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You seem like somebody who I could ask some (2) questions on the topic. 1. First of, I'm still not sure what to make of Gödel's theorems philosophically. In the "pop-culture", so to speak, there seems to be a notion that Gödel's theorems are, in essence, some grand, weighty statement about fundamental "inaccessibility of truth". That since Peano's arithmetic exemplifies very simple (compared to our general needs) mathematical system, no matter how we further develop math, one day there will be a useful, meaningful statement which cannot be proved or disproved. What's perhaps even worse, some people, even some with names I feel uncomfortable to argue against (Penrose, for example, seems to be of this opinion) try to use it as a transition to the idea that "human mind is more than a computer", which I always implicitly assumed to be just a manifestation of anthropocentric hubris. The key to their reasoning is the observation, that some expressions unprovable within some formal system are "obviously true" (and they often kinda are, and they are provable within a higher-order formal system). So, the story goes, since the Turing machine couldn't see it (because of Gödel's), but we see it, we are more than a Turing machine. And since that informal version of Gödel's findings was familiar to me long before I was acquainted with a formal version of it, I'm kinda used to the idea that "it must be right". However, when I'm looking at the formal explanation of Gödel's theorems, or even perhaps more interesting "applied" findings, based on them (like Goodstein's theorem), they all seem to be surprisingly "boring" and non-consequential clever tricks based on self-referencing. I mean, it's sure a very interesting finding about formal systems as such, but if we take a step back into the realms of common sense: isn't it rather quite intuitive, that a system cannot be proven to be consistent by the means of the same system? So it must be of no surprise that any formal system powerful enough to be able to express statements about consistency of itself must be "incomplete". So, my question is, am I missing something? Is there any truth to that pop-cultural image of Gödel's theorems? Because to me it seems like there actually isn't, just more of an "urban myth". 2. Continuum hypothesis. As I understand it, it is a matter of axiom, if 2^Aleph₀ ?= Aleph₁ and |R| could be as well Aleph₅ or whatever. Of course, we have most natural axiomatic system (with axiom of determinacy) where the former is true. But are there, like it was with Euclid's fifth postulate, any alternative, but still "interesting" constructs? Is there any use whatsoever in the assertion that continuum hypothesis is false? |
1. In short, this is a deeply philosophical question. My personal inclination follows yours that the incompleteness theorems are overhyped. I don't buy Penrose's argument. Like most other formal limiting results philosophically I view those results as representing fundamental limits of both human and machine reasoning. While a single unproveable but "obviously true" result generally point to an inadequacy of the formal system, if every formal system suffers from some inadequacy that is indicative of a limitation in the fundamental human faculties of comprehension rather than a limitation of formal systems per se (the assumption e.g. that our formal systems must use finitistic methods is one born of human limitations and, depending on one's thoughts about the universe, potentially fundamental physical ones).
There are philosophically defensible positions that try to claim the incompleteness theorems have implications for truth in the real world, but I go back and forth on whether I believe them and more to the point they are far more subtle than the usual pop culture presentations of Godel's incompleteness theorems.
2. The mathematical Platonists I've come across believe that the continuum hypothesis is in fact false. (As an aside it's interesting that you find the axiom of determinacy natural as it contradicts the axiom of choice.) This mainly hinges on your opinion of the reality of large cardinals (which perhaps count as your "interesting" constructs), which many Platonists for a variety of reasons believe to be real.