Is there any possible world in which 2+2=5?

Question

Gödel's incompleteness theorems show that arithmetic is either inconsistent or incomplete, and that arithmetic cannot prove its own consistency. It is useful to believe that arithmetic is consistent, and therefore also incomplete, but there are other points of view.

It seems to me that both views are compatible with the consistency of first-order logic itself. By this, I mean that we could live in a logically consistent world, using first-order logic as the language we formulate theories in, in which specifically the system of arithmetic is inconsistent.

Thus, I am wondering: What is the autopsy of the efforts to revisit logic in the wake of Gödel?

Is there any possible world in which 2+2=5, even if the shortest proof of it is gigabytes long, and individually we have no time to verify it without the use of a computer?

How do contemporary philosophers interpret Gödel as applied to this specific issue?

Answer 1

Gödel didn't show arithmetic inconsistent or incomplete. Rather, he showed that any formal system with expressive power adequate for expressing arithmetic was either inconsistent or incomplete, and that no such system could prove its own consistency.

There are infinitely many proofs that 2 + 2 = 5, and they can be as short as you like. But, unless there really is an undetected inconsistency in a system of arithmetic (say, first-order Peano arithmetic, "FOPA"), they will all be proofs in a system of "arithmetic" that has either false axioms or unsound rules of inference.

I don't know what a possible world in which 2 + 2 = 5 would be like. It is epistemically possible that 2 + 2 = 5 is provable in FOPA, but, since FOPA also can prove 2 + 2 = 4 and ¬(4 = 5), that would show FOPA inconsistent.

Gentzen did prove the consistency of Peano Arithmetic, but his proof relies upon mathematical induction up to ε₀, and is thus no use in persuading those sceptical of the consistency of PA. It is, however, an important contribution to proof theory, setting an upper bound on the principles one needs to show PA consistent. (By Gödel's second incompleteness theorem, that bound must be higher than the system of PA itself.)

Answer 2

Leibniz introduced a distinction between accidental impossibility and absolut impossibility, and used 2+2=5 as an example of an absolute impossibility i.e., a contradiction. Accidental impossibilities are entities such as negatives, imaginaries, and infinitesimals.

Other than that, 2+2=5 is easily shown to be equivalent to 0=1, which is the standard formula used to reference a contradiction in modern logic.

Answer 3

There is an extended meditation on arithmetic and determinacy which works out a strict/ultra-finitist perspective fairly well (da Silva Maia[23]), such that the largest finite number N, whatever it is, is styled as being its own successor N + 1. (Such a number is presented as a paradigmatic example of an inconsistent mathematical object in the IEP entry on the topic.⁰)

At any rate, generally, per Kant, let us suppose that all existence claims can be denied without contradiction. Then ∃4, or ∃5, or ∃6, or... can be denied without contradiction. A sort-of similar phenomenon in transfinite set theory occurs when we "truncate the universe of sets at height k" (for some cardinality k). So again per Kant, it would not be that 2 + 2 = 5 is possible as such, but that, if we deny ∃4, we would say something more like 2 + 2 = NaN (then 1 + 3 = NaN, etc.).

But if we go the inconsistent or the looping ultrafinitist routes, then let us declare 4 to be the largest finite number in some possible mathematical world such that 4 = 4 + 1 = 5. This might be a way to get 2 + 2 = 5 while also having 2 + 2 = 4 (so that 4 = 5).

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⁰The IEP entry on abstractionism in the philosophy of mathematics says that Frege's anti-zero is its own predecessor, however.

Answer 4

OUR world is a world in which 2+2=5! Put 2 plus two elephants into an enclosure in a zoo, feed them, and in 2 years, you will likely have 5 elephants.

Conservation of number, and A=A, neither are the case for objects in our universe, so the logic of arithmetic mathematics is only approximately true here.

Put two containers worth of water, plus two more containers worth of water into another container, and one will then only have one container of water.

Put two plus two fertilized eggs into an incubator, and in a week or two one will end up with zero fertilized eggs.

Et cetera.

Math and logic are not aspects of our physical universe. They exist in a logic space, and whether a particular logic approximates, and to what degree, to some aspect or another of the physical world is an empirical question.

Answer 5

As I understand it, Gödel's incompleteness theorems refer only to the infinite sets. In other words, there are true but unprovable facts about infinite sets (say, sequences). There can be some patterns in the sequences, which hold but cannot be proven.

But when concerning the predicates about finite sets, all of them can be proven in Peano arithmetic, including that 2+2=4.