Limitations

(Analysis of the system proposed in this post)

The idea of creating axiom systems describing Tarski's hierarchy in terms of ordinals, and ordinals in terms of Tarski's hierarchy, was discussed previously. It seems that the circularity would help prove the existence of lots of ordinals (and lots of Tarskian levels), hopefully without leading to paradox. So, if this gets us a lot of math, what is missing?

The obvious answer is "a truth predicate for that system". This doesn't lend much insight, though.

My bet is that we don't get anything uncountable by such a method. Where would it come from? We're constructing truth-predicates for countable languages, and (initially) countable sets of countable languages. Sure, the set of all countable ordinals is uncountable. But there is no reason to believe that we get the set of all countable ordinals!

My feeling is that if I studied more about the hyperarithmetical hierarchy, there would be an obvious mapping between some portion of it and the system(s) under consideration.

In some ways, the notion of "uncountable" seems to come from nowhere. All the formal constructions for "real number" and related entities seem to rely at some point on some other uncountable set. It's a bit like the (obviously illicit) method of defining a really big ordinal: "The ordering of all ordinals that can be defined without reference to this ordinal".

Yet I do feel that the notion is meaningful! So, where might it come from?

I have several ideas, none of them entirely convincing.

One idea is that uncountable sets can be motivated by considering the space of possible sequences of input from the external environment, assuming a countably infinite amount of time. I've seen similar ideas elsewhere. One might counter that all structures die eventually, so all sequences of input in real-world situations are finite; this makes the set into a mere countable infinity. On the other hand, one might say that this is merely the case in practice; the idea of an infinitely long-lived structure is still sound, and even physically possible (if not probable). But even so, I don't feel like this is a really complete justification.

Another attempt would be to claim that we need to allow for the possibility of infinitely long statements, despite not being able to actually make and manipulate such statements. (Universal statements and such may abbreviate infinitely many claims, but they are not literally infinitely long.) This idea is motivated by the following consideration: a nice way of getting a theory of set theory from a non-set-theoretic foundational logic is to think of a set as a statement with an unfilled slot into which entity-names can be put to make complete statements. Claiming that a set contains an element is thought of as claiming that the statement is true of that object. At first, this might seem to fully justify naive set theory: a set exists for each statement-with-hole. But, this can only work if the theory contains its own truth predicate, so that we can make arbitrary talk about whether a statement is true when we fill a slot with a given element. The amount of set theory that gets captured by this method depends basically on the extent to which the theory is capapble of self-reference; the naive theory of truth corresponds to naive set theory.

This is interesting by itself, but the point I'm making here is that if we want to have an uncountable number of sets (for example if we believe in the uncountable powerset of the real numbers), then we'll want a logic that acts as if infinite statements exist. What this means is an interesting question; we can't actually use these infinite statements, so what's the difference with the logic?

One difference is that I don't automatically have a way of interpreting talk about turing machines and numbers and other equivalent things anymore. I was justifying this referential power via talk about statements: they provide an immediate example of such entities. If we posit infinite statements that are "out there", somewhere in our set of sentences, we lose that quick way of grounding the idea of "finite". (This could be taken to show that such a method of grounding is not very real in the first place. :P)

Semantically, then, we think of the base-language as an infinitary logic, rather than regular first-order logic. The language formed by adding the first truth predicate is then thought of as already containing talk about uncountable sets. (This changes the axoims that we'd be justified in using to manipulate the truth predicate.)

All in all, I think this direction is mathematically interesting, but I'm not sure that it is really a route to justify uncountables.

A relevant question is: why do mathematicians think that uncountables exist? The proof is given by taking the powerset of some countably infinite set, which is defined as the set of all subsets of the countably infinite set. It's then shown that no function exists that maps the powerset onto the countable set. This can be done even in systems that does not really have any uncountable sets: the set of subsets of a countably infinite set will map onto the original set, but not by a function within the system. So from inside the system, it will look as if there are uncountables.

If this explains what mathematicians are doing, then mathematicians are being fooled into thinking there are real uncountables... but how can I say that? I'm just being fooled into thinking that there is a difference, a "real" and "not real", right?

I think it is plausible that this weirdness would go away, or at least change significantly, in a logic that resolves other foundational problems. We might have a much better story for why the function that would map the uncountable set onto the countable set doesn't exist, so that it becomes implausible to claim that it exists from the outside but not from the inside. (But would that make the uncountables "real"?)