It's an ordinal measure of something like how complex it is to tell if a given property is true in the natural numbers. For first order sentences, you can show that something holds in N if and only if its truth complexity is finite.
But it seems to be defined so as to allow ordinals between omega and omega_1 as values, but I can't see when it would ever be anywhere in that range.
@MaliceVidrine I assume you're looking at this, in which Theorem 5.4.9 seems to suggest that only computable ordinals are relevant, and in fact nowhere near ω[1]^CK.
That is what I'm looking at (and part of the chapter before it) But I am having a hard time seeing how you'd ever get omega, omega+omega, or virtually any infinite ordinal as the truth complexity of anything.
It looks like you'd always end up with either finite complexity, or the built in maximum (omega_1 for first order sentences, omega^{CK}_1 for Pi^1_1 sentences)
But then it would be silly to set the upper bound anywhere other than omega, so I'm certainly missing something.
If there is no finite upper bound on the ordinal needed for each G∈CS(F), then you need at least ω.
Unfortunately, I can't tell you much more than that because I know nothing about "truth complexity". I'm just looking at where the ordinal can increase. =P
I suppose the point is that truth complexity for arithmetical sentences bottoms out at a quantifier-free predicate that is trivially true or false for any given parameters, whereas for second-order sentences with a set parameter the 'search tree' does not bottom out because even after removing quantifiers we can still have membership in a set parameter, for which we need infinitely many 'tests'.
Case (∨) makes the ordinal grow by at most 1 because we only need a single witness.
I have a solemn obligation to acknowledge the distinction between what's good and what I like. How can I criticize other people's bad taste if I can't acknowledge my own? :P
It is written in the SEP:
Identity is often said to be a relation each thing bears to itself and to no other thing (e.g., Zalabardo 2000). This characterization is clearly circular (“no other thing”)'.
The question is: what is circular here?
Deutsch, Harry and Garbacz, Pawel, "Relativ...
A bit more on the philosophical side when I was pondering the meaning of x=x
Fitch rules defines equality as the inference rule:
=intro:
|
|------- (where E is an object expression)
|E=E.
=elim:
|E=F.
|P(E).
|-------- (where every unused variable appearing in F must not appear in P)
|P(F).
We also knew an equivalence relation is reflexive, transitive and symmetric
But philosophically, like what the link is discussed there, it left the curious question on, what do we mean by x=x non circularly speaking?
We cannot say x=x means there is a y such that x=y, because that presupposes the notion of "other" and "different" and hence you risk quantifying over the notion of "identity" itself
More importantly, how do we use this inference rule to tell apart "=" from other equivalence relations that obey the same inference rules plus some possibly extra ones?
We also cannot seemed to define identity as the shortest circular sentence because e.g. $x\neq x$, $x < x$, $x @ x$ is also circular and it has the same length as $x=x$
Most set theories, such as ZFC, require an underlying knowledge of first-order logic formulas (as strings of symbols). This means that they require acceptance of facts of string manipulations (which is essentially equivalent to accepting arithmetic on natural numbers!) First-order logic does not ...
> There are two main parts to the 'circularity' in Mathematics (which is in fact a sociohistorical construct). The first is the understanding of logic, including the conditional and equality. If you do not understand what "if" means, no one can explain it to you because any purported explanation will be circular. Likewise for "same". (There are many types of equality that philosophy talks about.)
> The second is the understanding of the arithmetic on the natural numbers including induction. This boils down to the understanding of "repeat". If you do not know the meaning of "repeat" or "again" or other forms, no explanation can pin it down.
> Now there arises the interesting question of how we could learn these basic undefinable concepts in the first place. We do so because we have an innate ability to recognize similarity in function.
> When people use words in some ways consistently, we can (unconsciously) learn the functions of those words by seeing how they are used and abstracting out the similarities in the contexts, word order, grammatical structure and so on. So we learn the meaning of "same" and things like that automatically.
@Secret Your last sentence makes no sense at all. You cannot define anything in a vacuum. Nothing else to say. Worse still, you cannot really define "circular", so your claim isn't even well-defined.
hmm, I thought a "circular definition" can be defined as X := Y where Y is a sentence which one of its symbols is X
But sure, I guess I am starting to understand why the notion of a Self for individuals is such a hard concept to think without, because we humans innately think that way to the point it becomes second nature
I yet again recommend you finish learning basic FOL first before attempting any sort of philosophical thinking, otherwise it will be a hopeless task to get any insight. FOL is designed to capture certain natural concepts including equality. It is very wrong to think of FOL's equality as an equivalence relation. Firstly, you're confusing between syntax and semantics. Secondly, the =elim rule clearly represents identicalness. Most equivalence relations fail such substitution rules!
@Secret That sometimes works, but arguably that's inaccurate. Either way, that 'definition' of 'circular' does not even apply to what you were saying; none of what you said has the form "X:=Y"!
@Secret It's not a hard concept. Don't be like some philosophers who want to confuse themselves over it. We say "E = F" to mean that "E" refers to the same object as "F". Doesn't matter whether you can determine it or not; that's just what it means.
