: the natural numbers do not have to be unambiguously identified, since much ambiguity is harmless. Models of natural numbers can be multiple if they are isomorphic, and even if they are not, provided they are non-standard in the standard sense of the term. Literally speaking, ambiguity is unavoidable since no satisfiable set of sentences has exactly one model (Boolos, Jeffrey 1980, p. 191).

Sloman is also concerned with the ambiguity between standard and (what he calls) non-standard models of F, in which G(F) is false. Sloman almost seems to be asking: "how do we know exactly what the standard model is if the best way of specifying it, namely F itself, is ambiguous?" Since F has both standard and (what Sloman calls) non-standard models, it cannot be used to specify the standard, intended model. Sloman concludes that 'F can be "seen" to be true only if there is some means, other than F, of specifying which model is in question' (p. 385). 'Until it is demonstrated that we do have some way of completely specifying exactly which infinite set we are talking about as a model of F it is not the case that we can claim to have seen that G(F) is true: for it will actually be false in some models of F and we have no basis for saying that our grasp of the "intended" model rules this out' (p. 388). But we do have some basis for saying this, and Sloman himself mentions it when he talks about 'our intuitive grasp of the natural number series, containing 0 and all its successors' (p. 387). The non-finite numbers which appear in those models of F in which G(F) is false are not natural according to this informal specification, since they are not successors of 0 in any intuitive, but only in a formal sense. However, such numbers appear already in non-standard models proper, those which are not isomorphic to the standard one but nevertheless preserve the truth-values of all statements (Boolos, Jeffrey 1980, p. 195). Thus, our concept of the intended model and those which are much less or even not intended is better than Sloman thinks, and does not have the weakness which he thinks it has ("too many models"). But it is conceivable that it has another, opposite weakness ("too few models"), which would even, on the face of it, support Sloman's position, though for a very different reason: it may be that we can't say that G(F) is true in the intended model because that model may not actually be a model. That would be the case if every consistent formalization of number theory were w-inconsistent (Hofstadter 1979, p. 459).

Therefore, Sloman's theses that the Gödel sentence does not mean what it seems to mean and that we can't say that it is true in the intended model are misguided. If Sloman's view of Gödel's sentence is confronted with the standard view, this view explains away the arguments on which Sloman's view is based.

It might seem that we have just refuted one of the weak-AI hypothesis in favour of classical formal interpretations. But we have just shown that there is nothing wrong with mathematics, Church-Turing theses or Gödel's sentence as lon in formal time-independent domains. Problems arise in other domains and other worlds.