You are getting at infinitary (first-order) logic. While a well-defined (and natural, and extensively studied) concept, it is quite different from ordinary first-order logic. Before diving into more detail, let me point out the main issue with your suggestion: There are "infinitely long" expr...

The answer to your question is a resounding: IT DEPENDS! This takes us into some set theory, which - while interesting - is a bit off the beaten track perhaps for CS; to address this, I've made the first part of my answer strictly about the computability theory, and then added a separate section ...

Relatively recent discoveries yield a number of so-called 'natural independence' results that provide much more natural examples of independence than does Gödel's example based upon the liar paradox (or other syntactic diagonalizations). As an example of such results, I'll sketch a simple example...

While the OP specifically asks about $\mathbb{R}$, I'm going to say a bit about the more general problem of additional axioms for mathematics in general. The case of $\mathbb{R}$ plays a special role here of course, but I think it helps to view it in the broader context. Because this answer is r...

Let me begin by linking to the current standard textbook on infinitary logic: Lectures on infinitary logic by Dave Marker. You may also be interested in his primer on the topic. It's also worth pointing out the huge collection Model-theoretic logics - one of the seminal texts in abstract model t...

The Turing jump $0^{(\alpha)}$ is defined for ordinals $\alpha<\omega_1^{\mathit{CK}}$ with $0^{(0)} = \varnothing$, $0^{(\alpha+1)}$ is the diagonal halting problem using $0^{(\alpha)}$ as an oracle, $0^{(\lambda)}$ for a limit $\lambda$ is the effective join of the $0^{(\lambda_n)}$ for a comp...

Below, $T$ is a complete first-order theory in a finite language with no finite models. Also asked at MO. Question Suppose $T$ has continuum-many countable models. We define two sets of Turing degrees associated to $T$ via second-order logic: $SecTh(T)$ is the set of Turing degrees of second-or...

I'll focus only on the second-order situation here, since my answer applies a fortiori to the higher orders. It essentially$^1$ requires us to introduce a new notation, to the point that - in my opinion - true second-order arithmetic (which I'll call "$TA_2$") is fundamentally impossible to desc...

There's an interesting situation here: while the question itself isn't a duplicate, a previous answer of mine I think resolves it. I don't quite know what to do in this case, so I've written a same-spirit, different-phrasing below, and not marked this question as a duplicate; to avoid "double-dip...

Your question as it stands is quite unclear, but let me take a stab at it; based on your previous questions, if nothing else I think you'll find this interesting. The simplest interpretation of your question is to look for an analogue of the Busy Beaver function for iterates of the Turing jump t...

There can indeed be ways of doing this, although I don't know of a natural one which works assuming only ZFC. I would say that they fall into two categories (and I suggest Kanamori's book as a great source on this sort of thing if you're interested): Fine structural (ZFC + "restricted universe"...

Below I've addressed your specific questions. However, based on your multiple questions about this I think it might be more useful to give a list of good sources, so I'll do that first. On "gaps" in the constructible universe: Marek/Srebrny, Gaps in the constructible universe. The introduction ...