Does Gödel’s incompleteness theorems imply the necessity of an infinite recursive hierarchy of “proofs”, and that any “proof” is relative?

Question

My understanding is that Gödel’s incompleteness theorems state that the consistency of any axiomatic system cannot be proven within that axiomatic system, and requires a stronger axiomatic system in order to prove its consistency. Let’s say that we have an axiomatic system and another, stronger axiomatic system that is capable of proving the first axiomatic system’s consistency. But doesn’t this second axiomatic system also require a stronger axiomatic system to prove its consistency in order for us to trust its result/proof of consistency for the first axiomatic system? And so on and so forth in a hierarchical, recursive fashion for all possible axiomatic systems, ad infinitum?

If so, then wouldn’t this reasoning show that we cannot actually (formally) “prove” anything, in an absolute sense, or that any “proof” is simply relative to some axiomatic system, since there exists this kind of fundamental hierarchical/recursive “dependency relation”? In other words, there is no “absolute anchoring”, so to speak — it’s all relative! At the risk of making a fool of myself, as I am not a physicist, it seems analogous to the physics space-time “frame of reference” situation, no?

Answer 1

Two good "semi-technical" introduction to the issue:

• T.Franzen, Gödel's Theorem. An Incomplete Guide to Its Use and Abuse (2005) and

• F.Berto, There's Something About Gödel (2009).

A complete treatment:

• P.Smith, An Introduction to Gödel's Theorems (2013).

Regarding the never-ending hierarchy of theories, see:

• T.Franzen, Inexhaustibility: A Non-Exhaustive Treatment (2004): Introduction, page 1:

Gödel presented two results in his 1931 paper, usually referred to as the first and the second incompleteness theorem. The proof of the first incompleteness theorem shows that for every consistent formal axiomatic theory in a wide class of such theories, there is at least one statement which can be formulated in the language of the theory but can neither be proved nor disproved in the theory. [...] The second theorem states that for a wide class of such theories T, if T is consistent, the consistency of T cannot be proved using only the axioms of the theory T itself.

The second incompleteness theorem also has a positive aspect, which was emphasized by Gödel (Collected Works, vol. III, p.309, italics in the original): "It is this theorem [the second incompleteness theorem] which makes the incompletability of mathematics particularly evident. For, it makes it impossible that someone should set up a certain well-defined system of axioms and rules and consistently make the following assertion about it: All of these axioms and rules I perceive (with mathematical certitude) to be correct, and moreover I believe that they contain all of mathematics. If somebody makes such a statement he contradicts himself. For if he perceives the axioms under consideration to be correct, he also perceives (with the same certainty) that they are consistent. Hence he has a mathematical insight not derivable from his axioms."

Answer 2

Does Gödel’s incompleteness theorems imply the necessity of an infinite recursive hierarchy of “proofs”, and that any “proof” is relative?

Gödel's proofs are based on a false logic so it doesn't prove anything. The more general point that an "axiomatic system cannot be proven within that axiomatic system, and requires a stronger axiomatic system" is trivial. Axioms are assumed true, but if we want to prove them, we need to assume another set of axioms, and so on ad infinitum. This is just how logic works.

However, even this is of little practical consequence, for reality decides which axioms are really true. We don't have to go into an infinite regress to prove our axioms. We only have to try and understand which is true of the reality and which is not.

Mathematicians are not minded to pay attention to what is really true and what is not. They prefer to fathom the logic of their axioms and this is all that seems to matter to them. Suit yourself.

In other words, there is no “absolute anchoring”, so to speak — it’s all relative! At the risk of making a fool of myself, as I am not a physicist, it seems analogous to the physics space-time “frame of reference” situation, no?

Not quite, no. Physicists still seem to favour assessing the truth of their theories against reality, and reality is nothing if not an "absolute anchoring".

Physicists take their measures to be relative to the frame of reference, but only because they take the speed of light be . . . absolute.