Search Results

Raymond Smullyan has several works, including several books, that explore diverse generalizations of the kind of fixed-points that your question is about. Run the following search on MathSciNet: ti:(s …

You haven't said which theory $T$ you intend to use, but probably you have in mind something like PA. You are proposing to define exponentiation $x^y$ essentially as "the smallest $z$ for which $T$ ca …

Set theory provides a good example. It is often convenient in set theory to work with the concept of "classes" and treat them as mathematical objects of their own kind. The standard axiomatization of …

Even if you're looking at extensions of ZFC, the answer is still yes. Consider the theory of ZFC plus the scheme asserting that there is no definable global choice function. That is, the formulas "the …

You must read the charming essay lampooning this notation, while also giving a thorough logical analysis of it, by Adrian Mathias. Adrian Mathias, A Term of Length 4,523,659,424,929, Synthese 133 (20 …

Cantor's Attic, which I founded with Victoria Gitman, is a compendium of information on the consistency strength hierarchy in set theory, spanning the range from ZFC (which in this context is viewed a …

The jump inversion theorem (Friedburg 1957) shows that any Turing degree $d$ above the halting problem is the jump of another degree $d=b'$, which means that $d$ is Turing equivalent to the halting pr …

The assertion that there is (or is not) a truth predicate is expressible in the second-order language of set theory, but assuming consistency, not by any first-order assertion. Second-order. In the …

The usual conception of truth predicate, or satisfaction class, is formulated in the way that you describe, allowing arguments. It doesn't handle only sentences (no free variables), but formulas, with …

In this answer, let me assume as you indicated in the comments that you are working in a second-order set theory with a truth-predicate for first-order truth. Such a theory goes strictly beyond ZFC in …

There are several subtle issues with your post. It is not in general possible to express the notion of "definable", because it leads to contradictions. For example, the class $D$ is not definable in …

When one studies a set with structures that do not interact with each other, then really all that matters about the set is its cardinality. And so the situation of your question can be viewed as arisi …

There can be no such theory $T$, even if you weaken the requirement to $T$ being merely arithmetically definable, rather than insisting it must be effective. To see this, consider the theory T+TA, wh …

The answer is yes, provided these theories are consistent. For example, PA proves that there is a number $n$, such that if there is no proof of a contradiction in PA of size at most $n$, then there is …

Your proposed treatment of having machines use binary input with the alphabet $\{0,1\}$, where $0$ counts as a blank symbol (so that the input is padded with infinitely many additional $0$s, is not Tu …